Positional notation
Encyclopedia
Positional notation or place-value notation is a method of representing or encoding number
s. Positional notation is distinguished from other notations (such as Roman numerals
) for its use of the same symbol for the different orders of magnitude (for example, the "ones place", "tens place", "hundreds place"). This and greatly simplified arithmetic
led to the quick spread of the notation across the world.
With the use of a radix point
, the notation can be extended to include fraction
s and the numeric expansions of real number
s. The Hindu-Arabic numeral system
is an example for a positional notation, based on the number 10.
) system, which is likely motivated by counting with the ten finger
s, is ubiquitous. Other bases have been used in the past however, and some continue to be used today. For example, the Babylonian numeral system
, credited as the first positional number system, was base 60. Counting rods
and most abacus
es have been used to represent numbers in a positional numeral system. Before positional notation became standard, simple additive systems (sign-value notation
) such as Roman Numerals
were used, and accountants in ancient Rome and during the Middle Ages used the abacus
or stone counters to do arithmetic.
With counting rods or abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly. For four centuries (from the 13th to the 16th) there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the latter continues to be used in Japan and other Asian countries.
Georges Ifrah
concludes in his Universal History of Numbers:
Aryabhata
stated "sthānam sthānam daśa guṇam" meaning "From place to place, ten times in value". Indian mathematicians and astronomers also developed Sanskrit positional number words to describe astronomical facts or algorithms using poetic sutras. A key argument against the positional system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing (e.g.) 100 into 5100, or 100 into 1000. Modern cheque
s require a natural language spelling of an amount, as well as the decimal amount itself, to prevent such fraud. For the same reason the Chinese also use natural language numerals, for example 100 is written as 壹佰,which can never be forged into 壹仟(1000) or 伍仟壹佰(5100).
, the base or radix is usually the number of unique digits
, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is 10, because it uses the 10 digits from 0 through 9. When a number 'hits' 9, the next number will not be another different symbol, but a '1' followed by a '0'. In binary, the radix is 2, since after it hits '1', instead of '2' or another written symbol, it jumps straight to '10', followed by '11' and '100'.
The highest symbol of a positional numeral system usually has the value one less than the value of the base of that numeral system. The standard positional numeral systems differ from one another only in the base they use.
The base is an integer that is greater than 1 (or less than negative 1), since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with a negative radix, numbers may have many different possible representations.
(In certain non-standard positional numeral systems
, including bijective numeration, the definition of the base or the allowed digits deviates from the above.)
In base-10 (decimal) positional notation, there are 10 decimal digits and the number
.
In base-16 (hexadecimal
), there are 16 hexadecimal digits (0–9 and A–F) and the number
(where B represents the number eleven as a single symbol)
In general, in base-b, there are b digits and the number
(Note that 
represents a sequence of digits, not multiplication
)
indicates that the number 23 is expressed in base 8 (and is therefore equivalent in value to the decimal number 19). This notation will be used in this article.
When describing base in mathematical notation
, the letter b is generally used as a symbol
for this concept, so, for a binary
system, b equals 2. Another common way of expressing the base is writing it as a decimal subscript after the number that is being represented. 11110112 implies that the number 1111011 is a base 2 number, equal to 12310 (a decimal notation representation), 1738 (octal
) and 7B16 (hexadecimal
). When using the written abbreviations of number bases, the base is not printed: Binary 1111011 is the same as 11110112.
The base b may also be indicated by the phrase "base b". So binary numbers are "base 2"; octal numbers are "base 8"; decimal numbers are "base 10"; and so on.
Numbers of a given radix b have digits {0, 1, ..., b-2, b-1}. Thus, binary numbers have digits {0, 1}; decimal numbers have digits {0, 1, 2, ..., 8, 9}; and so on. Thus the following are notational errors and do not make sense: 522, 22, 1A9. (In all cases, one or more digits is not in the set of allowed digits for the given base.)
of the base. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the nth power, where n is the number of other digits between a given digit and the radix point
. If a given digit is on the left hand side of the radix point (i.e. its value is an integer
) then n is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) then n is negative.
As an example of usage, the number 465 in its respective base 'b' (which must be at least base 7 because the highest digit in it is 6) is equal to:
If the number 465 was in base 10, then it would equal:
(46510 = 46510)
If however, the number were in base 7, then it would equal:
(4657 = 24310)
10b = b for any base b, since 10b = 1×b1 + 0×b0. For example 102 = 2; 103 = 3; 1016 = 1610. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.
Numbers that are not integer
s use places beyond a radix point
. For every position behind this point (and thus after the units digit), the power n decreases by 1. For example, the number 2.35 is equal to:
This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base b, then a group of objects is created with b objects. When the number of these groups exceeds b, then a group of these groups of objects is created with b groups of b objects; and so on. Thus the same number in different bases will have different values:
241 in base 5:
2 groups of 52 (25) 4 groups of 5 1 group of 1
ooooo ooooo
ooooo ooooo ooooo ooooo
ooooo ooooo + + o
ooooo ooooo ooooo ooooo
ooooo ooooo
241 in base 8:
2 groups of 82 (64) 4 groups of 8 1 group of 1
oooooooo oooooooo
oooooooo oooooooo
oooooooo oooooooo oooooooo oooooooo
oooooooo oooooooo + + o
oooooooo oooooooo
oooooooo oooooooo oooooooo oooooooo
oooooooo oooooooo
oooooooo oooooooo
The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one real number
and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.
The distinction between a digit and a numeral is most pronounced in the context of a number base.
A non-zero numeral with more than one digit position will mean a different number in a different number base, but in general, the digits will mean the same.
The base-8 numeral 238 contains two digits, "2" and "3",
and with a base number (subscripted) "8", means 19.
In our notation here, the subscript "8" of the numeral 238 is part of the numeral,
but this may not always be the case. Imagine the numeral "23" as having an ambiguous base number.
Then "23" could likely be any base, base-4 through base-60. In base-4 "23" means 11, and in base-60 it means the number 123.
The numeral "23" then, in this case, corresponds to the set of numbers {11, 13, 15, 17, 19, 21, 23, ..., 121, 123}
while it's digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" three.
In certain applications when a numeral with a fixed number of positions
needs to represent a greater number, a higher number-base with more digits per position can be used.
A three-digit, decimal numeral can represent only up to 999.
But if the number-base is increased to 11, say, by adding the digit "A",
then the same three positions, maximized to "AAA", can represent a number as great as 1330.
We could increase the number base again and assign "B" to 11, and so on (but
there is also a possible encryption between number and digit in the number-digit-numeral hierarchy).
A three-digit numeral "ZZZ" in base-60 could mean 215999.
If we use the entire collection of our alphanumerics
we could ultimately serve a base-62 numeral system,
but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0".
We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see Sexagesimal system below.)
The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16).
In binary
only digits "0" and "1" are in the numerals. In the octal
numerals, are the eight digits 0–7.
Hex
is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits.
The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16".
241 in base 5:
2 groups of 5² 4 groups of 5 1 group of 1
ooooo ooooo
ooooo ooooo ooooo ooooo
ooooo ooooo + + o
ooooo ooooo ooooo ooooo
ooooo ooooo
is equal to 107 in base 8:
1 group of 8² 0 groups of 8 7 groups of 1
oooooooo
oooooooo o o
oooooooo
oooooooo + + o o o
oooooooo
oooooooo o o
oooooooo
oooooooo
There is, however, a shorter method which is basically the above method calculated mathematically. Because we work in base ten normally, it is easier to think of numbers in this way and therefore easier to convert them to base ten first, though it is possible (but difficult) to convert straight between non-decimal bases without using this intermediate step.
A number anan-1...a2a1a0 where a0, a1... an are all digits in a base b (note that here, the subscript does not refer to the base number; it refers to different objects), the number can be represented in any other base, including decimal, by:
Thus, in the example above:
To convert from decimal to another base one must simply start dividing by the value of the other base, then dividing the result of the first division and overlooking the remainder, and so on until the base is larger than the result (so the result of the division would be a zero). Then the number in the desired base is the remainders being the most significant value the one corresponding to the last division and the least significant value is the remainder of the first division.
Example #1 decimal to septal:
Example #2 decimal to octal:
The most common example is that of changing from decimal to binary.
:
Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (...) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a bar across the repeating block:
For base 10 it is called a recurring decimal or repeating decimal.
An irrational number
has an infinite non-repeating representation in all integer bases. Whether a rational number
has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:
For integers p and q with gcd
(p, q) = 1, the fraction
p/q has a finite representation in base b if and only if each prime factor
of q is also a prime factor of b.
For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations:
In the decimal
(base-10) Hindu-Arabic numeral system
, each position starting from the right is a higher power of 10. The first position represents 100 (1), the second position 101 (10), the third position 102 (10 × 10 or 100), the fourth position 103 (10 × 10 × 10 or 1000), and so on.
Fraction
al values are indicated by a separator
, which varies by locale
. Usually this separator is a period or full stop
, or a comma
. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates 10-1 (0.1), the second position 10-2 (0.01), and so on for each successive position.
As an example, the number 2674 in a base 10 numeral system is :
+ ( 6 × 102 ) + ( 7 × 101 ) + ( 4 × 100 )
or
+ ( 6 × 100 ) + ( 7 × 10 ) + ( 4 × 1 ).
and other mesopotamian systems, by Hellenistic astronomers using Greek numerals
for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional.
Modern time separates each position by a colon or point. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be 10°25'59" (10 degrees 25 minutes 59 seconds). In both cases, only minutes and seconds use sexagesimal notation — angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second. This contrasts with the numbers used by Hellenistic and Renaissance
astronomers, who used thirds, fourths, etc. for finer increments. Where we might write 10°25'59.392", they would have written10°25′59″23‴31'12 or 10°25I59II23III31IV12V.
Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans.
In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon to separate the integral and fractional portions of the number and using a comma to separate the positions within each portion. For example, the mean synodic month used by both Babylonian and Hellenistic astronomers and still used in the Hebrew calendar
is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.
, the binary
(base 2) and hexadecimal
(base 16) bases are used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as 'shorthand' for binary - every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f).
The octal
numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit.
or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many factors
. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 102, hundred, commerce developed a word for 122, gross. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency Pound Sterling
(GBP) partially used base-12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly, £sd.
The Maya civilization
and other civilizations of pre-Columbian
Mesoamerica
used base-20 (vigesimal
), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western Africa
.
Remnants of a Gaulish
base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is soixante-cinq (literally, "sixty [and] five"), while seventy-five is soixante-quinze (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty (somewhat similar to the archaic English manner of speaking of "scores
", probably originating from the same underlying Celtic system). For example, eighty-two is quatre-vingt-deux (literally, four twenty[s] [and] two), while ninety-two is quatre-vingt-douze (literally, four twenty[s] [and] twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties [and] thirteen, and so on.
The Irish language
also used base-20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A remnant of this system may be seen in the modern word for 40, daoichead.
Danish numerals display a similar base-20
structure.
The Maori language of New Zealand also has evidence of an underlying base-20 system as seen in the terms "Te Hokowhitu a Tu" referring to a war party (literally "the seven 20s of Tu") and "Tama-hokotahi", referring to a great warrior ("the one man equal to 20").
The binary system
was used in the Egyptian Old Kingdom, 3,000 BCE to 2,050 BCE. It was cursive by rounding off rational numbers smaller than 1 to , with a 1/64 term thrown away (the system was called the Eye of Horus).
A number of Australian Aboriginal languages
employ binary or binary-like counting systems. For example, in Kala Lagaw Ya
, the numbers one through six are urapon, ukasar, ukasar-urapon, ukasar-ukasar, ukasar-ukasar-urapon, ukasar-ukasar-ukasar.
North and Central American natives used base 4 (quaternary
) to represent the four cardinal directions. Mesoamericans tended to add a second base 5 system to create a modified base 20 system.
A base-5 system (quinary
) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base 10, base 20, and base 60.
A base-8 system (octal
) was devised by the Yuki tribe
of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base 8 system (or a system which could only count up to 8) with a base 10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for 'new', newo-, suggesting that the number 9 had been recently invented and called the 'new number'.
Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages
the word for five is the same as "hand" or "fist" (Dyola language
of Guinea-Bissau
, Banda language
of Central Africa
). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan
region.
The Telefol language
, spoken in Papua New Guinea
, is notable for possessing a base-27 numeral system.
Balanced ternary
uses a base of 3 but the digit set is
Positional notation or place-value notation is a method of representing or encoding number
s. Positional notation is distinguished from other notations (such as Roman numerals
) for its use of the same symbol for the different orders of magnitude (for example, the "ones place", "tens place", "hundreds place"). This and greatly simplified arithmetic
led to the quick spread of the notation across the world.
With the use of a radix point
, the notation can be extended to include fraction
s and the numeric expansions of real number
s. The Hindu-Arabic numeral system
is an example for a positional notation, based on the number 10.
) system, which is likely motivated by counting with the ten finger
s, is ubiquitous. Other bases have been used in the past however, and some continue to be used today. For example, the Babylonian numeral system
, credited as the first positional number system, was base 60. Counting rods
and most abacus
es have been used to represent numbers in a positional numeral system. Before positional notation became standard, simple additive systems (sign-value notation
) such as Roman Numerals
were used, and accountants in ancient Rome and during the Middle Ages used the abacus
or stone counters to do arithmetic.
With counting rods or abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly. For four centuries (from the 13th to the 16th) there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the latter continues to be used in Japan and other Asian countries.
Georges Ifrah
concludes in his Universal History of Numbers:
Aryabhata
stated "sthānam sthānam daśa guṇam" meaning "From place to place, ten times in value". Indian mathematicians and astronomers also developed Sanskrit positional number words to describe astronomical facts or algorithms using poetic sutras. A key argument against the positional system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing (e.g.) 100 into 5100, or 100 into 1000. Modern cheque
s require a natural language spelling of an amount, as well as the decimal amount itself, to prevent such fraud. For the same reason the Chinese also use natural language numerals, for example 100 is written as 壹佰,which can never be forged into 壹仟(1000) or 伍仟壹佰(5100).
, the base or radix is usually the number of unique digits
, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is 10, because it uses the 10 digits from 0 through 9. When a number 'hits' 9, the next number will not be another different symbol, but a '1' followed by a '0'. In binary, the radix is 2, since after it hits '1', instead of '2' or another written symbol, it jumps straight to '10', followed by '11' and '100'.
The highest symbol of a positional numeral system usually has the value one less than the value of the base of that numeral system. The standard positional numeral systems differ from one another only in the base they use.
The base is an integer that is greater than 1 (or less than negative 1), since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with a negative radix, numbers may have many different possible representations.
(In certain non-standard positional numeral systems
, including bijective numeration, the definition of the base or the allowed digits deviates from the above.)
In base-10 (decimal) positional notation, there are 10 decimal digits and the number
.
In base-16 (hexadecimal
), there are 16 hexadecimal digits (0–9 and A–F) and the number
(where B represents the number eleven as a single symbol)
In general, in base-b, there are b digits and the number
(Note that 
represents a sequence of digits, not multiplication
)
indicates that the number 23 is expressed in base 8 (and is therefore equivalent in value to the decimal number 19). This notation will be used in this article.
When describing base in mathematical notation
, the letter b is generally used as a symbol
for this concept, so, for a binary
system, b equals 2. Another common way of expressing the base is writing it as a decimal subscript after the number that is being represented. 11110112 implies that the number 1111011 is a base 2 number, equal to 12310 (a decimal notation representation), 1738 (octal
) and 7B16 (hexadecimal
). When using the written abbreviations of number bases, the base is not printed: Binary 1111011 is the same as 11110112.
The base b may also be indicated by the phrase "base b". So binary numbers are "base 2"; octal numbers are "base 8"; decimal numbers are "base 10"; and so on.
Numbers of a given radix b have digits {0, 1, ..., b-2, b-1}. Thus, binary numbers have digits {0, 1}; decimal numbers have digits {0, 1, 2, ..., 8, 9}; and so on. Thus the following are notational errors and do not make sense: 522, 22, 1A9. (In all cases, one or more digits is not in the set of allowed digits for the given base.)
of the base. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the nth power, where n is the number of other digits between a given digit and the radix point
. If a given digit is on the left hand side of the radix point (i.e. its value is an integer
) then n is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) then n is negative.
As an example of usage, the number 465 in its respective base 'b' (which must be at least base 7 because the highest digit in it is 6) is equal to:
If the number 465 was in base 10, then it would equal:
(46510 = 46510)
If however, the number were in base 7, then it would equal:
(4657 = 24310)
10b = b for any base b, since 10b = 1×b1 + 0×b0. For example 102 = 2; 103 = 3; 1016 = 1610. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.
Numbers that are not integer
s use places beyond a radix point
. For every position behind this point (and thus after the units digit), the power n decreases by 1. For example, the number 2.35 is equal to:
This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base b, then a group of objects is created with b objects. When the number of these groups exceeds b, then a group of these groups of objects is created with b groups of b objects; and so on. Thus the same number in different bases will have different values:
241 in base 5:
2 groups of 52 (25) 4 groups of 5 1 group of 1
ooooo ooooo
ooooo ooooo ooooo ooooo
ooooo ooooo + + o
ooooo ooooo ooooo ooooo
ooooo ooooo
241 in base 8:
2 groups of 82 (64) 4 groups of 8 1 group of 1
oooooooo oooooooo
oooooooo oooooooo
oooooooo oooooooo oooooooo oooooooo
oooooooo oooooooo + + o
oooooooo oooooooo
oooooooo oooooooo oooooooo oooooooo
oooooooo oooooooo
oooooooo oooooooo
The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one real number
and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.
The distinction between a digit and a numeral is most pronounced in the context of a number base.
A non-zero numeral with more than one digit position will mean a different number in a different number base, but in general, the digits will mean the same.
The base-8 numeral 238 contains two digits, "2" and "3",
and with a base number (subscripted) "8", means 19.
In our notation here, the subscript "8" of the numeral 238 is part of the numeral,
but this may not always be the case. Imagine the numeral "23" as having an ambiguous base number.
Then "23" could likely be any base, base-4 through base-60. In base-4 "23" means 11, and in base-60 it means the number 123.
The numeral "23" then, in this case, corresponds to the set of numbers {11, 13, 15, 17, 19, 21, 23, ..., 121, 123}
while it's digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" three.
In certain applications when a numeral with a fixed number of positions
needs to represent a greater number, a higher number-base with more digits per position can be used.
A three-digit, decimal numeral can represent only up to 999.
But if the number-base is increased to 11, say, by adding the digit "A",
then the same three positions, maximized to "AAA", can represent a number as great as 1330.
We could increase the number base again and assign "B" to 11, and so on (but
there is also a possible encryption between number and digit in the number-digit-numeral hierarchy).
A three-digit numeral "ZZZ" in base-60 could mean 215999.
If we use the entire collection of our alphanumerics
we could ultimately serve a base-62 numeral system,
but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0".
We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see Sexagesimal system below.)
The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16).
In binary
only digits "0" and "1" are in the numerals. In the octal
numerals, are the eight digits 0–7.
Hex
is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits.
The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16".
241 in base 5:
2 groups of 5² 4 groups of 5 1 group of 1
ooooo ooooo
ooooo ooooo ooooo ooooo
ooooo ooooo + + o
ooooo ooooo ooooo ooooo
ooooo ooooo
is equal to 107 in base 8:
1 group of 8² 0 groups of 8 7 groups of 1
oooooooo
oooooooo o o
oooooooo
oooooooo + + o o o
oooooooo
oooooooo o o
oooooooo
oooooooo
There is, however, a shorter method which is basically the above method calculated mathematically. Because we work in base ten normally, it is easier to think of numbers in this way and therefore easier to convert them to base ten first, though it is possible (but difficult) to convert straight between non-decimal bases without using this intermediate step.
A number anan-1...a2a1a0 where a0, a1... an are all digits in a base b (note that here, the subscript does not refer to the base number; it refers to different objects), the number can be represented in any other base, including decimal, by:
Thus, in the example above:
To convert from decimal to another base one must simply start dividing by the value of the other base, then dividing the result of the first division and overlooking the remainder, and so on until the base is larger than the result (so the result of the division would be a zero). Then the number in the desired base is the remainders being the most significant value the one corresponding to the last division and the least significant value is the remainder of the first division.
Example #1 decimal to septal:
Example #2 decimal to octal:
The most common example is that of changing from decimal to binary.
:
Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (...) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a bar across the repeating block:
For base 10 it is called a recurring decimal or repeating decimal.
An irrational number
has an infinite non-repeating representation in all integer bases. Whether a rational number
has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:
For integers p and q with gcd
(p, q) = 1, the fraction
p/q has a finite representation in base b if and only if each prime factor
of q is also a prime factor of b.
For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations:
In the decimal
(base-10) Hindu-Arabic numeral system
, each position starting from the right is a higher power of 10. The first position represents 100 (1), the second position 101 (10), the third position 102 (10 × 10 or 100), the fourth position 103 (10 × 10 × 10 or 1000), and so on.
Fraction
al values are indicated by a separator
, which varies by locale
. Usually this separator is a period or full stop
, or a comma
. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates 10-1 (0.1), the second position 10-2 (0.01), and so on for each successive position.
As an example, the number 2674 in a base 10 numeral system is :
+ ( 6 × 102 ) + ( 7 × 101 ) + ( 4 × 100 )
or
+ ( 6 × 100 ) + ( 7 × 10 ) + ( 4 × 1 ).
and other mesopotamian systems, by Hellenistic astronomers using Greek numerals
for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional.
Modern time separates each position by a colon or point. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be 10°25'59" (10 degrees 25 minutes 59 seconds). In both cases, only minutes and seconds use sexagesimal notation — angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second. This contrasts with the numbers used by Hellenistic and Renaissance
astronomers, who used thirds, fourths, etc. for finer increments. Where we might write 10°25'59.392", they would have written10°25′59″23‴31'12 or 10°25I59II23III31IV12V.
Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans.
In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon to separate the integral and fractional portions of the number and using a comma to separate the positions within each portion. For example, the mean synodic month used by both Babylonian and Hellenistic astronomers and still used in the Hebrew calendar
is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.
, the binary
(base 2) and hexadecimal
(base 16) bases are used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as 'shorthand' for binary - every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f).
The octal
numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit.
or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many factors
. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 102, hundred, commerce developed a word for 122, gross. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency Pound Sterling
(GBP) partially used base-12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly, £sd.
The Maya civilization
and other civilizations of pre-Columbian
Mesoamerica
used base-20 (vigesimal
), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western Africa
.
Remnants of a Gaulish
base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is soixante-cinq (literally, "sixty [and] five"), while seventy-five is soixante-quinze (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty (somewhat similar to the archaic English manner of speaking of "scores
", probably originating from the same underlying Celtic system). For example, eighty-two is quatre-vingt-deux (literally, four twenty[s] [and] two), while ninety-two is quatre-vingt-douze (literally, four twenty[s] [and] twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties [and] thirteen, and so on.
The Irish language
also used base-20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A remnant of this system may be seen in the modern word for 40, daoichead.
Danish numerals display a similar base-20
structure.
The Maori language of New Zealand also has evidence of an underlying base-20 system as seen in the terms "Te Hokowhitu a Tu" referring to a war party (literally "the seven 20s of Tu") and "Tama-hokotahi", referring to a great warrior ("the one man equal to 20").
The binary system
was used in the Egyptian Old Kingdom, 3,000 BCE to 2,050 BCE. It was cursive by rounding off rational numbers smaller than 1 to , with a 1/64 term thrown away (the system was called the Eye of Horus).
A number of Australian Aboriginal languages
employ binary or binary-like counting systems. For example, in Kala Lagaw Ya
, the numbers one through six are urapon, ukasar, ukasar-urapon, ukasar-ukasar, ukasar-ukasar-urapon, ukasar-ukasar-ukasar.
North and Central American natives used base 4 (quaternary
) to represent the four cardinal directions. Mesoamericans tended to add a second base 5 system to create a modified base 20 system.
A base-5 system (quinary
) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base 10, base 20, and base 60.
A base-8 system (octal
) was devised by the Yuki tribe
of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base 8 system (or a system which could only count up to 8) with a base 10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for 'new', newo-, suggesting that the number 9 had been recently invented and called the 'new number'.
Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages
the word for five is the same as "hand" or "fist" (Dyola language
of Guinea-Bissau
, Banda language
of Central Africa
). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan
region.
The Telefol language
, spoken in Papua New Guinea
, is notable for possessing a base-27 numeral system.
Balanced ternary
uses a base of 3 but the digit set is
Positional notation or place-value notation is a method of representing or encoding number
s. Positional notation is distinguished from other notations (such as Roman numerals
) for its use of the same symbol for the different orders of magnitude (for example, the "ones place", "tens place", "hundreds place"). This and greatly simplified arithmetic
led to the quick spread of the notation across the world.
With the use of a radix point
, the notation can be extended to include fraction
s and the numeric expansions of real number
s. The Hindu-Arabic numeral system
is an example for a positional notation, based on the number 10.
) system, which is likely motivated by counting with the ten finger
s, is ubiquitous. Other bases have been used in the past however, and some continue to be used today. For example, the Babylonian numeral system
, credited as the first positional number system, was base 60. Counting rods
and most abacus
es have been used to represent numbers in a positional numeral system. Before positional notation became standard, simple additive systems (sign-value notation
) such as Roman Numerals
were used, and accountants in ancient Rome and during the Middle Ages used the abacus
or stone counters to do arithmetic.
With counting rods or abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly. For four centuries (from the 13th to the 16th) there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the latter continues to be used in Japan and other Asian countries.
Georges Ifrah
concludes in his Universal History of Numbers:
Aryabhata
stated "sthānam sthānam daśa guṇam" meaning "From place to place, ten times in value". Indian mathematicians and astronomers also developed Sanskrit positional number words to describe astronomical facts or algorithms using poetic sutras. A key argument against the positional system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing (e.g.) 100 into 5100, or 100 into 1000. Modern cheque
s require a natural language spelling of an amount, as well as the decimal amount itself, to prevent such fraud. For the same reason the Chinese also use natural language numerals, for example 100 is written as 壹佰,which can never be forged into 壹仟(1000) or 伍仟壹佰(5100).
, the base or radix is usually the number of unique digits
, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is 10, because it uses the 10 digits from 0 through 9. When a number 'hits' 9, the next number will not be another different symbol, but a '1' followed by a '0'. In binary, the radix is 2, since after it hits '1', instead of '2' or another written symbol, it jumps straight to '10', followed by '11' and '100'.
The highest symbol of a positional numeral system usually has the value one less than the value of the base of that numeral system. The standard positional numeral systems differ from one another only in the base they use.
The base is an integer that is greater than 1 (or less than negative 1), since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with a negative radix, numbers may have many different possible representations.
(In certain non-standard positional numeral systems
, including bijective numeration, the definition of the base or the allowed digits deviates from the above.)
In base-10 (decimal) positional notation, there are 10 decimal digits and the number
.
In base-16 (hexadecimal
), there are 16 hexadecimal digits (0–9 and A–F) and the number
(where B represents the number eleven as a single symbol)
In general, in base-b, there are b digits and the number
(Note that 
represents a sequence of digits, not multiplication
)
indicates that the number 23 is expressed in base 8 (and is therefore equivalent in value to the decimal number 19). This notation will be used in this article.
When describing base in mathematical notation
, the letter b is generally used as a symbol
for this concept, so, for a binary
system, b equals 2. Another common way of expressing the base is writing it as a decimal subscript after the number that is being represented. 11110112 implies that the number 1111011 is a base 2 number, equal to 12310 (a decimal notation representation), 1738 (octal
) and 7B16 (hexadecimal
). When using the written abbreviations of number bases, the base is not printed: Binary 1111011 is the same as 11110112.
The base b may also be indicated by the phrase "base b". So binary numbers are "base 2"; octal numbers are "base 8"; decimal numbers are "base 10"; and so on.
Numbers of a given radix b have digits {0, 1, ..., b-2, b-1}. Thus, binary numbers have digits {0, 1}; decimal numbers have digits {0, 1, 2, ..., 8, 9}; and so on. Thus the following are notational errors and do not make sense: 522, 22, 1A9. (In all cases, one or more digits is not in the set of allowed digits for the given base.)
of the base. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the nth power, where n is the number of other digits between a given digit and the radix point
. If a given digit is on the left hand side of the radix point (i.e. its value is an integer
) then n is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) then n is negative.
As an example of usage, the number 465 in its respective base 'b' (which must be at least base 7 because the highest digit in it is 6) is equal to:
If the number 465 was in base 10, then it would equal:
(46510 = 46510)
If however, the number were in base 7, then it would equal:
(4657 = 24310)
10b = b for any base b, since 10b = 1×b1 + 0×b0. For example 102 = 2; 103 = 3; 1016 = 1610. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.
Numbers that are not integer
s use places beyond a radix point
. For every position behind this point (and thus after the units digit), the power n decreases by 1. For example, the number 2.35 is equal to:
This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base b, then a group of objects is created with b objects. When the number of these groups exceeds b, then a group of these groups of objects is created with b groups of b objects; and so on. Thus the same number in different bases will have different values:
241 in base 5:
2 groups of 52 (25) 4 groups of 5 1 group of 1
ooooo ooooo
ooooo ooooo ooooo ooooo
ooooo ooooo + + o
ooooo ooooo ooooo ooooo
ooooo ooooo
241 in base 8:
2 groups of 82 (64) 4 groups of 8 1 group of 1
oooooooo oooooooo
oooooooo oooooooo
oooooooo oooooooo oooooooo oooooooo
oooooooo oooooooo + + o
oooooooo oooooooo
oooooooo oooooooo oooooooo oooooooo
oooooooo oooooooo
oooooooo oooooooo
The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one real number
and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.
The distinction between a digit and a numeral is most pronounced in the context of a number base.
A non-zero numeral with more than one digit position will mean a different number in a different number base, but in general, the digits will mean the same.
The base-8 numeral 238 contains two digits, "2" and "3",
and with a base number (subscripted) "8", means 19.
In our notation here, the subscript "8" of the numeral 238 is part of the numeral,
but this may not always be the case. Imagine the numeral "23" as having an ambiguous base number.
Then "23" could likely be any base, base-4 through base-60. In base-4 "23" means 11, and in base-60 it means the number 123.
The numeral "23" then, in this case, corresponds to the set of numbers {11, 13, 15, 17, 19, 21, 23, ..., 121, 123}
while it's digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" three.
In certain applications when a numeral with a fixed number of positions
needs to represent a greater number, a higher number-base with more digits per position can be used.
A three-digit, decimal numeral can represent only up to 999.
But if the number-base is increased to 11, say, by adding the digit "A",
then the same three positions, maximized to "AAA", can represent a number as great as 1330.
We could increase the number base again and assign "B" to 11, and so on (but
there is also a possible encryption between number and digit in the number-digit-numeral hierarchy).
A three-digit numeral "ZZZ" in base-60 could mean 215999.
If we use the entire collection of our alphanumerics
we could ultimately serve a base-62 numeral system,
but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0".
We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see Sexagesimal system below.)
The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16).
In binary
only digits "0" and "1" are in the numerals. In the octal
numerals, are the eight digits 0–7.
Hex
is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits.
The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16".
241 in base 5:
2 groups of 5² 4 groups of 5 1 group of 1
ooooo ooooo
ooooo ooooo ooooo ooooo
ooooo ooooo + + o
ooooo ooooo ooooo ooooo
ooooo ooooo
is equal to 107 in base 8:
1 group of 8² 0 groups of 8 7 groups of 1
oooooooo
oooooooo o o
oooooooo
oooooooo + + o o o
oooooooo
oooooooo o o
oooooooo
oooooooo
There is, however, a shorter method which is basically the above method calculated mathematically. Because we work in base ten normally, it is easier to think of numbers in this way and therefore easier to convert them to base ten first, though it is possible (but difficult) to convert straight between non-decimal bases without using this intermediate step.
A number anan-1...a2a1a0 where a0, a1... an are all digits in a base b (note that here, the subscript does not refer to the base number; it refers to different objects), the number can be represented in any other base, including decimal, by:
Thus, in the example above:
To convert from decimal to another base one must simply start dividing by the value of the other base, then dividing the result of the first division and overlooking the remainder, and so on until the base is larger than the result (so the result of the division would be a zero). Then the number in the desired base is the remainders being the most significant value the one corresponding to the last division and the least significant value is the remainder of the first division.
Example #1 decimal to septal:
Example #2 decimal to octal:
The most common example is that of changing from decimal to binary.
:
Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (...) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a bar across the repeating block:
For base 10 it is called a recurring decimal or repeating decimal.
An irrational number
has an infinite non-repeating representation in all integer bases. Whether a rational number
has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:
For integers p and q with gcd
(p, q) = 1, the fraction
p/q has a finite representation in base b if and only if each prime factor
of q is also a prime factor of b.
For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations:
In the decimal
(base-10) Hindu-Arabic numeral system
, each position starting from the right is a higher power of 10. The first position represents 100 (1), the second position 101 (10), the third position 102 (10 × 10 or 100), the fourth position 103 (10 × 10 × 10 or 1000), and so on.
Fraction
al values are indicated by a separator
, which varies by locale
. Usually this separator is a period or full stop
, or a comma
. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates 10-1 (0.1), the second position 10-2 (0.01), and so on for each successive position.
As an example, the number 2674 in a base 10 numeral system is :
+ ( 6 × 102 ) + ( 7 × 101 ) + ( 4 × 100 )
or
+ ( 6 × 100 ) + ( 7 × 10 ) + ( 4 × 1 ).
and other mesopotamian systems, by Hellenistic astronomers using Greek numerals
for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional.
Modern time separates each position by a colon or point. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be 10°25'59" (10 degrees 25 minutes 59 seconds). In both cases, only minutes and seconds use sexagesimal notation — angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second. This contrasts with the numbers used by Hellenistic and Renaissance
astronomers, who used thirds, fourths, etc. for finer increments. Where we might write 10°25'59.392", they would have written10°25′59″23‴31'12 or 10°25I59II23III31IV12V.
Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans.
In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon to separate the integral and fractional portions of the number and using a comma to separate the positions within each portion. For example, the mean synodic month used by both Babylonian and Hellenistic astronomers and still used in the Hebrew calendar
is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.
, the binary
(base 2) and hexadecimal
(base 16) bases are used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as 'shorthand' for binary - every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f).
The octal
numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit.
or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many factors
. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 102, hundred, commerce developed a word for 122, gross. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency Pound Sterling
(GBP) partially used base-12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly, £sd.
The Maya civilization
and other civilizations of pre-Columbian
Mesoamerica
used base-20 (vigesimal
), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western Africa
.
Remnants of a Gaulish
base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is soixante-cinq (literally, "sixty [and] five"), while seventy-five is soixante-quinze (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty (somewhat similar to the archaic English manner of speaking of "scores
", probably originating from the same underlying Celtic system). For example, eighty-two is quatre-vingt-deux (literally, four twenty[s] [and] two), while ninety-two is quatre-vingt-douze (literally, four twenty[s] [and] twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties [and] thirteen, and so on.
The Irish language
also used base-20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A remnant of this system may be seen in the modern word for 40, daoichead.
Danish numerals display a similar base-20
structure.
The Maori language of New Zealand also has evidence of an underlying base-20 system as seen in the terms "Te Hokowhitu a Tu" referring to a war party (literally "the seven 20s of Tu") and "Tama-hokotahi", referring to a great warrior ("the one man equal to 20").
The binary system
was used in the Egyptian Old Kingdom, 3,000 BCE to 2,050 BCE. It was cursive by rounding off rational numbers smaller than 1 to , with a 1/64 term thrown away (the system was called the Eye of Horus).
A number of Australian Aboriginal languages
employ binary or binary-like counting systems. For example, in Kala Lagaw Ya
, the numbers one through six are urapon, ukasar, ukasar-urapon, ukasar-ukasar, ukasar-ukasar-urapon, ukasar-ukasar-ukasar.
North and Central American natives used base 4 (quaternary
) to represent the four cardinal directions. Mesoamericans tended to add a second base 5 system to create a modified base 20 system.
A base-5 system (quinary
) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base 10, base 20, and base 60.
A base-8 system (octal
) was devised by the Yuki tribe
of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base 8 system (or a system which could only count up to 8) with a base 10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for 'new', newo-, suggesting that the number 9 had been recently invented and called the 'new number'.
Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages
the word for five is the same as "hand" or "fist" (Dyola language
of Guinea-Bissau
, Banda language
of Central Africa
). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan
region.
The Telefol language
, spoken in Papua New Guinea
, is notable for possessing a base-27 numeral system.
Balanced ternary
uses a base of 3 but the digit set is
,0,1} instead of {0,1,2}. The "" has an equivalent value of −1.
The negation of a number is easily formed by switching the on the 1s.
This system can be used to solve the balance problem, which requires finding a minimal set of known counter-weights to determine an unknown weight.
Weights of 1, 3, 9, ... 3n known units can be used to determine any unknown weight up to 1 + 3 + ... + 3n units.
A weight can be used on either side of the balance or not at all.
Weights used on the balance pan with the unknown weight are designated with , with 1 if used on the empty pan, and with 0 if not used.
If an unknown weight W is balanced with 3 (31) on its pan and 1 and 27 (30 and 33) on the other, then its weight in decimal is 25 or 101 in balanced base 3.
(1013 = 1 × 33 + 0 × 32 − 1 × 31 + 1 × 30 = 25).
The factorial number system uses a varying radix, giving factorial
s as place values; they are related to Chinese remainder theorem
and Residue number system
enumerations. This system effectively enumerates permutations. A derivative of this uses the Towers of Hanoi puzzle configuration as a counting system. The configuration of the towers can be put into 1 to 1 correspondence with the decimal count of the step at which the configuration occurs and vice versa.
:Category:Positional numeral systems
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s. Positional notation is distinguished from other notations (such as Roman numerals
Roman numerals
The numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...
) for its use of the same symbol for the different orders of magnitude (for example, the "ones place", "tens place", "hundreds place"). This and greatly simplified arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
led to the quick spread of the notation across the world.
With the use of a radix point
Radix point
In mathematics and computing, a radix point is the symbol used in numerical representations to separate the integer part of a number from its fractional part . "Radix point" is a general term that applies to all number bases...
, the notation can be extended to include fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
s and the numeric expansions of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. The Hindu-Arabic numeral system
Hindu-Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
is an example for a positional notation, based on the number 10.
History
Today, the base 10 (decimalDecimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
) system, which is likely motivated by counting with the ten finger
Finger
A finger is a limb of the human body and a type of digit, an organ of manipulation and sensation found in the hands of humans and other primates....
s, is ubiquitous. Other bases have been used in the past however, and some continue to be used today. For example, the Babylonian numeral system
Babylonian numerals
Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record....
, credited as the first positional number system, was base 60. Counting rods
Counting rods
Counting rods are small bars, typically 3–14 cm long, used by mathematicians for calculation in China, Japan, Korea, and Vietnam. They are placed either horizontally or vertically to represent any number and any fraction....
and most abacus
Abacus
The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of...
es have been used to represent numbers in a positional numeral system. Before positional notation became standard, simple additive systems (sign-value notation
Sign-value notation
A sign-value notation represents numbers by a series of numeric signs that added together equal the number represented. In Roman numerals for example, X means ten and L means fifty. Hence LXXX means eighty . There is no need for zero in sign-value notation...
) such as Roman Numerals
Roman numerals
The numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...
were used, and accountants in ancient Rome and during the Middle Ages used the abacus
Abacus
The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of...
or stone counters to do arithmetic.
With counting rods or abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly. For four centuries (from the 13th to the 16th) there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the latter continues to be used in Japan and other Asian countries.
Georges Ifrah
Georges Ifrah
Georges Ifrah is a French author and historian of mathematics, especially numerals. He was formerly a teacher of mathematics....
concludes in his Universal History of Numbers:
Aryabhata
Aryabhata
Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...
stated "sthānam sthānam daśa guṇam" meaning "From place to place, ten times in value". Indian mathematicians and astronomers also developed Sanskrit positional number words to describe astronomical facts or algorithms using poetic sutras. A key argument against the positional system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing (e.g.) 100 into 5100, or 100 into 1000. Modern cheque
Cheque
A cheque is a document/instrument See the negotiable cow—itself a fictional story—for discussions of cheques written on unusual surfaces. that orders a payment of money from a bank account...
s require a natural language spelling of an amount, as well as the decimal amount itself, to prevent such fraud. For the same reason the Chinese also use natural language numerals, for example 100 is written as 壹佰,which can never be forged into 壹仟(1000) or 伍仟壹佰(5100).
Base of the numeral system
In mathematical numeral systemsNumeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
, the base or radix is usually the number of unique digits
Numerical digit
A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is 10, because it uses the 10 digits from 0 through 9. When a number 'hits' 9, the next number will not be another different symbol, but a '1' followed by a '0'. In binary, the radix is 2, since after it hits '1', instead of '2' or another written symbol, it jumps straight to '10', followed by '11' and '100'.
The highest symbol of a positional numeral system usually has the value one less than the value of the base of that numeral system. The standard positional numeral systems differ from one another only in the base they use.
The base is an integer that is greater than 1 (or less than negative 1), since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with a negative radix, numbers may have many different possible representations.
(In certain non-standard positional numeral systems
Non-standard positional numeral systems
Non-standard positional numeral systems here designates numeral systems that may be denoted positional systems, but that deviate in one way or another from the following description of standard positional systems:...
, including bijective numeration, the definition of the base or the allowed digits deviates from the above.)
In base-10 (decimal) positional notation, there are 10 decimal digits and the number
.In base-16 (hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
), there are 16 hexadecimal digits (0–9 and A–F) and the number
(where B represents the number eleven as a single symbol)In general, in base-b, there are b digits and the number
(Note that represents a sequence of digits, not multiplicationMultiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
)
Notation
Sometimes, a subscript notation is used where the base number is written in subscript after the number represented. For example, indicates that the number 23 is expressed in base 8 (and is therefore equivalent in value to the decimal number 19). This notation will be used in this article.When describing base in mathematical notation
Mathematical notation
Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...
, the letter b is generally used as a symbol
Symbol
A symbol is something which represents an idea, a physical entity or a process but is distinct from it. The purpose of a symbol is to communicate meaning. For example, a red octagon may be a symbol for "STOP". On a map, a picture of a tent might represent a campsite. Numerals are symbols for...
for this concept, so, for a binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
system, b equals 2. Another common way of expressing the base is writing it as a decimal subscript after the number that is being represented. 11110112 implies that the number 1111011 is a base 2 number, equal to 12310 (a decimal notation representation), 1738 (octal
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
) and 7B16 (hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
). When using the written abbreviations of number bases, the base is not printed: Binary 1111011 is the same as 11110112.
The base b may also be indicated by the phrase "base b". So binary numbers are "base 2"; octal numbers are "base 8"; decimal numbers are "base 10"; and so on.
Numbers of a given radix b have digits {0, 1, ..., b-2, b-1}. Thus, binary numbers have digits {0, 1}; decimal numbers have digits {0, 1, 2, ..., 8, 9}; and so on. Thus the following are notational errors and do not make sense: 522, 22, 1A9. (In all cases, one or more digits is not in the set of allowed digits for the given base.)
Exponentiation
Positional number systems work using exponentiationExponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
of the base. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the nth power, where n is the number of other digits between a given digit and the radix point
Radix point
In mathematics and computing, a radix point is the symbol used in numerical representations to separate the integer part of a number from its fractional part . "Radix point" is a general term that applies to all number bases...
. If a given digit is on the left hand side of the radix point (i.e. its value is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
) then n is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) then n is negative.
As an example of usage, the number 465 in its respective base 'b' (which must be at least base 7 because the highest digit in it is 6) is equal to:
If the number 465 was in base 10, then it would equal:
(46510 = 46510)
If however, the number were in base 7, then it would equal:
(4657 = 24310)
10b = b for any base b, since 10b = 1×b1 + 0×b0. For example 102 = 2; 103 = 3; 1016 = 1610. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.
Numbers that are not integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s use places beyond a radix point
Radix point
In mathematics and computing, a radix point is the symbol used in numerical representations to separate the integer part of a number from its fractional part . "Radix point" is a general term that applies to all number bases...
. For every position behind this point (and thus after the units digit), the power n decreases by 1. For example, the number 2.35 is equal to:
This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base b, then a group of objects is created with b objects. When the number of these groups exceeds b, then a group of these groups of objects is created with b groups of b objects; and so on. Thus the same number in different bases will have different values:
241 in base 5:
2 groups of 52 (25) 4 groups of 5 1 group of 1
ooooo ooooo
ooooo ooooo ooooo ooooo
ooooo ooooo + + o
ooooo ooooo ooooo ooooo
ooooo ooooo
241 in base 8:
2 groups of 82 (64) 4 groups of 8 1 group of 1
oooooooo oooooooo
oooooooo oooooooo
oooooooo oooooooo oooooooo oooooooo
oooooooo oooooooo + + o
oooooooo oooooooo
oooooooo oooooooo oooooooo oooooooo
oooooooo oooooooo
oooooooo oooooooo
The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.
Digits and numerals
A digit is what is used as a position in place-value notation, and a numeral is one or more digits. Today's most common digits are the decimal digits "0", "1", "2", "3", "4", "5", "6", "7", "8" , and "9". The earliest sexigesimal digit place values were a block of tightly packed 1's, 5's or 10's glyphs, with blocks of one or more spaces between, where the first grapheme for zero developed.The distinction between a digit and a numeral is most pronounced in the context of a number base.
A non-zero numeral with more than one digit position will mean a different number in a different number base, but in general, the digits will mean the same.
The base-8 numeral 238 contains two digits, "2" and "3",
and with a base number (subscripted) "8", means 19.
In our notation here, the subscript "8" of the numeral 238 is part of the numeral,
but this may not always be the case. Imagine the numeral "23" as having an ambiguous base number.
Then "23" could likely be any base, base-4 through base-60. In base-4 "23" means 11, and in base-60 it means the number 123.
The numeral "23" then, in this case, corresponds to the set of numbers {11, 13, 15, 17, 19, 21, 23, ..., 121, 123}
while it's digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" three.
In certain applications when a numeral with a fixed number of positions
needs to represent a greater number, a higher number-base with more digits per position can be used.
A three-digit, decimal numeral can represent only up to 999.
But if the number-base is increased to 11, say, by adding the digit "A",
then the same three positions, maximized to "AAA", can represent a number as great as 1330.
We could increase the number base again and assign "B" to 11, and so on (but
there is also a possible encryption between number and digit in the number-digit-numeral hierarchy).
A three-digit numeral "ZZZ" in base-60 could mean 215999.
If we use the entire collection of our alphanumerics
we could ultimately serve a base-62 numeral system,
but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0".
We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see Sexagesimal system below.)
The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16).
In binary
Binary
- Mathematics :* Binary numeral system, a representation for numbers using only two digits * Binary function, a function in mathematics that takes two arguments- Computing :* Binary file, composed of something other than human-readable text...
only digits "0" and "1" are in the numerals. In the octal
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
numerals, are the eight digits 0–7.
Hex
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits.
The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16".
Base conversion
Bases can be converted between each other by drawing the diagram above and rearranging the objects to conform to the new base, for example:241 in base 5:
2 groups of 5² 4 groups of 5 1 group of 1
ooooo ooooo
ooooo ooooo ooooo ooooo
ooooo ooooo + + o
ooooo ooooo ooooo ooooo
ooooo ooooo
is equal to 107 in base 8:
1 group of 8² 0 groups of 8 7 groups of 1
oooooooo
oooooooo o o
oooooooo
oooooooo + + o o o
oooooooo
oooooooo o o
oooooooo
oooooooo
There is, however, a shorter method which is basically the above method calculated mathematically. Because we work in base ten normally, it is easier to think of numbers in this way and therefore easier to convert them to base ten first, though it is possible (but difficult) to convert straight between non-decimal bases without using this intermediate step.
A number anan-1...a2a1a0 where a0, a1... an are all digits in a base b (note that here, the subscript does not refer to the base number; it refers to different objects), the number can be represented in any other base, including decimal, by:
Thus, in the example above:
To convert from decimal to another base one must simply start dividing by the value of the other base, then dividing the result of the first division and overlooking the remainder, and so on until the base is larger than the result (so the result of the division would be a zero). Then the number in the desired base is the remainders being the most significant value the one corresponding to the last division and the least significant value is the remainder of the first division.
Example #1 decimal to septal:
Example #2 decimal to octal:
The most common example is that of changing from decimal to binary.
Infinite representations
The representation of non-integers can be extended to allow an infinite string of digits beyond the point. For example 1.12112111211112 ... base 3 represents the sum of the infinite seriesSeries (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
:
Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (...) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a bar across the repeating block:
For base 10 it is called a recurring decimal or repeating decimal.
An irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
has an infinite non-repeating representation in all integer bases. Whether a rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:
-
- or, with the base implied:
For integers p and q with gcd
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
(p, q) = 1, the fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
p/q has a finite representation in base b if and only if each prime factor
Prime factor
In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...
of q is also a prime factor of b.
For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations:
- 1. A finite or infinite number of zeroes can be appended:
- 2. The last non-zero digit can be reduced by one and an infinite string of digits, each corresponding to one less than the base, are appended (or replace any following zero digits):
Decimal system
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
(base-10) Hindu-Arabic numeral system
Hindu-Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
, each position starting from the right is a higher power of 10. The first position represents 100 (1), the second position 101 (10), the third position 102 (10 × 10 or 100), the fourth position 103 (10 × 10 × 10 or 1000), and so on.
Fraction
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
al values are indicated by a separator
Decimal separator
Different symbols have been and are used for the decimal mark. The choice of symbol for the decimal mark affects the choice of symbol for the thousands separator used in digit grouping. Consequently the latter is treated in this article as well....
, which varies by locale
Locale
In computing, locale is a set of parameters that defines the user's language, country and any special variant preferences that the user wants to see in their user interface...
. Usually this separator is a period or full stop
Full stop
A full stop is the punctuation mark commonly placed at the end of sentences. In American English, the term used for this punctuation is period. In the 21st century, it is often also called a dot by young people...
, or a comma
Comma (punctuation)
The comma is a punctuation mark. It has the same shape as an apostrophe or single closing quotation mark in many typefaces, but it differs from them in being placed on the baseline of the text. Some typefaces render it as a small line, slightly curved or straight but inclined from the vertical, or...
. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates 10-1 (0.1), the second position 10-2 (0.01), and so on for each successive position.
As an example, the number 2674 in a base 10 numeral system is :
+ ( 6 × 102 ) + ( 7 × 101 ) + ( 4 × 100 )
or
+ ( 6 × 100 ) + ( 7 × 10 ) + ( 4 × 1 ).
Sexagesimal system
The sexagesimal or base sixty system was used for the integral and fractional portions of Babylonian numeralsBabylonian numerals
Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record....
and other mesopotamian systems, by Hellenistic astronomers using Greek numerals
Greek numerals
Greek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Ionian numerals, Milesian numerals , Alexandrian numerals, or alphabetic numerals...
for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional.
Modern time separates each position by a colon or point. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be 10°25'59" (10 degrees 25 minutes 59 seconds). In both cases, only minutes and seconds use sexagesimal notation — angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second. This contrasts with the numbers used by Hellenistic and Renaissance
Renaissance
The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...
astronomers, who used thirds, fourths, etc. for finer increments. Where we might write 10°25'59.392", they would have written
Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans.
In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon to separate the integral and fractional portions of the number and using a comma to separate the positions within each portion. For example, the mean synodic month used by both Babylonian and Hellenistic astronomers and still used in the Hebrew calendar
Hebrew calendar
The Hebrew calendar , or Jewish calendar, is a lunisolar calendar used today predominantly for Jewish religious observances. It determines the dates for Jewish holidays and the appropriate public reading of Torah portions, yahrzeits , and daily Psalm reading, among many ceremonial uses...
is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.
Computing
In computingComputing
Computing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...
, the binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
(base 2) and hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
(base 16) bases are used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as 'shorthand' for binary - every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f).
The octal
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit.
Other bases in human language
Base-12 systems (duodecimalDuodecimal
The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written as 'A', 'T' or 'X', and the number eleven as 'B' or 'E'...
or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many factors
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 102, hundred, commerce developed a word for 122, gross. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency Pound Sterling
Pound sterling
The pound sterling , commonly called the pound, is the official currency of the United Kingdom, its Crown Dependencies and the British Overseas Territories of South Georgia and the South Sandwich Islands, British Antarctic Territory and Tristan da Cunha. It is subdivided into 100 pence...
(GBP) partially used base-12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly, £sd.
The Maya civilization
Maya civilization
The Maya is a Mesoamerican civilization, noted for the only known fully developed written language of the pre-Columbian Americas, as well as for its art, architecture, and mathematical and astronomical systems. Initially established during the Pre-Classic period The Maya is a Mesoamerican...
and other civilizations of pre-Columbian
Pre-Columbian
The pre-Columbian era incorporates all period subdivisions in the history and prehistory of the Americas before the appearance of significant European influences on the American continents, spanning the time of the original settlement in the Upper Paleolithic period to European colonization during...
Mesoamerica
Mesoamerica
Mesoamerica is a region and culture area in the Americas, extending approximately from central Mexico to Belize, Guatemala, El Salvador, Honduras, Nicaragua, and Costa Rica, within which a number of pre-Columbian societies flourished before the Spanish colonization of the Americas in the 15th and...
used base-20 (vigesimal
Vigesimal
The vigesimal or base 20 numeral system is based on twenty .- Places :...
), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western Africa
Africa
Africa is the world's second largest and second most populous continent, after Asia. At about 30.2 million km² including adjacent islands, it covers 6% of the Earth's total surface area and 20.4% of the total land area...
.
Remnants of a Gaulish
Gaulish language
The Gaulish language is an extinct Celtic language that was spoken by the Gauls, a people who inhabited the region known as Gaul from the Iron Age through the Roman period...
base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is soixante-cinq (literally, "sixty [and] five"), while seventy-five is soixante-quinze (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty (somewhat similar to the archaic English manner of speaking of "scores
20 (number)
20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.-In mathematics:*20 is the basis for vigesimal number systems....
", probably originating from the same underlying Celtic system). For example, eighty-two is quatre-vingt-deux (literally, four twenty[s] [and] two), while ninety-two is quatre-vingt-douze (literally, four twenty[s] [and] twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties [and] thirteen, and so on.
The Irish language
Irish language
Irish , also known as Irish Gaelic, is a Goidelic language of the Indo-European language family, originating in Ireland and historically spoken by the Irish people. Irish is now spoken as a first language by a minority of Irish people, as well as being a second language of a larger proportion of...
also used base-20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A remnant of this system may be seen in the modern word for 40, daoichead.
Danish numerals display a similar base-20
Vigesimal
The vigesimal or base 20 numeral system is based on twenty .- Places :...
structure.
The Maori language of New Zealand also has evidence of an underlying base-20 system as seen in the terms "Te Hokowhitu a Tu" referring to a war party (literally "the seven 20s of Tu") and "Tama-hokotahi", referring to a great warrior ("the one man equal to 20").
The binary system
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
was used in the Egyptian Old Kingdom, 3,000 BCE to 2,050 BCE. It was cursive by rounding off rational numbers smaller than 1 to , with a 1/64 term thrown away (the system was called the Eye of Horus).
A number of Australian Aboriginal languages
Australian Aboriginal languages
The Australian Aboriginal languages comprise several language families and isolates native to the Australian Aborigines of Australia and a few nearby islands, but by convention excluding the languages of Tasmania and the Torres Strait Islanders...
employ binary or binary-like counting systems. For example, in Kala Lagaw Ya
Kala Lagaw Ya
Kala Lagaw Ya is a language belonging to all the western and central Torres Strait Islands, Queensland, Australia. On some islands it has now largely been replaced by Brokan...
, the numbers one through six are urapon, ukasar, ukasar-urapon, ukasar-ukasar, ukasar-ukasar-urapon, ukasar-ukasar-ukasar.
North and Central American natives used base 4 (quaternary
Quaternary numeral system
Quaternary is the base- numeral system. It uses the digits 0, 1, 2 and 3 to represent any real number.It shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation and the characteristics of the representations of...
) to represent the four cardinal directions. Mesoamericans tended to add a second base 5 system to create a modified base 20 system.
A base-5 system (quinary
Quinary
Quinary is a numeral system with five as the base. A possible origination of a quinary system is that there are five fingers on either hand. The base five is stated from 0-4...
) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base 10, base 20, and base 60.
A base-8 system (octal
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
) was devised by the Yuki tribe
Yuki tribe
The Yuki are a Native American people from the zone of Round Valley, in what today is part of the territory of Mendocino County, Northern California. Yuki tribes are thought to have settled as far south as Hood Mountain in present-day Sonoma County...
of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base 8 system (or a system which could only count up to 8) with a base 10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for 'new', newo-, suggesting that the number 9 had been recently invented and called the 'new number'.
Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages
African languages
There are over 2100 and by some counts over 3000 languages spoken natively in Africa in several major language families:*Afro-Asiatic spread throughout the Middle East, North Africa, the Horn of Africa, and parts of the Sahel...
the word for five is the same as "hand" or "fist" (Dyola language
Dyola language
Jola or Diola, also called Jola-Fonyi, is a language spoken by half a million people in the Casamance region of Senegal, and neighboring countries. The French spelling is Diola and Diola Fogny. Jola people themselves say that a person is ajoola and they speak joola....
of Guinea-Bissau
Guinea-Bissau
The Republic of Guinea-Bissau is a country in West Africa. It is bordered by Senegal to the north, and Guinea to the south and east, with the Atlantic Ocean to its west....
, Banda language
Banda languages
Banda is a family of Ubangian languages spoken by the Banda people of Central Africa.-Languages:Olson classfies the Banda family as follows :*Central**Central Banda...
of Central Africa
Central Africa
Central Africa is a core region of the African continent which includes Burundi, the Central African Republic, Chad, the Democratic Republic of the Congo, and Rwanda....
). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan
Sudan
Sudan , officially the Republic of the Sudan , is a country in North Africa, sometimes considered part of the Middle East politically. It is bordered by Egypt to the north, the Red Sea to the northeast, Eritrea and Ethiopia to the east, South Sudan to the south, the Central African Republic to the...
region.
The Telefol language
Telefol language
Telefol is a language spoken by the Telefol people in Papua New Guinea, notable for possessing a base-27 numeral system.-History:The Iligimin people also spoke Telefol, but they were defeated by the Telefol proper.-Orthography:...
, spoken in Papua New Guinea
Papua New Guinea
Papua New Guinea , officially the Independent State of Papua New Guinea, is a country in Oceania, occupying the eastern half of the island of New Guinea and numerous offshore islands...
, is notable for possessing a base-27 numeral system.
Non-standard positional numeral systems
Interesting properties exist when the base is not fixed or positive and when the digit symbol sets denote negative values. There are many more variations. These systems are of practical and theoretic value to computer scientists.Balanced ternary
Balanced ternary
Balanced ternary is a non-standard positional numeral system , useful for comparison logic. It is a ternary system, but unlike the standard ternary system, the digits have the values −1, 0, and 1...
uses a base of 3 but the digit set is
Positional notation or place-value notation is a method of representing or encoding number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s. Positional notation is distinguished from other notations (such as Roman numerals
Roman numerals
The numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...
) for its use of the same symbol for the different orders of magnitude (for example, the "ones place", "tens place", "hundreds place"). This and greatly simplified arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
led to the quick spread of the notation across the world.
With the use of a radix point
Radix point
In mathematics and computing, a radix point is the symbol used in numerical representations to separate the integer part of a number from its fractional part . "Radix point" is a general term that applies to all number bases...
, the notation can be extended to include fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
s and the numeric expansions of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. The Hindu-Arabic numeral system
Hindu-Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
is an example for a positional notation, based on the number 10.
History
Today, the base 10 (decimalDecimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
) system, which is likely motivated by counting with the ten finger
Finger
A finger is a limb of the human body and a type of digit, an organ of manipulation and sensation found in the hands of humans and other primates....
s, is ubiquitous. Other bases have been used in the past however, and some continue to be used today. For example, the Babylonian numeral system
Babylonian numerals
Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record....
, credited as the first positional number system, was base 60. Counting rods
Counting rods
Counting rods are small bars, typically 3–14 cm long, used by mathematicians for calculation in China, Japan, Korea, and Vietnam. They are placed either horizontally or vertically to represent any number and any fraction....
and most abacus
Abacus
The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of...
es have been used to represent numbers in a positional numeral system. Before positional notation became standard, simple additive systems (sign-value notation
Sign-value notation
A sign-value notation represents numbers by a series of numeric signs that added together equal the number represented. In Roman numerals for example, X means ten and L means fifty. Hence LXXX means eighty . There is no need for zero in sign-value notation...
) such as Roman Numerals
Roman numerals
The numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...
were used, and accountants in ancient Rome and during the Middle Ages used the abacus
Abacus
The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of...
or stone counters to do arithmetic.
With counting rods or abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly. For four centuries (from the 13th to the 16th) there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the latter continues to be used in Japan and other Asian countries.
Georges Ifrah
Georges Ifrah
Georges Ifrah is a French author and historian of mathematics, especially numerals. He was formerly a teacher of mathematics....
concludes in his Universal History of Numbers:
Aryabhata
Aryabhata
Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...
stated "sthānam sthānam daśa guṇam" meaning "From place to place, ten times in value". Indian mathematicians and astronomers also developed Sanskrit positional number words to describe astronomical facts or algorithms using poetic sutras. A key argument against the positional system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing (e.g.) 100 into 5100, or 100 into 1000. Modern cheque
Cheque
A cheque is a document/instrument See the negotiable cow—itself a fictional story—for discussions of cheques written on unusual surfaces. that orders a payment of money from a bank account...
s require a natural language spelling of an amount, as well as the decimal amount itself, to prevent such fraud. For the same reason the Chinese also use natural language numerals, for example 100 is written as 壹佰,which can never be forged into 壹仟(1000) or 伍仟壹佰(5100).
Base of the numeral system
In mathematical numeral systemsNumeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
, the base or radix is usually the number of unique digits
Numerical digit
A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is 10, because it uses the 10 digits from 0 through 9. When a number 'hits' 9, the next number will not be another different symbol, but a '1' followed by a '0'. In binary, the radix is 2, since after it hits '1', instead of '2' or another written symbol, it jumps straight to '10', followed by '11' and '100'.
The highest symbol of a positional numeral system usually has the value one less than the value of the base of that numeral system. The standard positional numeral systems differ from one another only in the base they use.
The base is an integer that is greater than 1 (or less than negative 1), since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with a negative radix, numbers may have many different possible representations.
(In certain non-standard positional numeral systems
Non-standard positional numeral systems
Non-standard positional numeral systems here designates numeral systems that may be denoted positional systems, but that deviate in one way or another from the following description of standard positional systems:...
, including bijective numeration, the definition of the base or the allowed digits deviates from the above.)
In base-10 (decimal) positional notation, there are 10 decimal digits and the number
.In base-16 (hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
), there are 16 hexadecimal digits (0–9 and A–F) and the number
(where B represents the number eleven as a single symbol)In general, in base-b, there are b digits and the number
(Note that represents a sequence of digits, not multiplicationMultiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
)
Notation
Sometimes, a subscript notation is used where the base number is written in subscript after the number represented. For example, indicates that the number 23 is expressed in base 8 (and is therefore equivalent in value to the decimal number 19). This notation will be used in this article.When describing base in mathematical notation
Mathematical notation
Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...
, the letter b is generally used as a symbol
Symbol
A symbol is something which represents an idea, a physical entity or a process but is distinct from it. The purpose of a symbol is to communicate meaning. For example, a red octagon may be a symbol for "STOP". On a map, a picture of a tent might represent a campsite. Numerals are symbols for...
for this concept, so, for a binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
system, b equals 2. Another common way of expressing the base is writing it as a decimal subscript after the number that is being represented. 11110112 implies that the number 1111011 is a base 2 number, equal to 12310 (a decimal notation representation), 1738 (octal
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
) and 7B16 (hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
). When using the written abbreviations of number bases, the base is not printed: Binary 1111011 is the same as 11110112.
The base b may also be indicated by the phrase "base b". So binary numbers are "base 2"; octal numbers are "base 8"; decimal numbers are "base 10"; and so on.
Numbers of a given radix b have digits {0, 1, ..., b-2, b-1}. Thus, binary numbers have digits {0, 1}; decimal numbers have digits {0, 1, 2, ..., 8, 9}; and so on. Thus the following are notational errors and do not make sense: 522, 22, 1A9. (In all cases, one or more digits is not in the set of allowed digits for the given base.)
Exponentiation
Positional number systems work using exponentiationExponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
of the base. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the nth power, where n is the number of other digits between a given digit and the radix point
Radix point
In mathematics and computing, a radix point is the symbol used in numerical representations to separate the integer part of a number from its fractional part . "Radix point" is a general term that applies to all number bases...
. If a given digit is on the left hand side of the radix point (i.e. its value is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
) then n is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) then n is negative.
As an example of usage, the number 465 in its respective base 'b' (which must be at least base 7 because the highest digit in it is 6) is equal to:
If the number 465 was in base 10, then it would equal:
(46510 = 46510)
If however, the number were in base 7, then it would equal:
(4657 = 24310)
10b = b for any base b, since 10b = 1×b1 + 0×b0. For example 102 = 2; 103 = 3; 1016 = 1610. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.
Numbers that are not integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s use places beyond a radix point
Radix point
In mathematics and computing, a radix point is the symbol used in numerical representations to separate the integer part of a number from its fractional part . "Radix point" is a general term that applies to all number bases...
. For every position behind this point (and thus after the units digit), the power n decreases by 1. For example, the number 2.35 is equal to:
This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base b, then a group of objects is created with b objects. When the number of these groups exceeds b, then a group of these groups of objects is created with b groups of b objects; and so on. Thus the same number in different bases will have different values:
241 in base 5:
2 groups of 52 (25) 4 groups of 5 1 group of 1
ooooo ooooo
ooooo ooooo ooooo ooooo
ooooo ooooo + + o
ooooo ooooo ooooo ooooo
ooooo ooooo
241 in base 8:
2 groups of 82 (64) 4 groups of 8 1 group of 1
oooooooo oooooooo
oooooooo oooooooo
oooooooo oooooooo oooooooo oooooooo
oooooooo oooooooo + + o
oooooooo oooooooo
oooooooo oooooooo oooooooo oooooooo
oooooooo oooooooo
oooooooo oooooooo
The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.
Digits and numerals
A digit is what is used as a position in place-value notation, and a numeral is one or more digits. Today's most common digits are the decimal digits "0", "1", "2", "3", "4", "5", "6", "7", "8" , and "9". The earliest sexigesimal digit place values were a block of tightly packed 1's, 5's or 10's glyphs, with blocks of one or more spaces between, where the first grapheme for zero developed.The distinction between a digit and a numeral is most pronounced in the context of a number base.
A non-zero numeral with more than one digit position will mean a different number in a different number base, but in general, the digits will mean the same.
The base-8 numeral 238 contains two digits, "2" and "3",
and with a base number (subscripted) "8", means 19.
In our notation here, the subscript "8" of the numeral 238 is part of the numeral,
but this may not always be the case. Imagine the numeral "23" as having an ambiguous base number.
Then "23" could likely be any base, base-4 through base-60. In base-4 "23" means 11, and in base-60 it means the number 123.
The numeral "23" then, in this case, corresponds to the set of numbers {11, 13, 15, 17, 19, 21, 23, ..., 121, 123}
while it's digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" three.
In certain applications when a numeral with a fixed number of positions
needs to represent a greater number, a higher number-base with more digits per position can be used.
A three-digit, decimal numeral can represent only up to 999.
But if the number-base is increased to 11, say, by adding the digit "A",
then the same three positions, maximized to "AAA", can represent a number as great as 1330.
We could increase the number base again and assign "B" to 11, and so on (but
there is also a possible encryption between number and digit in the number-digit-numeral hierarchy).
A three-digit numeral "ZZZ" in base-60 could mean 215999.
If we use the entire collection of our alphanumerics
we could ultimately serve a base-62 numeral system,
but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0".
We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see Sexagesimal system below.)
The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16).
In binary
Binary
- Mathematics :* Binary numeral system, a representation for numbers using only two digits * Binary function, a function in mathematics that takes two arguments- Computing :* Binary file, composed of something other than human-readable text...
only digits "0" and "1" are in the numerals. In the octal
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
numerals, are the eight digits 0–7.
Hex
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits.
The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16".
Base conversion
Bases can be converted between each other by drawing the diagram above and rearranging the objects to conform to the new base, for example:241 in base 5:
2 groups of 5² 4 groups of 5 1 group of 1
ooooo ooooo
ooooo ooooo ooooo ooooo
ooooo ooooo + + o
ooooo ooooo ooooo ooooo
ooooo ooooo
is equal to 107 in base 8:
1 group of 8² 0 groups of 8 7 groups of 1
oooooooo
oooooooo o o
oooooooo
oooooooo + + o o o
oooooooo
oooooooo o o
oooooooo
oooooooo
There is, however, a shorter method which is basically the above method calculated mathematically. Because we work in base ten normally, it is easier to think of numbers in this way and therefore easier to convert them to base ten first, though it is possible (but difficult) to convert straight between non-decimal bases without using this intermediate step.
A number anan-1...a2a1a0 where a0, a1... an are all digits in a base b (note that here, the subscript does not refer to the base number; it refers to different objects), the number can be represented in any other base, including decimal, by:
Thus, in the example above:
To convert from decimal to another base one must simply start dividing by the value of the other base, then dividing the result of the first division and overlooking the remainder, and so on until the base is larger than the result (so the result of the division would be a zero). Then the number in the desired base is the remainders being the most significant value the one corresponding to the last division and the least significant value is the remainder of the first division.
Example #1 decimal to septal:
Example #2 decimal to octal:
The most common example is that of changing from decimal to binary.
Infinite representations
The representation of non-integers can be extended to allow an infinite string of digits beyond the point. For example 1.12112111211112 ... base 3 represents the sum of the infinite seriesSeries (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
:
Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (...) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a bar across the repeating block:
For base 10 it is called a recurring decimal or repeating decimal.
An irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
has an infinite non-repeating representation in all integer bases. Whether a rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:
-
- or, with the base implied:
For integers p and q with gcd
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
(p, q) = 1, the fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
p/q has a finite representation in base b if and only if each prime factor
Prime factor
In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...
of q is also a prime factor of b.
For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations:
- 1. A finite or infinite number of zeroes can be appended:
- 2. The last non-zero digit can be reduced by one and an infinite string of digits, each corresponding to one less than the base, are appended (or replace any following zero digits):
Decimal system
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
(base-10) Hindu-Arabic numeral system
Hindu-Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
, each position starting from the right is a higher power of 10. The first position represents 100 (1), the second position 101 (10), the third position 102 (10 × 10 or 100), the fourth position 103 (10 × 10 × 10 or 1000), and so on.
Fraction
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
al values are indicated by a separator
Decimal separator
Different symbols have been and are used for the decimal mark. The choice of symbol for the decimal mark affects the choice of symbol for the thousands separator used in digit grouping. Consequently the latter is treated in this article as well....
, which varies by locale
Locale
In computing, locale is a set of parameters that defines the user's language, country and any special variant preferences that the user wants to see in their user interface...
. Usually this separator is a period or full stop
Full stop
A full stop is the punctuation mark commonly placed at the end of sentences. In American English, the term used for this punctuation is period. In the 21st century, it is often also called a dot by young people...
, or a comma
Comma (punctuation)
The comma is a punctuation mark. It has the same shape as an apostrophe or single closing quotation mark in many typefaces, but it differs from them in being placed on the baseline of the text. Some typefaces render it as a small line, slightly curved or straight but inclined from the vertical, or...
. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates 10-1 (0.1), the second position 10-2 (0.01), and so on for each successive position.
As an example, the number 2674 in a base 10 numeral system is :
+ ( 6 × 102 ) + ( 7 × 101 ) + ( 4 × 100 )
or
+ ( 6 × 100 ) + ( 7 × 10 ) + ( 4 × 1 ).
Sexagesimal system
The sexagesimal or base sixty system was used for the integral and fractional portions of Babylonian numeralsBabylonian numerals
Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record....
and other mesopotamian systems, by Hellenistic astronomers using Greek numerals
Greek numerals
Greek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Ionian numerals, Milesian numerals , Alexandrian numerals, or alphabetic numerals...
for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional.
Modern time separates each position by a colon or point. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be 10°25'59" (10 degrees 25 minutes 59 seconds). In both cases, only minutes and seconds use sexagesimal notation — angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second. This contrasts with the numbers used by Hellenistic and Renaissance
Renaissance
The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...
astronomers, who used thirds, fourths, etc. for finer increments. Where we might write 10°25'59.392", they would have written
Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans.
In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon to separate the integral and fractional portions of the number and using a comma to separate the positions within each portion. For example, the mean synodic month used by both Babylonian and Hellenistic astronomers and still used in the Hebrew calendar
Hebrew calendar
The Hebrew calendar , or Jewish calendar, is a lunisolar calendar used today predominantly for Jewish religious observances. It determines the dates for Jewish holidays and the appropriate public reading of Torah portions, yahrzeits , and daily Psalm reading, among many ceremonial uses...
is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.
Computing
In computingComputing
Computing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...
, the binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
(base 2) and hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
(base 16) bases are used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as 'shorthand' for binary - every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f).
The octal
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit.
Other bases in human language
Base-12 systems (duodecimalDuodecimal
The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written as 'A', 'T' or 'X', and the number eleven as 'B' or 'E'...
or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many factors
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 102, hundred, commerce developed a word for 122, gross. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency Pound Sterling
Pound sterling
The pound sterling , commonly called the pound, is the official currency of the United Kingdom, its Crown Dependencies and the British Overseas Territories of South Georgia and the South Sandwich Islands, British Antarctic Territory and Tristan da Cunha. It is subdivided into 100 pence...
(GBP) partially used base-12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly, £sd.
The Maya civilization
Maya civilization
The Maya is a Mesoamerican civilization, noted for the only known fully developed written language of the pre-Columbian Americas, as well as for its art, architecture, and mathematical and astronomical systems. Initially established during the Pre-Classic period The Maya is a Mesoamerican...
and other civilizations of pre-Columbian
Pre-Columbian
The pre-Columbian era incorporates all period subdivisions in the history and prehistory of the Americas before the appearance of significant European influences on the American continents, spanning the time of the original settlement in the Upper Paleolithic period to European colonization during...
Mesoamerica
Mesoamerica
Mesoamerica is a region and culture area in the Americas, extending approximately from central Mexico to Belize, Guatemala, El Salvador, Honduras, Nicaragua, and Costa Rica, within which a number of pre-Columbian societies flourished before the Spanish colonization of the Americas in the 15th and...
used base-20 (vigesimal
Vigesimal
The vigesimal or base 20 numeral system is based on twenty .- Places :...
), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western Africa
Africa
Africa is the world's second largest and second most populous continent, after Asia. At about 30.2 million km² including adjacent islands, it covers 6% of the Earth's total surface area and 20.4% of the total land area...
.
Remnants of a Gaulish
Gaulish language
The Gaulish language is an extinct Celtic language that was spoken by the Gauls, a people who inhabited the region known as Gaul from the Iron Age through the Roman period...
base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is soixante-cinq (literally, "sixty [and] five"), while seventy-five is soixante-quinze (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty (somewhat similar to the archaic English manner of speaking of "scores
20 (number)
20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.-In mathematics:*20 is the basis for vigesimal number systems....
", probably originating from the same underlying Celtic system). For example, eighty-two is quatre-vingt-deux (literally, four twenty[s] [and] two), while ninety-two is quatre-vingt-douze (literally, four twenty[s] [and] twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties [and] thirteen, and so on.
The Irish language
Irish language
Irish , also known as Irish Gaelic, is a Goidelic language of the Indo-European language family, originating in Ireland and historically spoken by the Irish people. Irish is now spoken as a first language by a minority of Irish people, as well as being a second language of a larger proportion of...
also used base-20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A remnant of this system may be seen in the modern word for 40, daoichead.
Danish numerals display a similar base-20
Vigesimal
The vigesimal or base 20 numeral system is based on twenty .- Places :...
structure.
The Maori language of New Zealand also has evidence of an underlying base-20 system as seen in the terms "Te Hokowhitu a Tu" referring to a war party (literally "the seven 20s of Tu") and "Tama-hokotahi", referring to a great warrior ("the one man equal to 20").
The binary system
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
was used in the Egyptian Old Kingdom, 3,000 BCE to 2,050 BCE. It was cursive by rounding off rational numbers smaller than 1 to , with a 1/64 term thrown away (the system was called the Eye of Horus).
A number of Australian Aboriginal languages
Australian Aboriginal languages
The Australian Aboriginal languages comprise several language families and isolates native to the Australian Aborigines of Australia and a few nearby islands, but by convention excluding the languages of Tasmania and the Torres Strait Islanders...
employ binary or binary-like counting systems. For example, in Kala Lagaw Ya
Kala Lagaw Ya
Kala Lagaw Ya is a language belonging to all the western and central Torres Strait Islands, Queensland, Australia. On some islands it has now largely been replaced by Brokan...
, the numbers one through six are urapon, ukasar, ukasar-urapon, ukasar-ukasar, ukasar-ukasar-urapon, ukasar-ukasar-ukasar.
North and Central American natives used base 4 (quaternary
Quaternary numeral system
Quaternary is the base- numeral system. It uses the digits 0, 1, 2 and 3 to represent any real number.It shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation and the characteristics of the representations of...
) to represent the four cardinal directions. Mesoamericans tended to add a second base 5 system to create a modified base 20 system.
A base-5 system (quinary
Quinary
Quinary is a numeral system with five as the base. A possible origination of a quinary system is that there are five fingers on either hand. The base five is stated from 0-4...
) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base 10, base 20, and base 60.
A base-8 system (octal
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
) was devised by the Yuki tribe
Yuki tribe
The Yuki are a Native American people from the zone of Round Valley, in what today is part of the territory of Mendocino County, Northern California. Yuki tribes are thought to have settled as far south as Hood Mountain in present-day Sonoma County...
of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base 8 system (or a system which could only count up to 8) with a base 10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for 'new', newo-, suggesting that the number 9 had been recently invented and called the 'new number'.
Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages
African languages
There are over 2100 and by some counts over 3000 languages spoken natively in Africa in several major language families:*Afro-Asiatic spread throughout the Middle East, North Africa, the Horn of Africa, and parts of the Sahel...
the word for five is the same as "hand" or "fist" (Dyola language
Dyola language
Jola or Diola, also called Jola-Fonyi, is a language spoken by half a million people in the Casamance region of Senegal, and neighboring countries. The French spelling is Diola and Diola Fogny. Jola people themselves say that a person is ajoola and they speak joola....
of Guinea-Bissau
Guinea-Bissau
The Republic of Guinea-Bissau is a country in West Africa. It is bordered by Senegal to the north, and Guinea to the south and east, with the Atlantic Ocean to its west....
, Banda language
Banda languages
Banda is a family of Ubangian languages spoken by the Banda people of Central Africa.-Languages:Olson classfies the Banda family as follows :*Central**Central Banda...
of Central Africa
Central Africa
Central Africa is a core region of the African continent which includes Burundi, the Central African Republic, Chad, the Democratic Republic of the Congo, and Rwanda....
). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan
Sudan
Sudan , officially the Republic of the Sudan , is a country in North Africa, sometimes considered part of the Middle East politically. It is bordered by Egypt to the north, the Red Sea to the northeast, Eritrea and Ethiopia to the east, South Sudan to the south, the Central African Republic to the...
region.
The Telefol language
Telefol language
Telefol is a language spoken by the Telefol people in Papua New Guinea, notable for possessing a base-27 numeral system.-History:The Iligimin people also spoke Telefol, but they were defeated by the Telefol proper.-Orthography:...
, spoken in Papua New Guinea
Papua New Guinea
Papua New Guinea , officially the Independent State of Papua New Guinea, is a country in Oceania, occupying the eastern half of the island of New Guinea and numerous offshore islands...
, is notable for possessing a base-27 numeral system.
Non-standard positional numeral systems
Interesting properties exist when the base is not fixed or positive and when the digit symbol sets denote negative values. There are many more variations. These systems are of practical and theoretic value to computer scientists.Balanced ternary
Balanced ternary
Balanced ternary is a non-standard positional numeral system , useful for comparison logic. It is a ternary system, but unlike the standard ternary system, the digits have the values −1, 0, and 1...
uses a base of 3 but the digit set is
Positional notation or place-value notation is a method of representing or encoding number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s. Positional notation is distinguished from other notations (such as Roman numerals
Roman numerals
The numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...
) for its use of the same symbol for the different orders of magnitude (for example, the "ones place", "tens place", "hundreds place"). This and greatly simplified arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
led to the quick spread of the notation across the world.
With the use of a radix point
Radix point
In mathematics and computing, a radix point is the symbol used in numerical representations to separate the integer part of a number from its fractional part . "Radix point" is a general term that applies to all number bases...
, the notation can be extended to include fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
s and the numeric expansions of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. The Hindu-Arabic numeral system
Hindu-Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
is an example for a positional notation, based on the number 10.
History
Today, the base 10 (decimalDecimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
) system, which is likely motivated by counting with the ten finger
Finger
A finger is a limb of the human body and a type of digit, an organ of manipulation and sensation found in the hands of humans and other primates....
s, is ubiquitous. Other bases have been used in the past however, and some continue to be used today. For example, the Babylonian numeral system
Babylonian numerals
Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record....
, credited as the first positional number system, was base 60. Counting rods
Counting rods
Counting rods are small bars, typically 3–14 cm long, used by mathematicians for calculation in China, Japan, Korea, and Vietnam. They are placed either horizontally or vertically to represent any number and any fraction....
and most abacus
Abacus
The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of...
es have been used to represent numbers in a positional numeral system. Before positional notation became standard, simple additive systems (sign-value notation
Sign-value notation
A sign-value notation represents numbers by a series of numeric signs that added together equal the number represented. In Roman numerals for example, X means ten and L means fifty. Hence LXXX means eighty . There is no need for zero in sign-value notation...
) such as Roman Numerals
Roman numerals
The numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...
were used, and accountants in ancient Rome and during the Middle Ages used the abacus
Abacus
The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of...
or stone counters to do arithmetic.
With counting rods or abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly. For four centuries (from the 13th to the 16th) there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the latter continues to be used in Japan and other Asian countries.
Georges Ifrah
Georges Ifrah
Georges Ifrah is a French author and historian of mathematics, especially numerals. He was formerly a teacher of mathematics....
concludes in his Universal History of Numbers:
Aryabhata
Aryabhata
Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...
stated "sthānam sthānam daśa guṇam" meaning "From place to place, ten times in value". Indian mathematicians and astronomers also developed Sanskrit positional number words to describe astronomical facts or algorithms using poetic sutras. A key argument against the positional system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing (e.g.) 100 into 5100, or 100 into 1000. Modern cheque
Cheque
A cheque is a document/instrument See the negotiable cow—itself a fictional story—for discussions of cheques written on unusual surfaces. that orders a payment of money from a bank account...
s require a natural language spelling of an amount, as well as the decimal amount itself, to prevent such fraud. For the same reason the Chinese also use natural language numerals, for example 100 is written as 壹佰,which can never be forged into 壹仟(1000) or 伍仟壹佰(5100).
Base of the numeral system
In mathematical numeral systemsNumeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
, the base or radix is usually the number of unique digits
Numerical digit
A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is 10, because it uses the 10 digits from 0 through 9. When a number 'hits' 9, the next number will not be another different symbol, but a '1' followed by a '0'. In binary, the radix is 2, since after it hits '1', instead of '2' or another written symbol, it jumps straight to '10', followed by '11' and '100'.
The highest symbol of a positional numeral system usually has the value one less than the value of the base of that numeral system. The standard positional numeral systems differ from one another only in the base they use.
The base is an integer that is greater than 1 (or less than negative 1), since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with a negative radix, numbers may have many different possible representations.
(In certain non-standard positional numeral systems
Non-standard positional numeral systems
Non-standard positional numeral systems here designates numeral systems that may be denoted positional systems, but that deviate in one way or another from the following description of standard positional systems:...
, including bijective numeration, the definition of the base or the allowed digits deviates from the above.)
In base-10 (decimal) positional notation, there are 10 decimal digits and the number
.In base-16 (hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
), there are 16 hexadecimal digits (0–9 and A–F) and the number
(where B represents the number eleven as a single symbol)In general, in base-b, there are b digits and the number
(Note that represents a sequence of digits, not multiplicationMultiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
)
Notation
Sometimes, a subscript notation is used where the base number is written in subscript after the number represented. For example, indicates that the number 23 is expressed in base 8 (and is therefore equivalent in value to the decimal number 19). This notation will be used in this article.When describing base in mathematical notation
Mathematical notation
Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...
, the letter b is generally used as a symbol
Symbol
A symbol is something which represents an idea, a physical entity or a process but is distinct from it. The purpose of a symbol is to communicate meaning. For example, a red octagon may be a symbol for "STOP". On a map, a picture of a tent might represent a campsite. Numerals are symbols for...
for this concept, so, for a binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
system, b equals 2. Another common way of expressing the base is writing it as a decimal subscript after the number that is being represented. 11110112 implies that the number 1111011 is a base 2 number, equal to 12310 (a decimal notation representation), 1738 (octal
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
) and 7B16 (hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
). When using the written abbreviations of number bases, the base is not printed: Binary 1111011 is the same as 11110112.
The base b may also be indicated by the phrase "base b". So binary numbers are "base 2"; octal numbers are "base 8"; decimal numbers are "base 10"; and so on.
Numbers of a given radix b have digits {0, 1, ..., b-2, b-1}. Thus, binary numbers have digits {0, 1}; decimal numbers have digits {0, 1, 2, ..., 8, 9}; and so on. Thus the following are notational errors and do not make sense: 522, 22, 1A9. (In all cases, one or more digits is not in the set of allowed digits for the given base.)
Exponentiation
Positional number systems work using exponentiationExponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
of the base. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the nth power, where n is the number of other digits between a given digit and the radix point
Radix point
In mathematics and computing, a radix point is the symbol used in numerical representations to separate the integer part of a number from its fractional part . "Radix point" is a general term that applies to all number bases...
. If a given digit is on the left hand side of the radix point (i.e. its value is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
) then n is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) then n is negative.
As an example of usage, the number 465 in its respective base 'b' (which must be at least base 7 because the highest digit in it is 6) is equal to:
If the number 465 was in base 10, then it would equal:
(46510 = 46510)
If however, the number were in base 7, then it would equal:
(4657 = 24310)
10b = b for any base b, since 10b = 1×b1 + 0×b0. For example 102 = 2; 103 = 3; 1016 = 1610. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.
Numbers that are not integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s use places beyond a radix point
Radix point
In mathematics and computing, a radix point is the symbol used in numerical representations to separate the integer part of a number from its fractional part . "Radix point" is a general term that applies to all number bases...
. For every position behind this point (and thus after the units digit), the power n decreases by 1. For example, the number 2.35 is equal to:
This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base b, then a group of objects is created with b objects. When the number of these groups exceeds b, then a group of these groups of objects is created with b groups of b objects; and so on. Thus the same number in different bases will have different values:
241 in base 5:
2 groups of 52 (25) 4 groups of 5 1 group of 1
ooooo ooooo
ooooo ooooo ooooo ooooo
ooooo ooooo + + o
ooooo ooooo ooooo ooooo
ooooo ooooo
241 in base 8:
2 groups of 82 (64) 4 groups of 8 1 group of 1
oooooooo oooooooo
oooooooo oooooooo
oooooooo oooooooo oooooooo oooooooo
oooooooo oooooooo + + o
oooooooo oooooooo
oooooooo oooooooo oooooooo oooooooo
oooooooo oooooooo
oooooooo oooooooo
The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.
Digits and numerals
A digit is what is used as a position in place-value notation, and a numeral is one or more digits. Today's most common digits are the decimal digits "0", "1", "2", "3", "4", "5", "6", "7", "8" , and "9". The earliest sexigesimal digit place values were a block of tightly packed 1's, 5's or 10's glyphs, with blocks of one or more spaces between, where the first grapheme for zero developed.The distinction between a digit and a numeral is most pronounced in the context of a number base.
A non-zero numeral with more than one digit position will mean a different number in a different number base, but in general, the digits will mean the same.
The base-8 numeral 238 contains two digits, "2" and "3",
and with a base number (subscripted) "8", means 19.
In our notation here, the subscript "8" of the numeral 238 is part of the numeral,
but this may not always be the case. Imagine the numeral "23" as having an ambiguous base number.
Then "23" could likely be any base, base-4 through base-60. In base-4 "23" means 11, and in base-60 it means the number 123.
The numeral "23" then, in this case, corresponds to the set of numbers {11, 13, 15, 17, 19, 21, 23, ..., 121, 123}
while it's digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" three.
In certain applications when a numeral with a fixed number of positions
needs to represent a greater number, a higher number-base with more digits per position can be used.
A three-digit, decimal numeral can represent only up to 999.
But if the number-base is increased to 11, say, by adding the digit "A",
then the same three positions, maximized to "AAA", can represent a number as great as 1330.
We could increase the number base again and assign "B" to 11, and so on (but
there is also a possible encryption between number and digit in the number-digit-numeral hierarchy).
A three-digit numeral "ZZZ" in base-60 could mean 215999.
If we use the entire collection of our alphanumerics
we could ultimately serve a base-62 numeral system,
but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0".
We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see Sexagesimal system below.)
The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16).
In binary
Binary
- Mathematics :* Binary numeral system, a representation for numbers using only two digits * Binary function, a function in mathematics that takes two arguments- Computing :* Binary file, composed of something other than human-readable text...
only digits "0" and "1" are in the numerals. In the octal
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
numerals, are the eight digits 0–7.
Hex
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits.
The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16".
Base conversion
Bases can be converted between each other by drawing the diagram above and rearranging the objects to conform to the new base, for example:241 in base 5:
2 groups of 5² 4 groups of 5 1 group of 1
ooooo ooooo
ooooo ooooo ooooo ooooo
ooooo ooooo + + o
ooooo ooooo ooooo ooooo
ooooo ooooo
is equal to 107 in base 8:
1 group of 8² 0 groups of 8 7 groups of 1
oooooooo
oooooooo o o
oooooooo
oooooooo + + o o o
oooooooo
oooooooo o o
oooooooo
oooooooo
There is, however, a shorter method which is basically the above method calculated mathematically. Because we work in base ten normally, it is easier to think of numbers in this way and therefore easier to convert them to base ten first, though it is possible (but difficult) to convert straight between non-decimal bases without using this intermediate step.
A number anan-1...a2a1a0 where a0, a1... an are all digits in a base b (note that here, the subscript does not refer to the base number; it refers to different objects), the number can be represented in any other base, including decimal, by:
Thus, in the example above:
To convert from decimal to another base one must simply start dividing by the value of the other base, then dividing the result of the first division and overlooking the remainder, and so on until the base is larger than the result (so the result of the division would be a zero). Then the number in the desired base is the remainders being the most significant value the one corresponding to the last division and the least significant value is the remainder of the first division.
Example #1 decimal to septal:
Example #2 decimal to octal:
The most common example is that of changing from decimal to binary.
Infinite representations
The representation of non-integers can be extended to allow an infinite string of digits beyond the point. For example 1.12112111211112 ... base 3 represents the sum of the infinite seriesSeries (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
:
Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (...) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a bar across the repeating block:
For base 10 it is called a recurring decimal or repeating decimal.
An irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
has an infinite non-repeating representation in all integer bases. Whether a rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:
-
- or, with the base implied:
For integers p and q with gcd
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
(p, q) = 1, the fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
p/q has a finite representation in base b if and only if each prime factor
Prime factor
In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...
of q is also a prime factor of b.
For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations:
- 1. A finite or infinite number of zeroes can be appended:
- 2. The last non-zero digit can be reduced by one and an infinite string of digits, each corresponding to one less than the base, are appended (or replace any following zero digits):
Decimal system
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
(base-10) Hindu-Arabic numeral system
Hindu-Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
, each position starting from the right is a higher power of 10. The first position represents 100 (1), the second position 101 (10), the third position 102 (10 × 10 or 100), the fourth position 103 (10 × 10 × 10 or 1000), and so on.
Fraction
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
al values are indicated by a separator
Decimal separator
Different symbols have been and are used for the decimal mark. The choice of symbol for the decimal mark affects the choice of symbol for the thousands separator used in digit grouping. Consequently the latter is treated in this article as well....
, which varies by locale
Locale
In computing, locale is a set of parameters that defines the user's language, country and any special variant preferences that the user wants to see in their user interface...
. Usually this separator is a period or full stop
Full stop
A full stop is the punctuation mark commonly placed at the end of sentences. In American English, the term used for this punctuation is period. In the 21st century, it is often also called a dot by young people...
, or a comma
Comma (punctuation)
The comma is a punctuation mark. It has the same shape as an apostrophe or single closing quotation mark in many typefaces, but it differs from them in being placed on the baseline of the text. Some typefaces render it as a small line, slightly curved or straight but inclined from the vertical, or...
. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates 10-1 (0.1), the second position 10-2 (0.01), and so on for each successive position.
As an example, the number 2674 in a base 10 numeral system is :
+ ( 6 × 102 ) + ( 7 × 101 ) + ( 4 × 100 )
or
+ ( 6 × 100 ) + ( 7 × 10 ) + ( 4 × 1 ).
Sexagesimal system
The sexagesimal or base sixty system was used for the integral and fractional portions of Babylonian numeralsBabylonian numerals
Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record....
and other mesopotamian systems, by Hellenistic astronomers using Greek numerals
Greek numerals
Greek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Ionian numerals, Milesian numerals , Alexandrian numerals, or alphabetic numerals...
for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional.
Modern time separates each position by a colon or point. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be 10°25'59" (10 degrees 25 minutes 59 seconds). In both cases, only minutes and seconds use sexagesimal notation — angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second. This contrasts with the numbers used by Hellenistic and Renaissance
Renaissance
The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...
astronomers, who used thirds, fourths, etc. for finer increments. Where we might write 10°25'59.392", they would have written
Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans.
In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon to separate the integral and fractional portions of the number and using a comma to separate the positions within each portion. For example, the mean synodic month used by both Babylonian and Hellenistic astronomers and still used in the Hebrew calendar
Hebrew calendar
The Hebrew calendar , or Jewish calendar, is a lunisolar calendar used today predominantly for Jewish religious observances. It determines the dates for Jewish holidays and the appropriate public reading of Torah portions, yahrzeits , and daily Psalm reading, among many ceremonial uses...
is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.
Computing
In computingComputing
Computing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...
, the binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
(base 2) and hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
(base 16) bases are used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as 'shorthand' for binary - every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f).
The octal
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit.
Other bases in human language
Base-12 systems (duodecimalDuodecimal
The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written as 'A', 'T' or 'X', and the number eleven as 'B' or 'E'...
or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many factors
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 102, hundred, commerce developed a word for 122, gross. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency Pound Sterling
Pound sterling
The pound sterling , commonly called the pound, is the official currency of the United Kingdom, its Crown Dependencies and the British Overseas Territories of South Georgia and the South Sandwich Islands, British Antarctic Territory and Tristan da Cunha. It is subdivided into 100 pence...
(GBP) partially used base-12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly, £sd.
The Maya civilization
Maya civilization
The Maya is a Mesoamerican civilization, noted for the only known fully developed written language of the pre-Columbian Americas, as well as for its art, architecture, and mathematical and astronomical systems. Initially established during the Pre-Classic period The Maya is a Mesoamerican...
and other civilizations of pre-Columbian
Pre-Columbian
The pre-Columbian era incorporates all period subdivisions in the history and prehistory of the Americas before the appearance of significant European influences on the American continents, spanning the time of the original settlement in the Upper Paleolithic period to European colonization during...
Mesoamerica
Mesoamerica
Mesoamerica is a region and culture area in the Americas, extending approximately from central Mexico to Belize, Guatemala, El Salvador, Honduras, Nicaragua, and Costa Rica, within which a number of pre-Columbian societies flourished before the Spanish colonization of the Americas in the 15th and...
used base-20 (vigesimal
Vigesimal
The vigesimal or base 20 numeral system is based on twenty .- Places :...
), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western Africa
Africa
Africa is the world's second largest and second most populous continent, after Asia. At about 30.2 million km² including adjacent islands, it covers 6% of the Earth's total surface area and 20.4% of the total land area...
.
Remnants of a Gaulish
Gaulish language
The Gaulish language is an extinct Celtic language that was spoken by the Gauls, a people who inhabited the region known as Gaul from the Iron Age through the Roman period...
base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is soixante-cinq (literally, "sixty [and] five"), while seventy-five is soixante-quinze (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty (somewhat similar to the archaic English manner of speaking of "scores
20 (number)
20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.-In mathematics:*20 is the basis for vigesimal number systems....
", probably originating from the same underlying Celtic system). For example, eighty-two is quatre-vingt-deux (literally, four twenty[s] [and] two), while ninety-two is quatre-vingt-douze (literally, four twenty[s] [and] twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties [and] thirteen, and so on.
The Irish language
Irish language
Irish , also known as Irish Gaelic, is a Goidelic language of the Indo-European language family, originating in Ireland and historically spoken by the Irish people. Irish is now spoken as a first language by a minority of Irish people, as well as being a second language of a larger proportion of...
also used base-20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A remnant of this system may be seen in the modern word for 40, daoichead.
Danish numerals display a similar base-20
Vigesimal
The vigesimal or base 20 numeral system is based on twenty .- Places :...
structure.
The Maori language of New Zealand also has evidence of an underlying base-20 system as seen in the terms "Te Hokowhitu a Tu" referring to a war party (literally "the seven 20s of Tu") and "Tama-hokotahi", referring to a great warrior ("the one man equal to 20").
The binary system
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
was used in the Egyptian Old Kingdom, 3,000 BCE to 2,050 BCE. It was cursive by rounding off rational numbers smaller than 1 to , with a 1/64 term thrown away (the system was called the Eye of Horus).
A number of Australian Aboriginal languages
Australian Aboriginal languages
The Australian Aboriginal languages comprise several language families and isolates native to the Australian Aborigines of Australia and a few nearby islands, but by convention excluding the languages of Tasmania and the Torres Strait Islanders...
employ binary or binary-like counting systems. For example, in Kala Lagaw Ya
Kala Lagaw Ya
Kala Lagaw Ya is a language belonging to all the western and central Torres Strait Islands, Queensland, Australia. On some islands it has now largely been replaced by Brokan...
, the numbers one through six are urapon, ukasar, ukasar-urapon, ukasar-ukasar, ukasar-ukasar-urapon, ukasar-ukasar-ukasar.
North and Central American natives used base 4 (quaternary
Quaternary numeral system
Quaternary is the base- numeral system. It uses the digits 0, 1, 2 and 3 to represent any real number.It shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation and the characteristics of the representations of...
) to represent the four cardinal directions. Mesoamericans tended to add a second base 5 system to create a modified base 20 system.
A base-5 system (quinary
Quinary
Quinary is a numeral system with five as the base. A possible origination of a quinary system is that there are five fingers on either hand. The base five is stated from 0-4...
) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base 10, base 20, and base 60.
A base-8 system (octal
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
) was devised by the Yuki tribe
Yuki tribe
The Yuki are a Native American people from the zone of Round Valley, in what today is part of the territory of Mendocino County, Northern California. Yuki tribes are thought to have settled as far south as Hood Mountain in present-day Sonoma County...
of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base 8 system (or a system which could only count up to 8) with a base 10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for 'new', newo-, suggesting that the number 9 had been recently invented and called the 'new number'.
Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages
African languages
There are over 2100 and by some counts over 3000 languages spoken natively in Africa in several major language families:*Afro-Asiatic spread throughout the Middle East, North Africa, the Horn of Africa, and parts of the Sahel...
the word for five is the same as "hand" or "fist" (Dyola language
Dyola language
Jola or Diola, also called Jola-Fonyi, is a language spoken by half a million people in the Casamance region of Senegal, and neighboring countries. The French spelling is Diola and Diola Fogny. Jola people themselves say that a person is ajoola and they speak joola....
of Guinea-Bissau
Guinea-Bissau
The Republic of Guinea-Bissau is a country in West Africa. It is bordered by Senegal to the north, and Guinea to the south and east, with the Atlantic Ocean to its west....
, Banda language
Banda languages
Banda is a family of Ubangian languages spoken by the Banda people of Central Africa.-Languages:Olson classfies the Banda family as follows :*Central**Central Banda...
of Central Africa
Central Africa
Central Africa is a core region of the African continent which includes Burundi, the Central African Republic, Chad, the Democratic Republic of the Congo, and Rwanda....
). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan
Sudan
Sudan , officially the Republic of the Sudan , is a country in North Africa, sometimes considered part of the Middle East politically. It is bordered by Egypt to the north, the Red Sea to the northeast, Eritrea and Ethiopia to the east, South Sudan to the south, the Central African Republic to the...
region.
The Telefol language
Telefol language
Telefol is a language spoken by the Telefol people in Papua New Guinea, notable for possessing a base-27 numeral system.-History:The Iligimin people also spoke Telefol, but they were defeated by the Telefol proper.-Orthography:...
, spoken in Papua New Guinea
Papua New Guinea
Papua New Guinea , officially the Independent State of Papua New Guinea, is a country in Oceania, occupying the eastern half of the island of New Guinea and numerous offshore islands...
, is notable for possessing a base-27 numeral system.
Non-standard positional numeral systems
Interesting properties exist when the base is not fixed or positive and when the digit symbol sets denote negative values. There are many more variations. These systems are of practical and theoretic value to computer scientists.Balanced ternary
Balanced ternary
Balanced ternary is a non-standard positional numeral system , useful for comparison logic. It is a ternary system, but unlike the standard ternary system, the digits have the values −1, 0, and 1...
uses a base of 3 but the digit set is
,0,1} instead of {0,1,2}. The "" has an equivalent value of −1.
The negation of a number is easily formed by switching the on the 1s.
This system can be used to solve the balance problem, which requires finding a minimal set of known counter-weights to determine an unknown weight.
Weights of 1, 3, 9, ... 3n known units can be used to determine any unknown weight up to 1 + 3 + ... + 3n units.
A weight can be used on either side of the balance or not at all.
Weights used on the balance pan with the unknown weight are designated with , with 1 if used on the empty pan, and with 0 if not used.
If an unknown weight W is balanced with 3 (31) on its pan and 1 and 27 (30 and 33) on the other, then its weight in decimal is 25 or 101 in balanced base 3.
(1013 = 1 × 33 + 0 × 32 − 1 × 31 + 1 × 30 = 25).
The factorial number system uses a varying radix, giving factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
s as place values; they are related to Chinese remainder theorem
Chinese remainder theorem
The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...
and Residue number system
Residue number system
A residue number system represents a large integer using a set of smaller integers, so that computation may be performed more efficiently...
enumerations. This system effectively enumerates permutations. A derivative of this uses the Towers of Hanoi puzzle configuration as a counting system. The configuration of the towers can be put into 1 to 1 correspondence with the decimal count of the step at which the configuration occurs and vice versa.
| Decimal equivalents: | −3 | −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Balanced base 3: | 0 | 1 | 0 | 1 | 1 | 10 | 11 | 1 | 10 | 11 | 10 | |
| Base −2: | 1101 | 10 | 11 | 0 | 1 | 110 | 111 | 100 | 101 | 11010 | 11011 | 11000 |
| Factoroid: | 0 | 10 | 100 | 110 | 200 | 210 | 1000 | 1010 | 1100 |
Non-positional positions
Each position does not need to be positional itself. Babylonian sexagesimal numerals were positional, but in each position were groups of two kinds of wedges representing ones and tens (a narrow vertical wedge ( | ) and an open left pointing wedge (<)) — up to 14 symbols per position (5 tens (<<<<<) and 9 ones ( ||||||||| ) grouped into one or two near squares containing up to three tiers of symbols, or a place holder (\\) for the lack of a position). Hellenistic astronomers used one or two alphabetic Greek numerals for each position (one chosen from 5 letters representing 10–50 and/or one chosen from 9 letters representing 1–9, or a zero symbol).See also
- Numeral systemNumeral systemA numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
- Hindu-Arabic numeral systemHindu-Arabic numeral systemThe Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
- Non-standard positional numeral systemsNon-standard positional numeral systemsNon-standard positional numeral systems here designates numeral systems that may be denoted positional systems, but that deviate in one way or another from the following description of standard positional systems:...
:Category:Positional numeral systems
- Mixed radixMixed radixMixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a multiple of the next smaller one, but not by the same...
- AlgorismAlgorismAlgorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist...
- Subtractive notationSubtractive notationSubtractive notation is an early form of positional notation used with Roman numerals as a shorthand to replace four or five characters in a numeral representing a number with usually just two characters. Using subtractive notation the numeral VIIII becomes simply IX...
External links
- Accurate Base Conversion
- The Development of Hindu Arabic and Traditional Chinese Arithmetics
- Implementation of Base Conversion at cut-the-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
- Learn to count other bases on your fingers