 x Sexagesimal Overview

Sexagesimal is a 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....

with sixty
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:...

as its base
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...

. It originated with the ancient Sumerians in the 3rd millennium BC
3rd millennium BC
The 3rd millennium BC spans the Early to Middle Bronze Age.It represents a period of time in which imperialism, or the desire to conquer, grew to prominence, in the city states of the Middle East, but also throughout Eurasia, with Indo-European expansion to Anatolia, Europe and Central Asia. The...

, it was passed down to the ancient 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, and it is still used — in a modified form — for measuring 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....

, angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s, and the geographic coordinates
Geographic coordinate system
A geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position...

that are angles.

The number 60, 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....

, has twelve factors
Factorization
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...

, namely } of which two, three, and five are 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. With so many factors, many 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 involving sexagesimal numbers are simplified.  How would you go about calculating a number's square entirely in sexagesimal notation (i.e. base 60). For example, how would you calculate the square ... How can u identify to solve a sexagesimal? Discussions Sheena Encyclopedia
Sexagesimal is a 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....

with sixty
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:...

as its base
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...

. It originated with the ancient Sumerians in the 3rd millennium BC
3rd millennium BC
The 3rd millennium BC spans the Early to Middle Bronze Age.It represents a period of time in which imperialism, or the desire to conquer, grew to prominence, in the city states of the Middle East, but also throughout Eurasia, with Indo-European expansion to Anatolia, Europe and Central Asia. The...

, it was passed down to the ancient 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, and it is still used — in a modified form — for measuring 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....

, angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s, and the geographic coordinates
Geographic coordinate system
A geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position...

that are angles.

The number 60, 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....

, has twelve factors
Factorization
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...

, namely } of which two, three, and five are 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. With so many factors, many 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 involving sexagesimal numbers are simplified. For example, one hour can be divided evenly into sections of 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, six minutes, five minutes, etc. Sixty is the smallest number that is evenly divisible by every number from one to six. This is because .

In this article, all sexagesimal digits are represented as decimal numbers, except where otherwise noted. [For example, 10 means 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...

and 60 means sixty
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:...

.]

## Origin

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.

According to Neugebauer, the origins of the sixty-count was through a count of three twenties. The precursor to the later six-ten alternation was through symbols for the sixths, (ie 1/6, 2/6, 3/6, 4/6, 5/6), coupled with decimal numbers, lead to the same three-score count, and also to the division-system that the Sumerians were famous for. In normal use, numbers were a hap-hazard collection of units, tens, sixties, and hundreds. A number like 192, would be expressed uniformly in the tables as 3A2 (A as the symbol for the '10', would in the surrounding text, be given as XIxii ie, hundred (big 10), sixty (big 1), ten (little 10), two (little 1's).

### Babylonian mathematics

The sexagesimal system as used in ancient Mesopotamia
Mesopotamia
Mesopotamia is a toponym for the area of the Tigris–Euphrates river system, largely corresponding to modern-day Iraq, northeastern Syria, southeastern Turkey and southwestern Iran.Widely considered to be the cradle of civilization, Bronze Age Mesopotamia included Sumer and the...

was not a pure base-60 system, in the sense that it did not use 60 distinct symbols for its 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...

. Instead, the cuneiform digits used 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...

as a sub-base in the fashion of a 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...

: a sexagesimal digit was composed of a group of narrow, wedge-shaped marks representing units up to nine (Y, YY, YYY, YYYY, ... YYYYYYYYY) and a group of wide, wedge-shaped marks representing up to five tens (<, <<, <<<, <<<<, <<<<<). The value of the digit was the sum of the values of its component parts:
Numbers larger than 59 were indicated by multiple symbol blocks of this form in place value notation
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...

.

Because there was no symbol for zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...

in Sumerian or early Babylonian numbering systems, it is not always immediately obvious how a number should be interpreted, and its true value must sometimes have been determined by its context. Later Babylonian texts used a placeholder to represent zero, but only in the medial positions, and not on the right-hand side of the number, as we do in numbers like 13,200.

### Other historical usages

In the 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...

, a sexagenary cycle
Sexagenary cycle
The Chinese sexagenary cycle , also known as the Stems-and-Branches , is a cycle of sixty terms used for recording days or years. It appears, as a means of recording days, in the first Chinese written texts, the Shang dynasty oracle bones from the late second millennium BC. Its use to record years...

is commonly used, in which days or years are named by positions in a sequence of ten stems and in another sequence of 12 branches. The same stem and branch repeat every 60 steps through this cycle.

Ptolemy's
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...

, a treatise on mathematical astronomy written in the second century AD, uses base 60 for all numerals. In particular, his table of chords, which was essentially the only extensive trigonometric table for more than a millennium, is in base 60.

Base-60 number systems have also been used in some other cultures that are unrelated to the Sumerians, for by example the Ekagi people of Western New Guinea
Western New Guinea
West Papua informally refers to the Indonesian western half of the island of New Guinea and other smaller islands to its west. The region is officially administered as two provinces: Papua and West Papua. The eastern half of New Guinea is Papua New Guinea.The population of approximately 3 million...

.

### Modern usage

Unlike most other numeral systems, sexagesimal is not used so much in modern times as a means for general computations, or in logic, but rather, it is used in measuring angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s, geographic coordinates, and 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....

.

One hour
Hour
The hour is a unit of measurement of time. In modern usage, an hour comprises 60 minutes, or 3,600 seconds...

of time is divided into 60 minute
Minute
A minute is a unit of measurement of time or of angle. The minute is a unit of time equal to 1/60th of an hour or 60 seconds. In the UTC time scale, a minute on rare occasions has 59 or 61 seconds; see leap second. The minute is not an SI unit; however, it is accepted for use with SI units...

s, and one minute is divided into 60 seconds. Thus, a measurement of time such as "3:23:17" (three hours, 23 minutes, and 17 seconds) can be interpreted as a sexagesimal number, meaning 3×602 + 23×601 + 17. As with the ancient Babylonian sexagesimal system, however, each of the three sexagesimal digits in this number (3, 23, and 17) is written using the decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

system.

Similarly, the practical unit of angular measure is the degree
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

, of which there are 360
360 (number)
360 or three sixty is the natural number following 359 and preceding 361.-In mathematics:*Divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360. *360 makes a highly composite number...

in a circle. There are 60 minutes of arc
Minute of arc
A minute of arc, arcminute, or minute of angle , is a unit of angular measurement equal to one sixtieth of one degree. In turn, a second of arc or arcsecond is one sixtieth of one minute of arc....

in a degree, and 60 arcseconds in a minute.

In some usage systems, each position past the sexagesimal point was numbered, using Latin or French roots: prime or primus, seconde or secundus, tierce, quatre, quinte, etc. To this day we call the second-order part of an hour
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

or of a degree a "second". Until at least the 18th century, 1/60 of a second was called a "tierce" or "third".

### In popular culture

In Robert A. Heinlein
Robert A. Heinlein
Robert Anson Heinlein was an American science fiction writer. Often called the "dean of science fiction writers", he was one of the most influential and controversial authors of the genre. He set a standard for science and engineering plausibility and helped to raise the genre's standards of...

's science fiction
Science fiction
Science fiction is a genre of fiction dealing with imaginary but more or less plausible content such as future settings, futuristic science and technology, space travel, aliens, and paranormal abilities...

novel
Novel
A novel is a book of long narrative in literary prose. The genre has historical roots both in the fields of the medieval and early modern romance and in the tradition of the novella. The latter supplied the present generic term in the late 18th century....

, Methuselah's Children
Methuselah's Children
Methuselah's Children is a science fiction novel by Robert A. Heinlein, originally serialized in Astounding Science Fiction in the July, August, and September 1941 issues. It was expanded into a full-length novel in 1958....

, Heinlein described a future race of super-intelligent humans which uses a base-60 number system and alphabet of exactly sixty ideographs.

In Stel Pavlou
Stel Pavlou
Stelios Grant Pavlou is a British author and screenwriter.-Biography:Stel Pavlou was born in Gillingham, Kent in England, of Greek Cypriot descent. He grew up in Rochester and Chatham, Medway and attended Chatham Grammar School for Boys. The middle child of three, his younger brother is the...

's novel Decipher
Decipher (novel)
Decipher is a speculative fiction novel by Stel Pavlou , published in 2001 in England by Simon and Schuster and 2002 in the United States by St. Martin's Press. It is published in many languages with some significant title changes...

, this number system is the center of focus, as the buckyball carbon element is used in the book to store data, and only base 60 is found to be able to be successfully understood by the computers used in it. At least one popular book uses the spelling "sexigesimal" instead of "sexagesimal," with the latter being the more common spelling of the word.

Book VIII of Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...

's Republic involves an allegory of marriage centered on the number 604 = 12,960,000 and its divisors. This number has the particularly simple sexagesimal representation 1:0:0:0:0. Later scholars have invoked both Babylonian mathematics and music theory in an attempt to explain this passage.

## Fractions

In the sexagesimal system, any 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...

in which the denominator is a regular number
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...

(having only 2, 3, and 5 in its prime factorization) may be expressed exactly. The table below shows the sexagesimal representation of all fractions of this type in which the denominator is less than 60. The sexagesimal values in this table may be interpreted as giving the number of minutes and seconds in a given fraction of an hour; for instance, 1/9 of an hour is 6 minutes and 40 seconds. However, the representation of these fractions as sexagesimal numbers does not depend on such an interpretation.
 Fraction: Sexagesimal:  Fraction: Sexagesimal: Fraction: Sexagesimal: 1/2 1/3 1/4 1/5 1/6 1/8 1/9 1/10 30 20 15 12 10 7:30 6:40 6 1/12 1/15 1/16 1/18 1/20 1/24 1/25 1/27 5 4 3:45 3:20 3 2:30 2:24 2:13:20 1/30 1/32 1/36 1/40 1/45 1/48 1/50 1/54 2 1:52:30 1:40 1:30 1:20 1:15 1:12 1:6:40

However numbers that are not regular form more complicated repeating fractions. For example:
1/7 = 0:8:34:17:8:34:17 ... (with the sequence of sexagesimal digits 8:34:17 repeating infinitely often).

The fact in 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...

that the two numbers that are adjacent to 60, namely 59 and 61, are both prime numbers implies that simple repeating fractions that repeat with a period of one or two sexagesimal digits can only have 59 or 61 as their denominators, and that other non-regular primes have fractions that repeat with a longer period.

## Examples The square root of 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
Unit square
In mathematics, a unit square is a square whose sides have length 1. Often, "the" unit square refers specifically to the square in the Cartesian plane with corners at , , , and .-In the real plane:...

, was approximated by the Babylonians of the Old Babylonian Period (1900 BC – 1650 BC) as Because is 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....

, it cannot be expressed exactly in sexagesimal numbers, but its sexagesimal expansion does begin 1 : 24 : 51 : 10 : 7 : 46 : 6 : 4 : 44 ...

The length of the tropical year
Tropical year
A tropical year , for general purposes, is the length of time that the Sun takes to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice...

in Neo-Babylonian astronomy
Chaldea
Chaldea or Chaldaea , from Greek , Chaldaia; Akkadian ; Hebrew כשדים, Kaśdim; Aramaic: ܟܐܠܕܘ, Kaldo) was a marshy land located in modern-day southern Iraq which came to briefly rule Babylon...

(see Hipparchus
Hipparchus
Hipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...

), 365.24579... days, can be expressed in sexagesimal as 6:5:14:44:51 (6×60 ＋ 5 + 14/60 + 44/602 + 51/603) days. The average length of a year in the Gregorian calendar
Gregorian calendar
The Gregorian calendar, also known as the Western calendar, or Christian calendar, is the internationally accepted civil calendar. It was introduced by Pope Gregory XIII, after whom the calendar was named, by a decree signed on 24 February 1582, a papal bull known by its opening words Inter...

is exactly 6 : 5 : 14 : 33 in the same notation because the values 14 and 33 were the first two values for the tropical year from the Alfonsine tables
Alfonsine tables
The Alfonsine tables provided data for computing the position of the Sun, Moon and planets relative to the fixed stars....

, which were in sexagesimal notation.

The value of 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...

as used by the Greek
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...

mathematican and scientist Claudius Ptolemaeus (Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

) was 3.141666... ≈ 377/120 = 3:8:30 = 3 + 8/60 + 30/602. Jamshīd al-Kāshī
Jamshid al-Kashi
Ghiyāth al-Dīn Jamshīd Masʾūd al-Kāshī was a Persian astronomer and mathematician.-Biography:...

, a 15th-century Persian mathematician, calculated π in sexagesimal numbers to an accuracy of nine sexagesimal digits. 