Duodecimal
Encyclopedia
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' (another common notation, introduced by Sir Isaac Pitman
, is to use a rotated '2' for ten and a reversed '3' for eleven). The number twelve (that is, the number written as '12' in the base ten
numerical system) is instead written as '10' in duodecimal (meaning "1 dozen
and 0 units", instead of "1 ten and 0 units"), whereas the digit string '12' means "1 dozen and 2 units" (i.e. the same number that in decimal is written as '14'). Similarly, in duodecimal '100' means "1 gross
", '1000' means "1 great gross", and '0.1' means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth").
The number twelve, a highly composite number
, is the smallest number with four non-trivial factor
s (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the subitizing
range. As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers (whereas ten has only two non-trivial factors: 2 and 5, with neither 3 nor 4), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. Of its factors, 2 and 3 are prime
, which means the reciprocals
of all 3-smooth
numbers (such as 2, 3, 4, 6, 8, 9...) have a terminating representation in duodecimal. In particular, the five most elementary fractions (½, ⅓, ⅔, ¼ and ¾) all have a short terminating representation in duodecimal (0.6, 0.4, 0.8, 0.3 and 0.9, respectively), and twelve is the smallest radix with this feature (since it is the least common multiple
of 3 and 4). This all makes it a more convenient number system for computing fractions than most other number systems in common use, such as the decimal
, vigesimal
, binary
, octal
and hexadecimal
systems, although the sexagesimal system (where the reciprocals of all 5-smooth
numbers terminate) does better in this respect (but at the cost of an unwieldy multiplication table).
Languages using duodecimal number systems are uncommon. Languages in the Nigeria
n Middle Belt such as Janji, Gbiri-Niragu (Kahugu), the Nimbia dialect of Gwandara; the Chepang
language of Nepal
and the Mahl language of Minicoy Island in India
are known to use duodecimal numerals. In fiction, J. R. R. Tolkien
's Elvish languages
use a hybrid decimal-duodecimal system, primarily decimal but with special names for multiples of six.
Germanic languages
have special words for 11 and 12, such as eleven and twelve in English
, which are often misinterpreted as vestiges of a duodecimal system. However, they are considered to come from Proto-Germanic *ainlif and *twalif (respectively one left and two left), both of which were decimal.
Historically, units of time
in many civilization
s are duodecimal. There are twelve signs of the zodiac
, twelve months in a year, and the Babylonians had twelve hours in day (although at some point this was changed to 24). Traditional Chinese calendar
s, clocks, and compasses are based on the twelve Earthly Branches
.
Being a versatile denominator in fractions may explain why we have 12 inches in an imperial foot, 12 ounces in a troy
pound, 12 old British pence
in a shilling
, 24 (12×2) hours in a day, and many other items counted by the dozen
, gross
(144
, square
of 12) or great gross (1728
, cube
of 12). The Romans used a fraction system based on 12, including the uncia
which became both the English words ounce
and inch. Pre-decimalisation
, the United Kingdom
and Republic of Ireland
used a mixed duodecimal-vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the pound sterling
or Irish pound
), and Charlemagne
established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places.
The importance of 12 has been attributed to the number of lunar cycles in a year, and also to the fact that humans have 12 finger bones (phalanges) on one hand (three on each of four fingers). It is possible to count to 12 with your thumb acting as a pointer, touching each finger bone in turn. A traditional finger counting
system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. In this system, the one (usually right) hand counts repeatedly to 12, displaying the number of iterations on the other (usually left), until five dozens, i. e. the 60, are full.
can be written as A, eleven
can be written as B, and twelve is written as 10. For alternative symbols, see below.
According to this notation, duodecimal 50 expresses the same quantity as decimal 60
(= five times twelve), duodecimal 60 is equivalent to decimal 72
(= six times twelve = half a gross), duodecimal 100 has the same value as decimal 144
(= twelve times twelve = one gross), etc.
The number 12 has six factors, which are 1, 2, 3, 4, 6, and 12
, of which 2 and 3 are prime
. The decimal system has only four factors, which are 1, 2, 5, and 10
; of which 2 and 5 are prime. Vigesimal adds two factors to those of ten, namely 4 and 20
, but no additional prime factor. Although twenty has 6 factors, 2 of them prime, similarly to twelve, it is also a much larger base (i.e., the digit set and the multiplication table are much larger). Binary has only two factors, 1 and 2, the latter being prime. Hexadecimal has five factors, adding 4, 8 and 16
to those of 2, but no additional prime. Trigesimal
is the smallest system that has three different prime factors (all of the three smallest primes: 2, 3 and 5) and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30), the smallest system that has four different prime factors is Base 210 and the pattern follows the primorial
s. Sexagesimal -- which the ancient Sumerians and Babylonia
ns among others actually used—adds the four convenient factors 4, 12, 20, and 60 to this but no new prime factors.
123,456.78 = 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08
This decomposition works the same no matter what base the number is expressed in. Just isolate each non-zero digit, padding them with as many zeros as necessary to preserve their respective place values. If the digits in the given number include zeroes (for example, 102,304.05), these are, of course, left out in the digit decomposition (102,304.05 = 100,000 + 2,000 + 300 + 4 + 0.05). Then we use the digit conversion tables to obtain the equivalent value in the target base for each digit. If the given number is in dozenal and the target base is decimal, we get:
(dozenal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (decimal) 248,832 + 41,472 + 5,184 + 576 + 60 + 6 + 0.58333333333... + 0.05555555555...
Now, since the summands are already converted to base ten, we use the usual decimal arithmetic to perform the addition and recompose the number, arriving at the conversion result:
Dozenal -----> Decimal
100,000 = 248,832
20,000 = 41,472
3,000 = 5,184
400 = 576
50 = 60
+ 6 = + 6
0.7 = 0.58333333333...
0.08 = 0.05555555555...
--------------------------------------------
123,456.78 = 296,130.63888888888...
That is, (dozenal) 123,456.78 equals (decimal) 296,130.63888888888... ≈ 296,130.64
If the given number is in decimal and the target base is dozenal, the method is basically same. Using the digit conversion tables:
(decimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (dozenal) 49,A54 + B,6A8 + 1,8A0 + 294 + 42 + 6 + 0.84972497249724972497... + 0.0B62...
However, in order to do this sum and recompose the number, we now have to use the addition tables for dozenal, instead of the addition tables for decimal most people are already familiar with, because the summands are now in base twelve and so the arithmetic with them has to be in dozenal as well. In decimal, 6 + 6 equals 12, but in dozenal it equals 10; so if we used decimal arithmetic with dozenal numbers we would arrive at an incorrect result. Doing the arithmetic properly in dozenal, we get the result:
Decimal -----> Dozenal
100,000 = 49,A54
20,000 = B,6A8
3,000 = 1,8A0
400 = 294
50 = 42
+ 6 = + 6
0.7 = 0.84972497249724972497...
0.08 = 0.0B62...
--------------------------------------------------------
123,456.78 = 5B,540.943A...
That is, (decimal) 123,456.78 equals (dozenal) 5B,540.943A... ≈ 5B,540.94
s may be simple:
= 0.6 = 0.4 = 0.3 = 0.2 = 0.16 = 0.14
or complicated
= 0.24972497... recurring (easily rounded to 0.25) = 0.186A35186A35... recurring (easily rounded to 0.187) = 0.124972497... recurring (rounded to 0.125) = 0.11111... recurring (rounded to 0.11) = 0.0B0B... recurring (rounded to 0.0B)
As explained in recurring decimals, whenever an irreducible fraction
is written in radix point
notation in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factor
s of its denominator are also prime factors of the base. Thus, in base-ten (= 2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: = , = and = can be expressed exactly as 0.125, 0.05 and 0.002 respectively. and , however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, is exact; and recur because they include 5 as a factor; is exact; and recurs, just as it does in decimal.
problems than factors of 5 (or would be, were it not for the decimal system having influenced most cultures). Thus, in practical applications, the nuisance of recurring decimals is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.
However, when recurring fractions do occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because 12
(twelve) is between two prime number
s, 11
(eleven) and 13
(thirteen), whereas ten is adjacent to composite number
9. Nonetheless, having a shorter or longer period doesn't help the main inconvenience that one does not get a finite representation for such fractions in the given base (so rounding
, which introduces inexactitude, is necessary to handle them in calculations), and overall one is more likely to have to deal with infinite recurring digits when fractions are expressed in decimal than in duodecimal, because one out of every three consecutive numbers contains the prime factor 3 in its factorization, while only one out of every five contains the prime factor 5. All other prime factors, except 2, are not shared by either ten or twelve, so they do not
influence the relative likeliness of encountering recurring digits (any irreducible fraction that contains any of these other factors in its denominator will recur in either base). Also, the prime factor 2 appears twice in the factorization of twelve, while only once in the factorization of ten; which means that most fractions whose denominators are powers of two
will have a shorter, more convenient terminating representation in dozenal than in decimal (e.g., 1/(2^{2}) = 0.25 dec = 0.3 doz; 1/(2^{3}) = 0.125 dec = 0.16 doz; 1/(2^{4}) = 0.0625 dec = 0.09 doz; 1/(2^{5}) = 0.03125 dec = 0.046 doz; etc.).
s, none of them has a finite representation in any of the rational
-based positional number systems (such as the decimal and duodecimal ones); this is because a rational-based positional number system is essentially nothing but a way of expressing quantities as a sum of fractions whose denominators are powers of the base, and by definition no finite sum of rational numbers can ever result in an irrational number. For example, 123.456 = 1 × 1/10^{−2} + 2 × 1/10^{−1} + 3 × 1/10^{0} + 4 × 1/10^{1} + 5 × 1/10^{2} + 6 × 1/10^{3} (this is also the reason why fractions that contain prime factors in their denominator not in common with those of the base do not have a terminating representation in that base). Moreover, the infinite series of digits of an irrational number doesn't exhibit a pattern of repetition; instead, the different digits succeed in a seemingly random fashion. The following chart compares the first few digits of the decimal and duodecimal representation of several of the most important algebraic
and transcendental
irrational numbers. Some of these numbers may be perceived as having fortuitous patterns, making them easier to memorize, when represented in one base or the other.
The first few digits of the decimal and dozenal representation of another important number, the Euler-Mascheroni constant
(the status of which as a rational or irrational number is not yet known), are:
Rather than the symbols 'A' for ten and 'B' for eleven as used in hexadecimal
notation and vigesimal
notation (or 'T' and 'E' for ten and eleven), he suggested in his book and used a script X and a script E, (U+
1D4B3) and (U+2130), to represent the digits ten and eleven respectively, because, at least on a page of Roman script, these characters were distinct from any existing letters or numerals, yet were readily available in printers' fonts. He chose for its resemblance to the Roman numeral X, and as the first letter of the word "eleven".
Another popular notation, introduced by Sir Isaac Pitman
, is to use a rotated 2 (resembling a script τ for 'ten') to represent ten and a rotated or horizontally flipped 3 (which again resembles ε) to represent eleven. This is the convention commonly employed by the Dozenal Society of Great Britain and has the advantage of being easily recognizable as digits because of their resemblance in shape to existing digits. On the other hand, the Dozenal Society of America adopted for some years the convention of using an asterisk
* for ten and a hash
# for eleven. The reason was the symbol * resembles a struck-through X while # resembles a doubly-struck-through 11, and both symbols are already present in telephone
dial
s. However, critics pointed out these symbols do not look anything like digits. Some other systems write 10 as Φ (a combination of 1 and 0) and eleven as a cross of two lines (+, x, or † for example). Problems with these symbols are evident, most notably that most of them can not be represented in the seven-segment display
of most calculator
displays ( being an exception, although "E" is used on calculators to indicate an error message
). However, 10 and 11 do fit, both within a single digit (11 fits as is, while the 10 has to be tilted sideways, resulting in a character that resembles an O with a macron
, ō or o). A and B also fit (although B must be represented as lowercase "b" and as such, 6 must have a bar over it to distinguish the two figures) and are used on calculators for bases higher than ten.
In "Little Twelvetoes", American television series Schoolhouse Rock!
portrayed an alien child using base-twelve arithmetic, using "dek", "el" and "doh" as names for ten, eleven and twelve, and Andrews' script-X and script-E for the digit symbols. ("Dek" is from the prefix "deca", "el" being short for "eleven" and "doh" an apparent shortening of "dozen".)
The Dozenal Society of America and the Dozenal Society of Great Britain promote widespread adoption of the base-twelve system. They use the word dozenal instead of "duodecimal" because the latter comes from Latin roots that express twelve in base-ten terminology.
The renowned mathematician and mental calculator Alexander Craig Aitken
was an outspoken advocate of the advantages and superiority of duodecimal over decimal:
In Leo Frankowski
's Conrad Stargard
novels, Conrad introduces a duodecimal system of arithmetic at the suggestion of a merchant, who is accustomed to buying and selling goods in dozens and grosses, rather than tens or hundreds. He then invents an entire system of weights and measures in base twelve, including a clock with twelve hours in a day, rather than twenty-four hours.
In Lee Carroll
's Kryon: Alchemy of the Human Spirit, a chapter is dedicated to the advantages of the duodecimal system. The duodecimal system is supposedly suggested by Kryon (one of the widely popular New Age
channeled entities) for all-round use, aiming at better and more natural representation of nature of the Universe through mathematics. An individual article "Mathematica" by James D. Watt (included in the above publication) exposes a few of the unusual symmetry connections between the duodecimal system and the golden ratio
, as well as provides numerous number symmetry-based arguments for the universal nature of the base-12 number system.
Positional notation
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...
numeral system
Numeral 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....
using twelve
12 (number)
12 is the natural number following 11 and preceding 13.The word "twelve" is the largest number with a single-morpheme name in English. Etymology suggests that "twelve" arises from the Germanic compound twalif "two-leftover", so a literal translation would yield "two remaining [after having ten...
as its base
Radix
In mathematical numeral systems, the base or radix for the simplest case is 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 ten, because it uses the ten digits from 0 through 9.In any numeral...
. In this system, the number ten
10 (number)
10 is an even natural number following 9 and preceding 11.-In mathematics:Ten is a composite number, its proper divisors being , and...
may be written as 'A', 'T' or 'X', and the number eleven
11 (number)
11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...
as 'B' or 'E' (another common notation, introduced by Sir Isaac Pitman
Isaac Pitman
Sir Isaac Pitman , knighted in 1894, developed the most widely used system of shorthand, known now as Pitman shorthand. He first proposed this in Stenographic Soundhand in 1837. Pitman was a qualified teacher and taught at a private school he founded in Wotton-under-Edge...
, is to use a rotated '2' for ten and a reversed '3' for eleven). The number twelve (that is, the number written as '12' in the base ten
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
numerical system) is instead written as '10' in duodecimal (meaning "1 dozen
Dozen
A dozen is a grouping of approximately twelve. The dozen may be one of the earliest primitive groupings, perhaps because there are approximately a dozen cycles of the moon or months in a cycle of the sun or year...
and 0 units", instead of "1 ten and 0 units"), whereas the digit string '12' means "1 dozen and 2 units" (i.e. the same number that in decimal is written as '14'). Similarly, in duodecimal '100' means "1 gross
Gross (unit)
A gross is equal to a dozen dozen, i.e. 12 × 12 = 144.It can be used in duodecimal counting. The use of gross likely originated from the fact that 144 can be counted on the fingers using the fingertips and first two joints of each finger when marked by the thumb of one hand. The other hand...
", '1000' means "1 great gross", and '0.1' means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth").
The number twelve, a highly composite number
Highly composite number
A highly composite number is a positive integer with more divisors than any positive integer smaller than itself.The initial or smallest twenty-one highly composite numbers are listed in the table at right....
, is the smallest number with four non-trivial factor
Integer factorization
In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer....
s (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the subitizing
Subitizing and counting
Subitizing, coined in 1949 by E.L. Kaufman et al. refers to the rapid, accurate, and confident judgments of number performed for small numbers of items. The term is derived from the Latin adjective subitus and captures a feeling of immediately knowing how many items lie within the visual scene,...
range. As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers (whereas ten has only two non-trivial factors: 2 and 5, with neither 3 nor 4), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. Of its factors, 2 and 3 are prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, which means the reciprocals
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
of all 3-smooth
Smooth number
In number theory, a smooth number is an integer which factors completely into small prime numbers. The term seems to have been coined by Leonard Adleman. Smooth numbers are especially important in cryptography relying on factorization.-Definition:...
numbers (such as 2, 3, 4, 6, 8, 9...) have a terminating representation in duodecimal. In particular, the five most elementary fractions (½, ⅓, ⅔, ¼ and ¾) all have a short terminating representation in duodecimal (0.6, 0.4, 0.8, 0.3 and 0.9, respectively), and twelve is the smallest radix with this feature (since it is the least common multiple
Least common multiple
In arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b...
of 3 and 4). This all makes it a more convenient number system for computing fractions than most other number systems in common use, such as the decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
, vigesimal
Vigesimal
The vigesimal or base 20 numeral system is based on twenty .- Places :...
, 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...
, 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 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...
systems, although the sexagesimal system (where the reciprocals of all 5-smooth
Regular number
Regular numbers are numbers that evenly divide powers of 60. As an example, 602 = 3600 = 48 × 75, so both 48 and 75 are divisors of a power of 60...
numbers terminate) does better in this respect (but at the cost of an unwieldy multiplication table).
Origin
- In this section, numerals are based on decimal placesNumerical digitA 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...
. For example, 10 means ten10 (number)10 is an even natural number following 9 and preceding 11.-In mathematics:Ten is a composite number, its proper divisors being , and...
, 12 means twelve12 (number)12 is the natural number following 11 and preceding 13.The word "twelve" is the largest number with a single-morpheme name in English. Etymology suggests that "twelve" arises from the Germanic compound twalif "two-leftover", so a literal translation would yield "two remaining [after having ten...
.
Languages using duodecimal number systems are uncommon. Languages in the Nigeria
Nigeria
Nigeria , officially the Federal Republic of Nigeria, is a federal constitutional republic comprising 36 states and its Federal Capital Territory, Abuja. The country is located in West Africa and shares land borders with the Republic of Benin in the west, Chad and Cameroon in the east, and Niger in...
n Middle Belt such as Janji, Gbiri-Niragu (Kahugu), the Nimbia dialect of Gwandara; the Chepang
Chepang
Chepang is the commonly used name given to an indigenous ethnic group living in central and southern Nepal.The language is also known as Chepang but is called Chyo-bang by the people themselves. Some Bahun Chettri castes call these people the "Praja" meaning "political subjects"...
language of Nepal
Nepal
Nepal , officially the Federal Democratic Republic of Nepal, is a landlocked sovereign state located in South Asia. It is located in the Himalayas and bordered to the north by the People's Republic of China, and to the south, east, and west by the Republic of India...
and the Mahl language of Minicoy Island in India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
are known to use duodecimal numerals. In fiction, J. R. R. Tolkien
J. R. R. Tolkien
John Ronald Reuel Tolkien, CBE was an English writer, poet, philologist, and university professor, best known as the author of the classic high fantasy works The Hobbit, The Lord of the Rings, and The Silmarillion.Tolkien was Rawlinson and Bosworth Professor of Anglo-Saxon at Pembroke College,...
's Elvish languages
Elvish languages
J. R. R. Tolkien constructed many Elvish languages. These were the languages spoken by the tribes of his Elves. Tolkien was a philologist by profession, and spent much time on his constructed languages. The Elvish languages were the first thing he imagined for his secondary world. Tolkien said that...
use a hybrid decimal-duodecimal system, primarily decimal but with special names for multiples of six.
Germanic languages
Germanic languages
The Germanic languages constitute a sub-branch of the Indo-European language family. The common ancestor of all of the languages in this branch is called Proto-Germanic , which was spoken in approximately the mid-1st millennium BC in Iron Age northern Europe...
have special words for 11 and 12, such as eleven and twelve in English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...
, which are often misinterpreted as vestiges of a duodecimal system. However, they are considered to come from Proto-Germanic *ainlif and *twalif (respectively one left and two left), both of which were decimal.
Historically, units of time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
in many civilization
Civilization
Civilization is a sometimes controversial term that has been used in several related ways. Primarily, the term has been used to refer to the material and instrumental side of human cultures that are complex in terms of technology, science, and division of labor. Such civilizations are generally...
s are duodecimal. There are twelve signs of the zodiac
Zodiac
In astronomy, the zodiac is a circle of twelve 30° divisions of celestial longitude which are centred upon the ecliptic: the apparent path of the Sun across the celestial sphere over the course of the year...
, twelve months in a year, and the Babylonians had twelve hours in day (although at some point this was changed to 24). Traditional Chinese calendar
Chinese calendar
The Chinese calendar is a lunisolar calendar, incorporating elements of a lunar calendar with those of a solar calendar. It is not exclusive to China, but followed by many other Asian cultures as well...
s, clocks, and compasses are based on the twelve Earthly Branches
Earthly Branches
The Earthly Branches provide one Chinese system for reckoning time.This system was built from observations of the orbit of Jupiter. Chinese astronomers divided the celestial circle into 12 sections to follow the orbit of Suìxīng . Astronomers rounded the orbit of Suixing to 12 years...
.
Being a versatile denominator in fractions may explain why we have 12 inches in an imperial foot, 12 ounces in a troy
Troy weight
Troy weight is a system of units of mass customarily used for precious metals, gemstones, and black powder.There are 12 troy ounces per troy pound, rather than the 16 ounces per pound found in the more common avoirdupois system. The troy ounce is 480 grains, compared with the avoirdupois ounce,...
pound, 12 old British pence
British One Penny coin (pre-decimal)
The English Penny, originally a coin of 1.3 to 1.5 g pure silver, includes the penny introduced around the year 785 by King Offa of Mercia. However, his coins were similar in size and weight to the continental deniers of the period, and to the Anglo-Saxon sceats which had gone before it, which were...
in a shilling
Shilling
The shilling is a unit of currency used in some current and former British Commonwealth countries. The word shilling comes from scilling, an accounting term that dates back to Anglo-Saxon times where it was deemed to be the value of a cow in Kent or a sheep elsewhere. The word is thought to derive...
, 24 (12×2) hours in a day, and many other items counted by the dozen
Dozen
A dozen is a grouping of approximately twelve. The dozen may be one of the earliest primitive groupings, perhaps because there are approximately a dozen cycles of the moon or months in a cycle of the sun or year...
, gross
Gross (unit)
A gross is equal to a dozen dozen, i.e. 12 × 12 = 144.It can be used in duodecimal counting. The use of gross likely originated from the fact that 144 can be counted on the fingers using the fingertips and first two joints of each finger when marked by the thumb of one hand. The other hand...
(144
144 (number)
144 is the natural number following 143 and preceding 145. 144 is a dozen dozens, or one gross.-In mathematics:It is the twelfth Fibonacci number, and the largest one to also be a square, as the square of 12 , following 89 and preceding 233.144 is the smallest number with exactly 15 divisors.144 is...
, square
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
of 12) or great gross (1728
1728 (number)
1728 is the natural number following 1727 and preceding 1729. 1728 is a dozen gross, one great gross .-In mathematics:...
, cube
Cube (arithmetic)
In arithmetic and algebra, the cube of a number n is its third power — the result of the number multiplying by itself three times:...
of 12). The Romans used a fraction system based on 12, including the uncia
Uncia (length)
An ' is an ancient Roman unit of length that roughly corresponds to an inch. One uncia is 0.97 inches or 24.6 millimeters. There are twelve in one pes....
which became both the English words ounce
Ounce
The ounce is a unit of mass with several definitions, the most commonly used of which are equal to approximately 28 grams. The ounce is used in a number of different systems, including various systems of mass that form part of the imperial and United States customary systems...
and inch. Pre-decimalisation
Decimal Day
Decimal Day was the day the United Kingdom and Ireland decimalised their currencies.-Old system:Under the old currency of pounds, shillings and pence, the pound was made up of 240 pence , with 12 pence in a shilling and 20 shillings in a...
, the United Kingdom
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
and Republic of Ireland
Republic of Ireland
Ireland , described as the Republic of Ireland , is a sovereign state in Europe occupying approximately five-sixths of the island of the same name. Its capital is Dublin. Ireland, which had a population of 4.58 million in 2011, is a constitutional republic governed as a parliamentary democracy,...
used a mixed duodecimal-vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the 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...
or Irish pound
Irish pound
The Irish pound was the currency of Ireland until 2002. Its ISO 4217 code was IEP, and the usual notation was the prefix £...
), and Charlemagne
Charlemagne
Charlemagne was King of the Franks from 768 and Emperor of the Romans from 800 to his death in 814. He expanded the Frankish kingdom into an empire that incorporated much of Western and Central Europe. During his reign, he conquered Italy and was crowned by Pope Leo III on 25 December 800...
established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places.
The importance of 12 has been attributed to the number of lunar cycles in a year, and also to the fact that humans have 12 finger bones (phalanges) on one hand (three on each of four fingers). It is possible to count to 12 with your thumb acting as a pointer, touching each finger bone in turn. A traditional finger counting
Finger counting
Finger counting, or dactylonomy, is the art of counting along one's fingers. Though marginalized in modern societies by Arabic numerals, formerly different systems flourished in many cultures, including educated methods far more sophisticated than the one-by-one finger count taught today in...
system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. In this system, the one (usually right) hand counts repeatedly to 12, displaying the number of iterations on the other (usually left), until five dozens, i. e. the 60, are full.
Places
In a duodecimal place system, ten10 (number)
10 is an even natural number following 9 and preceding 11.-In mathematics:Ten is a composite number, its proper divisors being , and...
can be written as A, eleven
11 (number)
11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...
can be written as B, and twelve is written as 10. For alternative symbols, see below.
According to this notation, duodecimal 50 expresses the same quantity as decimal 60
60 (number)
60 is the natural number following 59 and preceding 61. Being three times twenty, 60 is called "three score" in some older literature.-In mathematics:...
(= five times twelve), duodecimal 60 is equivalent to decimal 72
72 (number)
72 is the natural number following 71 and preceding 73. It is half a gross or 6 dozen .-In mathematics:...
(= six times twelve = half a gross), duodecimal 100 has the same value as decimal 144
144 (number)
144 is the natural number following 143 and preceding 145. 144 is a dozen dozens, or one gross.-In mathematics:It is the twelfth Fibonacci number, and the largest one to also be a square, as the square of 12 , following 89 and preceding 233.144 is the smallest number with exactly 15 divisors.144 is...
(= twelve times twelve = one gross), etc.
Comparison to other numeral systems
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|
2 | 4 | 6 | 8 | A | 10 | 12 | 14 | 16 | 18 | 1A | 20 |
3 | 6 | 9 | 10 | 13 | 16 | 19 | 20 | 23 | 26 | 29 | 30 |
4 | 8 | 10 | 14 | 18 | 20 | 24 | 28 | 30 | 34 | 38 | 40 |
5 | A | 13 | 18 | 21 | 26 | 2B | 34 | 39 | 42 | 47 | 50 |
6 | 10 | 16 | 20 | 26 | 30 | 36 | 40 | 46 | 50 | 56 | 60 |
7 | 12 | 19 | 24 | 2B | 36 | 41 | 48 | 53 | 5A | 65 | 70 |
8 | 14 | 20 | 28 | 34 | 40 | 48 | 54 | 60 | 68 | 74 | 80 |
9 | 16 | 23 | 30 | 39 | 46 | 53 | 60 | 69 | 76 | 83 | 90 |
A | 18 | 26 | 34 | 42 | 50 | 5A | 68 | 76 | 84 | 92 | A0 |
B | 1A | 29 | 38 | 47 | 56 | 65 | 74 | 83 | 92 | A1 | B0 |
10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | A0 | B0 | 100 |
The number 12 has six factors, which are 1, 2, 3, 4, 6, and 12
12 (number)
12 is the natural number following 11 and preceding 13.The word "twelve" is the largest number with a single-morpheme name in English. Etymology suggests that "twelve" arises from the Germanic compound twalif "two-leftover", so a literal translation would yield "two remaining [after having ten...
, of which 2 and 3 are prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. The decimal system has only four factors, which are 1, 2, 5, and 10
10 (number)
10 is an even natural number following 9 and preceding 11.-In mathematics:Ten is a composite number, its proper divisors being , and...
; of which 2 and 5 are prime. Vigesimal adds two factors to those of ten, namely 4 and 20
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....
, but no additional prime factor. Although twenty has 6 factors, 2 of them prime, similarly to twelve, it is also a much larger base (i.e., the digit set and the multiplication table are much larger). Binary has only two factors, 1 and 2, the latter being prime. Hexadecimal has five factors, adding 4, 8 and 16
16 (number)
16 is the natural number following 15 and preceding 17. 16 is a composite number, and a square number, being 42 = 4 × 4. It is the smallest number with exactly five divisors, its proper divisors being , , and ....
to those of 2, but no additional prime. Trigesimal
Base 30
Base 30 or trigesimal is a positional numeral system using 30 as the radix. Digits in this base can be represented using the Arabic numerals 0-9 and the Latin letters A-T....
is the smallest system that has three different prime factors (all of the three smallest primes: 2, 3 and 5) and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30), the smallest system that has four different prime factors is Base 210 and the pattern follows the primorial
Primorial
In mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than multiplying successive positive integers, only successive prime numbers are multiplied...
s. Sexagesimal -- which the ancient Sumerians and Babylonia
Babylonia
Babylonia was an ancient cultural region in central-southern Mesopotamia , with Babylon as its capital. Babylonia emerged as a major power when Hammurabi Babylonia was an ancient cultural region in central-southern Mesopotamia (present-day Iraq), with Babylon as its capital. Babylonia emerged as...
ns among others actually used—adds the four convenient factors 4, 12, 20, and 60 to this but no new prime factors.
Conversion tables to and from decimal
To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under positional notation). Alternatively, one can use digit-conversion tables. The ones provided below can be used to convert any dozenal number between 0.01 and BBB,BBB.BB to decimal, or any decimal number between 0.01 and 999,999.99 to dozenal. To use them, we first decompose the given number into a sum of numbers with only one significant digit each. For example:123,456.78 = 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08
This decomposition works the same no matter what base the number is expressed in. Just isolate each non-zero digit, padding them with as many zeros as necessary to preserve their respective place values. If the digits in the given number include zeroes (for example, 102,304.05), these are, of course, left out in the digit decomposition (102,304.05 = 100,000 + 2,000 + 300 + 4 + 0.05). Then we use the digit conversion tables to obtain the equivalent value in the target base for each digit. If the given number is in dozenal and the target base is decimal, we get:
(dozenal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (decimal) 248,832 + 41,472 + 5,184 + 576 + 60 + 6 + 0.58333333333... + 0.05555555555...
Now, since the summands are already converted to base ten, we use the usual decimal arithmetic to perform the addition and recompose the number, arriving at the conversion result:
Dozenal -----> Decimal
100,000 = 248,832
20,000 = 41,472
3,000 = 5,184
400 = 576
50 = 60
+ 6 = + 6
0.7 = 0.58333333333...
0.08 = 0.05555555555...
--------------------------------------------
123,456.78 = 296,130.63888888888...
That is, (dozenal) 123,456.78 equals (decimal) 296,130.63888888888... ≈ 296,130.64
If the given number is in decimal and the target base is dozenal, the method is basically same. Using the digit conversion tables:
(decimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (dozenal) 49,A54 + B,6A8 + 1,8A0 + 294 + 42 + 6 + 0.84972497249724972497... + 0.0B62...
However, in order to do this sum and recompose the number, we now have to use the addition tables for dozenal, instead of the addition tables for decimal most people are already familiar with, because the summands are now in base twelve and so the arithmetic with them has to be in dozenal as well. In decimal, 6 + 6 equals 12, but in dozenal it equals 10; so if we used decimal arithmetic with dozenal numbers we would arrive at an incorrect result. Doing the arithmetic properly in dozenal, we get the result:
Decimal -----> Dozenal
100,000 = 49,A54
20,000 = B,6A8
3,000 = 1,8A0
400 = 294
50 = 42
+ 6 = + 6
0.7 = 0.84972497249724972497...
0.08 = 0.0B62...
--------------------------------------------------------
123,456.78 = 5B,540.943A...
That is, (decimal) 123,456.78 equals (dozenal) 5B,540.943A... ≈ 5B,540.94
Dozenal to decimal digit conversion
Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. |
100,000 | 248,832 | 10,000 | 20,736 | 1,000 | 1,728 | 100 | 144 | 10 | 12 | 1 | 1 | 0.1 | 0.08 | 0.01 | 0.0069 |
200,000 | 497,664 | 20,000 | 41,472 | 2,000 | 3,456 | 200 | 288 | 20 | 24 | 2 | 2 | 0.2 | 0.1 | 0.02 | 0.013 |
300,000 | 746,496 | 30,000 | 62,208 | 3,000 | 5,184 | 300 | 432 | 30 | 36 | 3 | 3 | 0.3 | 0.25 | 0.03 | 0.0208 |
400,000 | 995,328 | 40,000 | 82,944 | 4,000 | 6,912 | 400 | 576 | 40 | 48 | 4 | 4 | 0.4 | 0. | 0.04 | 0.02 |
500,000 | 1,244,160 | 50,000 | 103,680 | 5,000 | 8,640 | 500 | 720 | 50 | 60 | 5 | 5 | 0.5 | 0.41 | 0.05 | 0.0347 |
600,000 | 1,492,992 | 60,000 | 124,416 | 6,000 | 10,368 | 600 | 864 | 60 | 72 | 6 | 6 | 0.6 | 0.5 | 0.06 | 0.041 |
700,000 | 1,741,824 | 70,000 | 145,152 | 7,000 | 12,096 | 700 | 1008 | 70 | 84 | 7 | 7 | 0.7 | 0.58 | 0.07 | 0.0486 |
800,000 | 1,990,656 | 80,000 | 165,888 | 8,000 | 13,824 | 800 | 1152 | 80 | 96 | 8 | 8 | 0.8 | 0. | 0.08 | 0.0 |
900,000 | 2,239,488 | 90,000 | 186,624 | 9,000 | 15,552 | 900 | 1,296 | 90 | 108 | 9 | 9 | 0.9 | 0.75 | 0.09 | 0.0625 |
A00,000 | 2,488,320 | A0,000 | 207,360 | A,000 | 17,280 | A00 | 1,440 | A0 | 120 | A | 10 | 0.A | 0.8 | 0.0A | 0.069 |
B00,000 | 2,737,152 | B0,000 | 228,096 | B,000 | 19,008 | B00 | 1,584 | B0 | 132 | B | 11 | 0.B | 0.91 | 0.0B | 0.0763 |
Decimal to dozenal digit conversion
Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. |
100,000 | 49,A54 | 10,000 | 5,954 | 1,000 | 6B4 | 100 | 84 | 10 | A | 1 | 1 | 0.1 | 0.1 | 0.01 | 0.0 |
200,000 | 97,8A8 | 20,000 | B,6A8 | 2,000 | 1,1A8 | 200 | 148 | 20 | 18 | 2 | 2 | 0.2 | 0. | 0.02 | 0.0 |
300,000 | 125,740 | 30,000 | 15,440 | 3,000 | 1,8A0 | 300 | 210 | 30 | 26 | 3 | 3 | 0.3 | 0.3 | 0.03 | 0.0 |
400,000 | 173,594 | 40,000 | 1B,194 | 4,000 | 2,394 | 400 | 294 | 40 | 34 | 4 | 4 | 0.4 | 0. | 0.04 | 0.0 |
500,000 | 201,428 | 50,000 | 24,B28 | 5,000 | 2,A88 | 500 | 358 | 50 | 42 | 5 | 5 | 0.5 | 0.6 | 0.05 | 0.0 |
600,000 | 24B,280 | 60,000 | 2A,880 | 6,000 | 3,580 | 600 | 420 | 60 | 50 | 6 | 6 | 0.6 | 0. | 0.06 | 0.0 |
700,000 | 299,114 | 70,000 | 34,614 | 7,000 | 4,074 | 700 | 4A4 | 70 | 5A | 7 | 7 | 0.7 | 0.8 | 0.07 | 0.0 |
800,000 | 326,B68 | 80,000 | 3A,368 | 8,000 | 4,768 | 800 | 568 | 80 | 68 | 8 | 8 | 0.8 | 0. | 0.08 | 0. |
900,000 | 374,A00 | 90,000 | 44,100 | 9,000 | 5,260 | 900 | 630 | 90 | 76 | 9 | 9 | 0.9 | 0.A | 0.09 | 0.1 |
Conversion of powers
Exponent | Powers of 2 | Powers of 3 | Powers of 4 | Powers of 5 | Powers of 6 | Powers of 7 | ||||||
Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | |
^6 | 64 | 54 | 729 | 509 | 4,096 | 2454 | 15,625 | 9,061 | 46,656 | 23,000 | 117,649 | 58,101 |
^5 | 32 | 28 | 243 | 183 | 1,024 | 714 | 3,125 | 1,985 | 7,776 | 4,600 | 16,807 | 9,887 |
^4 | 16 | 14 | 81 | 69 | 256 | 194 | 625 | 441 | 1,296 | 900 | 2,401 | 1,481 |
^3 | 8 | 8 | 27 | 23 | 64 | 54 | 125 | A5 | 216 | 160 | 343 | 247 |
^2 | 4 | 4 | 9 | 9 | 16 | 14 | 25 | 21 | 36 | 30 | 49 | 41 |
^1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 7 | 7 |
^−1 | 0.5 | 0.6 | 0. | 0.4 | 0.25 | 0.3 | 0.2 | 0. | 0.1 | 0.2 | 0. | 0. |
^−2 | 0.25 | 0.3 | 0. | 0.14 | 0.0625 | 0.09 | 0.04 | 0. | 0.02 | 0.04 | 0. | 0. |
Exponent | Powers of 8 | Powers of 9 | Powers of 10 | Powers of 11 | Powers of 12 | |||||
Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | Dec. | Doz. | |
^6 | 262,144 | 107,854 | 531,441 | 217,669 | 1,000,000 | 402,854 | 1,771,561 | 715,261 | 2,985,984 | 1,000,000 |
^5 | 32,768 | 16,B68 | 59,049 | 2A,209 | 100,000 | 49,A54 | 161,051 | 79,24B | 248,832 | 100,000 |
^4 | 4,096 | 2,454 | 6,561 | 3,969 | 10,000 | 5,954 | 14,641 | 8,581 | 20,736 | 10,000 |
^3 | 512 | 368 | 729 | 509 | 1,000 | 6B4 | 1,331 | 92B | 1,728 | 1,000 |
^2 | 64 | 54 | 81 | 69 | 100 | 84 | 121 | A1 | 144 | 100 |
^1 | 8 | 8 | 9 | 9 | 10 | A | 11 | B | 12 | 10 |
^−1 | 0.125 | 0.16 | 0. | 0.14 | 0.1 | 0.1 | 0. | 0. | 0.08 | 0.1 |
^−2 | 0.015625 | 0.023 | 0. | 0.0194 | 0.01 | 0.0 | 0. | 0. | 0.0069 | 0.01 |
Fractions
Duodecimal fractionFraction (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 may be simple:
= 0.6 = 0.4 = 0.3 = 0.2 = 0.16 = 0.14
or complicated
= 0.24972497... recurring (easily rounded to 0.25) = 0.186A35186A35... recurring (easily rounded to 0.187) = 0.124972497... recurring (rounded to 0.125) = 0.11111... recurring (rounded to 0.11) = 0.0B0B... recurring (rounded to 0.0B)
Examples in duodecimal | Decimal equivalent |
1 × = 0.76 | 1 × = 0.625 |
100 × = 76 | 144 × = 90 |
= 76 | = 90 |
= 54 | = 64 |
1A.6 + 7.6 = 26 | 22.5 + 7.5 = 30 |
As explained in recurring decimals, whenever an irreducible fraction
Irreducible fraction
An irreducible fraction is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent vulgar fraction...
is written in 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...
notation in any base, the fraction can be expressed exactly (terminates) if and only if all the 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...
s of its denominator are also prime factors of the base. Thus, in base-ten (= 2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: = , = and = can be expressed exactly as 0.125, 0.05 and 0.002 respectively. and , however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, is exact; and recur because they include 5 as a factor; is exact; and recurs, just as it does in decimal.
Recurring digits
Arguably, factors of 3 are more commonly encountered in real-life divisionDivision (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
problems than factors of 5 (or would be, were it not for the decimal system having influenced most cultures). Thus, in practical applications, the nuisance of recurring decimals is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.
However, when recurring fractions do occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because 12
12 (number)
12 is the natural number following 11 and preceding 13.The word "twelve" is the largest number with a single-morpheme name in English. Etymology suggests that "twelve" arises from the Germanic compound twalif "two-leftover", so a literal translation would yield "two remaining [after having ten...
(twelve) is between two prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, 11
11 (number)
11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...
(eleven) and 13
13 (number)
13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....
(thirteen), whereas ten is adjacent to composite number
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
9. Nonetheless, having a shorter or longer period doesn't help the main inconvenience that one does not get a finite representation for such fractions in the given base (so rounding
Rounding
Rounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing $23.4476 with $23.45, or the fraction 312/937 with 1/3, or the expression √2 with 1.414.Rounding is often done on purpose to...
, which introduces inexactitude, is necessary to handle them in calculations), and overall one is more likely to have to deal with infinite recurring digits when fractions are expressed in decimal than in duodecimal, because one out of every three consecutive numbers contains the prime factor 3 in its factorization, while only one out of every five contains the prime factor 5. All other prime factors, except 2, are not shared by either ten or twelve, so they do not
influence the relative likeliness of encountering recurring digits (any irreducible fraction that contains any of these other factors in its denominator will recur in either base). Also, the prime factor 2 appears twice in the factorization of twelve, while only once in the factorization of ten; which means that most fractions whose denominators are powers of two
Power of two
In mathematics, a power of two means a number of the form 2n where n is an integer, i.e. the result of exponentiation with as base the number two and as exponent the integer n....
will have a shorter, more convenient terminating representation in dozenal than in decimal (e.g., 1/(2^{2}) = 0.25 dec = 0.3 doz; 1/(2^{3}) = 0.125 dec = 0.16 doz; 1/(2^{4}) = 0.0625 dec = 0.09 doz; 1/(2^{5}) = 0.03125 dec = 0.046 doz; etc.).
Prime factors of the base: 2, 5 |
Duodecim>l / Dozenal base Prime factors of the base: 2, 3 |
||||
Fraction | Prime factors of the denominator |
Positional representation | Positional representation | Prime factors of the denominator |
Fraction |
1/2 | 2 | 0.5 | 0.6 | 2 | 1/2 |
1/3 | 3 | 0.3333... = 0. | 0.4 | 3 | 1/3 |
1/4 | 2 | 0.25 | 0.3 | 2 | 1/4 |
1/5 | 5 | 0.2 | 0.24972497... = 0. | 5 | 1/5 |
1/6 | 2, 3 | 0.1 | 0.2 | 2, 3 | 1/6 |
1/7 | 7 | 0. | 0. | 7 | 1/7 |
1/8 | 2 | 0.125 | 0.16 | 2 | 1/8 |
1/9 | 3 | 0. | 0.14 | 3 | 1/9 |
1/10 | 2, 5 | 0.1 | 0.1 | 2, 5 | 1/A |
1/11 | 11 | 0. | 0. | B | 1/B |
1/12 | 2, 3 | 0.08 | 0.1 | 2, 3 | 1/10 |
1/13 | 13 | 0. | 0. | 11 | 1/11 |
1/14 | 2, 7 | 0.0 | 0.0 | 2, 7 | 1/12 |
1/15 | 3, 5 | 0.0 | 0.0 | 3, 5 | 1/13 |
1/16 | 2 | 0.0625 | 0.09 | 2 | 1/14 |
1/17 | 17 | 0. | 0. | 15 | 1/15 |
1/18 | 2, 3 | 0.0 | 0.08 | 2, 3 | 1/16 |
1/19 | 19 | 0. | 0. | 17 | 1/17 |
1/20 | 2, 5 | 0.05 | 0.0 | 2, 5 | 1/18 |
1/21 | 3, 7 | 0. | 0.0 | 3, 7 | 1/19 |
1/22 | 2, 11 | 0.0 | 0.0 | 2, B | 1/1A |
1/23 | 23 | 0. | 0. | 1B | 1/1B |
1/24 | 2, 3 | 0.041 | 0.06 | 2, 3 | 1/20 |
1/25 | 5 | 0.04 | 0. | 5 | 1/21 |
1/26 | 2, 13 | 0.0 | 0.0 | 2, 11 | 1/22 |
1/27 | 3 | 0. | 0.054 | 3 | 1/23 |
1/28 | 2, 7 | 0.03 | 0.0 | 2, 7 | 1/24 |
1/29 | 29 | 0. | 0. | 25 | 1/25 |
1/30 | 2, 3, 5 | 0.0 | 0.0 | 2, 3, 5 | 1/26 |
1/31 | 31 | 0. | 0. | 27 | 1/27 |
1/32 | 2 | 0.03125 | 0.046 | 2 | 1/28 |
1/33 | 3, 11 | 0. | 0.0 | 3, B | 1/29 |
1/34 | 2, 17 | 0.0 | 0.0 | 2, 15 | 1/2A |
1/35 | 5, 7 | 0.0 | 0. | 5, 7 | 1/2B |
1/36 | 2, 3 | 0.02 | 0.04 | 2, 3 | 1/30 |
Irrational numbers
As for irrational numberIrrational 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....
s, none of them has a finite representation in any of the rational
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...
-based positional number systems (such as the decimal and duodecimal ones); this is because a rational-based positional number system is essentially nothing but a way of expressing quantities as a sum of fractions whose denominators are powers of the base, and by definition no finite sum of rational numbers can ever result in an irrational number. For example, 123.456 = 1 × 1/10^{−2} + 2 × 1/10^{−1} + 3 × 1/10^{0} + 4 × 1/10^{1} + 5 × 1/10^{2} + 6 × 1/10^{3} (this is also the reason why fractions that contain prime factors in their denominator not in common with those of the base do not have a terminating representation in that base). Moreover, the infinite series of digits of an irrational number doesn't exhibit a pattern of repetition; instead, the different digits succeed in a seemingly random fashion. The following chart compares the first few digits of the decimal and duodecimal representation of several of the most important algebraic
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
and transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
irrational numbers. Some of these numbers may be perceived as having fortuitous patterns, making them easier to memorize, when represented in one base or the other.
Algebraic irrational number | In decimal | In duodecimal / dozenal |
√2 Square root of 2 The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the... (the length of the diagonal Diagonal A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a... of a unit square Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles... ) |
1.41421356237309... (≈ 1.414) | 1.4B79170A07B857... (≈ 1.5) |
√3 Square root of 3 The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property... (the length of the diagonal of a unit cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... , or twice the height Height Height is the measurement of vertical distance, but has two meanings in common use. It can either indicate how "tall" something is, or how "high up" it is. For example "The height of the building is 50 m" or "The height of the airplane is 10,000 m"... of an equilateral triangle of unit side) |
1.73205080756887... (≈ 1.732) | 1.894B97BB968704... (≈ 1.895) |
√5 (the length of the diagonal Diagonal A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a... of a 1×2 rectangle Rectangle In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle... ) |
2.2360679774997... (≈ 2.236) | 2.29BB132540589... (≈ 2.2A) |
φ Golden ratio In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989... (phi, the golden ratio = ) |
1.6180339887498... (≈ 1.618) | 1.74BB6772802A4... (≈ 1.75) |
Transcendental irrational number | In decimal | In duodecimal / dozenal |
π Pi ' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants... (pi, the ratio of circumference Circumference The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it.... to diameter Diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle... ) |
3.1415926535897932384626433 8327950288419716939937510... (≈ 3.1416) |
3.184809493B918664573A6211B B151551A05729290A7809A492... (≈ 3.1848) |
e E (mathematical constant) The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base... (the base of the natural logarithm Natural logarithm The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828... ) |
2.718281828459045... (≈ 2.718) | 2.8752360698219B8... (≈ 2.875) |
The first few digits of the decimal and dozenal representation of another important number, the Euler-Mascheroni constant
Euler-Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
(the status of which as a rational or irrational number is not yet known), are:
Number | In decimal | In duodecimal / dozenal |
γ Euler-Mascheroni constant The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter .... (the limiting difference between the harmonic series Harmonic series (mathematics) In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength... and the natural logarithm) |
0.57721566490153... (~ 0.577) | 0.6B15188A6760B3... (~ 0.7) |
Advocacy and "dozenalism"
The case for the duodecimal system was put forth at length in F. Emerson Andrews' 1935 book New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics. Emerson noted that, due to the prevalence of factors of twelve in many traditional units of weight and measure, many of the computational advantages claimed for the metric system could be realized either by the adoption of ten-based weights and measure or by the adoption of the duodecimal number system.Rather than the symbols 'A' for ten and 'B' for eleven as used in 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...
notation and vigesimal
Vigesimal
The vigesimal or base 20 numeral system is based on twenty .- Places :...
notation (or 'T' and 'E' for ten and eleven), he suggested in his book and used a script X and a script E, (U+
Unicode
Unicode is a computing industry standard for the consistent encoding, representation and handling of text expressed in most of the world's writing systems...
1D4B3) and (U+2130), to represent the digits ten and eleven respectively, because, at least on a page of Roman script, these characters were distinct from any existing letters or numerals, yet were readily available in printers' fonts. He chose for its resemblance to the Roman numeral X, and as the first letter of the word "eleven".
Another popular notation, introduced by Sir Isaac Pitman
Isaac Pitman
Sir Isaac Pitman , knighted in 1894, developed the most widely used system of shorthand, known now as Pitman shorthand. He first proposed this in Stenographic Soundhand in 1837. Pitman was a qualified teacher and taught at a private school he founded in Wotton-under-Edge...
, is to use a rotated 2 (resembling a script τ for 'ten') to represent ten and a rotated or horizontally flipped 3 (which again resembles ε) to represent eleven. This is the convention commonly employed by the Dozenal Society of Great Britain and has the advantage of being easily recognizable as digits because of their resemblance in shape to existing digits. On the other hand, the Dozenal Society of America adopted for some years the convention of using an asterisk
Asterisk
An asterisk is a typographical symbol or glyph. It is so called because it resembles a conventional image of a star. Computer scientists and mathematicians often pronounce it as star...
* for ten and a hash
Number sign
Number sign is a name for the symbol #, which is used for a variety of purposes including, in some countries, the designation of a number...
# for eleven. The reason was the symbol * resembles a struck-through X while # resembles a doubly-struck-through 11, and both symbols are already present in telephone
Telephone
The telephone , colloquially referred to as a phone, is a telecommunications device that transmits and receives sounds, usually the human voice. Telephones are a point-to-point communication system whose most basic function is to allow two people separated by large distances to talk to each other...
dial
Rotary dial
The rotary dial is a device mounted on or in a telephone or switchboard that is designed to send electrical pulses, known as pulse dialing, corresponding to the number dialed. The early form of the rotary dial used lugs on a finger plate instead of holes. Almon Brown Strowger filed the first patent...
s. However, critics pointed out these symbols do not look anything like digits. Some other systems write 10 as Φ (a combination of 1 and 0) and eleven as a cross of two lines (+, x, or † for example). Problems with these symbols are evident, most notably that most of them can not be represented in the seven-segment display
Seven-segment display
A seven-segment display , or seven-segment indicator, is a form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot-matrix displays...
of most calculator
Calculator
An electronic calculator is a small, portable, usually inexpensive electronic device used to perform the basic operations of arithmetic. Modern calculators are more portable than most computers, though most PDAs are comparable in size to handheld calculators.The first solid-state electronic...
displays ( being an exception, although "E" is used on calculators to indicate an error message
Error message
An error message is information displayed when an unexpected condition occurs, usually on a computer or other device. On modern operating systems with graphical user interfaces, error messages are often displayed using dialog boxes...
). However, 10 and 11 do fit, both within a single digit (11 fits as is, while the 10 has to be tilted sideways, resulting in a character that resembles an O with a macron
Macron
A macron, from the Greek , meaning "long", is a diacritic placed above a vowel . It was originally used to mark a long or heavy syllable in Greco-Roman metrics, but now marks a long vowel...
, ō or o). A and B also fit (although B must be represented as lowercase "b" and as such, 6 must have a bar over it to distinguish the two figures) and are used on calculators for bases higher than ten.
In "Little Twelvetoes", American television series Schoolhouse Rock!
Schoolhouse Rock!
Schoolhouse Rock! is an American interstitial programming series of animated musical educational short films that aired during the Saturday morning children's programming on the U.S. television network ABC. The topics covered included grammar, science, economics, history, mathematics, and civics...
portrayed an alien child using base-twelve arithmetic, using "dek", "el" and "doh" as names for ten, eleven and twelve, and Andrews' script-X and script-E for the digit symbols. ("Dek" is from the prefix "deca", "el" being short for "eleven" and "doh" an apparent shortening of "dozen".)
The Dozenal Society of America and the Dozenal Society of Great Britain promote widespread adoption of the base-twelve system. They use the word dozenal instead of "duodecimal" because the latter comes from Latin roots that express twelve in base-ten terminology.
The renowned mathematician and mental calculator Alexander Craig Aitken
Alexander Aitken
Alexander Craig Aitken was one of New Zealand's greatest mathematicians. He studied for a PhD at the University of Edinburgh, where his dissertation, "Smoothing of Data", was considered so impressive that he was awarded a DSc in 1926, and was elected a fellow of the Royal Society of Edinburgh...
was an outspoken advocate of the advantages and superiority of duodecimal over decimal:
In Leo Frankowski
Leo Frankowski
Leo Frankowski was an American writer of science fiction novels. He lived in Russia for four years with his now ex-wife and adopted teenage daughter, but...
's Conrad Stargard
Conrad Stargard
Conrad Stargard is the protagonist and title character in a series of time travel novels written by Leo Frankowski. In them, a Polish engineer named Conrad Schwartz is sent back in time to the 13th century where he has to establish himself and cope with various crises including the eventual...
novels, Conrad introduces a duodecimal system of arithmetic at the suggestion of a merchant, who is accustomed to buying and selling goods in dozens and grosses, rather than tens or hundreds. He then invents an entire system of weights and measures in base twelve, including a clock with twelve hours in a day, rather than twenty-four hours.
In Lee Carroll
Lee Carroll
Lee Carroll is an American channeller, speaker and author.Originally an audio engineer, Carroll claims that he began to channel communication with an entity from a higher dimension called Kryon in 1989...
's Kryon: Alchemy of the Human Spirit, a chapter is dedicated to the advantages of the duodecimal system. The duodecimal system is supposedly suggested by Kryon (one of the widely popular New Age
New Age
The New Age movement is a Western spiritual movement that developed in the second half of the 20th century. Its central precepts have been described as "drawing on both Eastern and Western spiritual and metaphysical traditions and then infusing them with influences from self-help and motivational...
channeled entities) for all-round use, aiming at better and more natural representation of nature of the Universe through mathematics. An individual article "Mathematica" by James D. Watt (included in the above publication) exposes a few of the unusual symmetry connections between the duodecimal system and the golden ratio
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
, as well as provides numerous number symmetry-based arguments for the universal nature of the base-12 number system.
See also
- SenarySenaryIn mathematics, a senary numeral system is a base- numeral system.Senary may be considered useful in the study of prime numbers since all primes other than 2 and 3, when expressed in base-six, have 1 or 5 as the final digit...
(base 6) - QuadrovigesimalBase 24The base- system is a numeral system with 24 as its base.There are 24 hours in a nychthemeron , so solar time includes a base-24 component.See also base 12. Decimal Equivalent...
(base 24) - HexatridecimalBase 36Base 36 is a positional numeral system using 36 as the radix. The choice of 36 is convenient in that the digits can be represented using the Arabic numerals 0-9 and the Latin letters A-Z...
(base 36) - Sexagesimal (base 60)
- Babylonian numeralsBabylonian numeralsBabylonian 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....