# Sexagesimal

Sexagesimal (base 60) is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinates.

The number 60, a superior highly composite number, has twelve factors, namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, of which 2, 3, and 5 are prime numbers. With so many factors, many fractions 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, 6 minutes, 5 minutes, 4 minutes, 3 minutes, 2 minutes, and 1 minute. 60 is the smallest number that is divisible by every number from 1 to 6; that is, it is the lowest common multiple of 1, 2, 3, 4, 5, and 6.

In this article, all sexagesimal digits are represented as decimal numbers, except where otherwise noted. For example, 10 means ten and 60 means sixty.

## Origin

It is possible for people to count on their fingers to 12 using one hand only, with the thumb pointing to each finger bone on the four fingers in turn. A traditional 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, one hand (usually right) counts repeatedly to 12, displaying the number of iterations on the other (usually left), until five dozens, i. e. the 60, are full.[1][2]

According to Otto Neugebauer,[3] the origins of sexagesimal are not as simple, consistent, or singular in time as they are often portrayed.[4] Throughout their many centuries of use, which continues today for specialized topics such as time, angles, and astronomical coordinate systems, sexagesimal notations have always contained a strong undercurrent of decimal notation, such as in how sexigesimal digits are written.[5] Their use has also always included (and continues to include) inconsistencies in where and how various bases are to represent numbers even within a single text.[6]

The most powerful driver for rigorous, fully self-consistent use of sexagesimal has always been its mathematical advantages for writing and calculating fractions. In ancient texts this shows up in the fact that sexigesimal is used most uniformly and consistently in mathematical tables of data.[7] Another practical factor that helped expand the use of sexagesimal in the past even if less consistently that in mathematical tables, was its decided advantages to merchants and buyers for making everyday financial transactions easier when they involved bargaining for and dividing up larger quantities of goods. The early shekel in particular was one-sixtieth of a mana,[8] though the Greeks later coerced this relationship into the more base-10 compatible ratio of a shekel being one-fiftieth of a mina.

Apart from mathematical tables, the inconsistencies in how numbers were represented within most texts extended all the way down to the most basic Cuneiform symbols used to represent numeric quantities.[9] For example, the Cuneiform symbol for 1 was an ellipse made by applying the rounded end of the stylus at an angle to the clay, while the sexagesimal symbol for 60 was a larger oval or "big 1". But within the same texts in which these symbols were used, the number 10 was represented as a circle made by applying the round end of the style perpendicular to the clay, and a larger circle or "big 10" was used to represent 100. Such multi-base numeric quantity symbols could be mixed with each other and with abbreviations, even within a single number. The detailed and even the magnitudes implied (since zero was not used consistently) were idiomatic to the particular time periods, cultures, and quantities or concepts being represented. While such context-dependent representations of numeric quantities are easy to critique in retrospect, in modern time we still have "dozens" of regularly used examples (some quite "gross") of topic-dependent base mixing, including the particularly ironic recent innovation of adding decimal fractions to sexagesimal astronomical coordinates.[10]

## Usage

### Babylonian mathematics

The sexagesimal system as used in ancient Mesopotamia was not a pure base-60 system, in the sense that it did not use 60 distinct symbols for its digits. Instead, the cuneiform digits used ten as a sub-base in the fashion of a 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.

Because there was no symbol for zero 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. Without context, this system was fairly ambiguous. For example, the symbols for 1 and 60 are identical.[11][12] 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 13200.[12]

### Other historical usages

In the Chinese calendar, a sexagenary cycle 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.

Book VIII of Plato's Republic involves an allegory of marriage centered on the number 604 = 12960000 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.[13]

Ptolemy's Almagest, a treatise on mathematical astronomy written in the second century AD, uses base 60 to express the fractional parts of numbers. In particular, his table of chords, which was essentially the only extensive trigonometric table for more than a millennium, has fractional parts in base 60.

The sexagesimal number system continued to be frequently used by European astronomers for performing calculations as late as 1671.[14]

In the late eighteenth and early nineteenth century Tamil astronomers were found to make astronomical calculations, reckoning with shells using a mixture of decimal and sexagesimal notations developed by Hellenistic astronomers.[15]

Base-60 number systems have also been used in some other cultures that are unrelated to the Sumerians, for example by the Ekari people of Western New Guinea.[16][17]

### Notation

In Hellenistic Greek astronomical texts, such as the writings of Ptolemy, sexagesimal numbers were written using the Greek alphabetic numerals, with each sexagesimal digit being treated as a distinct number. The Greeks limited their use of sexagesimal numbers to the fractional part of a number and employed a variety of markers to indicate a zero.[18]

In medieval Latin texts, sexagesimal numbers were written using Hindu-Arabic numerals; the different levels of fractions were denoted minuta (i.e., fraction), minuta secunda, minuta tertia, etc. By the seventeenth century it became common to denote the integer part of sexagesimal numbers by a superscripted zero, and the various fractional parts by one or more accent marks. John Wallis, in his Mathesis universalis, generalized this notation to include higher multiples of 60; giving as an example the number 49‵‵‵‵,36‵‵‵,25‵‵,15‵,1°,15′,25′′,36′′′,49′′′′; where the numbers to the left are multiplied by higher powers of 60, the numbers to the right are divided by powers of 60, and the number marked with the superscripted zero is multiplied by 1.[19] This notation leads to the modern signs for degrees, minutes, and seconds. The same minute and second nomenclature is also used for units of time, and the modern notation for time with hours, minutes, and seconds written in decimal and separated from each other by colons may be interpreted as a form of sexagesimal notation.

In modern studies of ancient mathematics and astronomy it is customary to write sexagesimal numbers with each sexagesimal digit represented in standard decimal notation as a number from 0 to 59, and with each digit separated by a comma. When appropriate, the fractional part of the sexagesimal number is separated from the whole number part by a semicolon rather than a comma, although in many cases this distinction may not appear in the original historical document and must be taken as an interpretation of the text.[20] Using this notation the square root of two, which in decimal notation appears as 1.41421... appears in modern sexagesimal notation as 1;24,51,10....[21] This notation is used in this article.

### Modern usage

Further information: Sexagesimal degree

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 angles, geographic coordinates, and time.

One hour of time is divided into 60 minutes, 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 × 600 seconds. 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 system.

Similarly, the practical unit of angular measure is the degree, of which there are 360 (six sixties) in a circle. There are 60 minutes 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 or of a degree a "second". Until at least the 18th century, 1/60 of a second was called a "tierce" or "third".[22][23]

## Fractions

In the sexagesimal system, any fraction in which the denominator is a regular number (having only 2, 3, and 5 in its prime factorization) may be expressed exactly.[24] 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 many times) = 0;8,34,17
1/11 = 0;5,27,16,21,49
1/13 = 0;4,36,55,23
1/14 = 0;4,17,8,34
1/17 = 0;3,31,45,52,56,28,14,7
1/19 = 0;3,9,28,25,15,47,22,6,18,56,50,31,34,44,12,37,53,41

The fact in arithmetic that the two numbers that are adjacent to sixty, 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 (1/59 = 0;1; 1/61 = 0;0,59), and that other non-regular primes have fractions that repeat with a longer period.

## Examples

Babylonian tablet YBC 7289 showing the sexagesimal number 1;24,51,10 approximating 2

The square root of 2, the length of the diagonal of a unit square, was approximated by the Babylonians of the Old Babylonian Period (1900 BC – 1650 BC) as

[25]

Because 2  1.41421356... is an irrational number, it cannot be expressed exactly in sexagesimal (or indeed any integer-base system), but its sexagesimal expansion does begin 1;24,51,10,7,46,6,4,44...

The length of the tropical year in Neo-Babylonian astronomy (see Hipparchus), 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 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, which were in sexagesimal notation.

The value of π as used by the Greek mathematician and scientist Claudius Ptolemaeus (Ptolemy) was 3;8,30 = 3 + 8/60 + 30/602 = 377/1203.141666....[26] Jamshīd al-Kāshī, a 15th-century Persian mathematician, calculated π in sexagesimal numbers to an accuracy of nine sexagesimal digits; his value for 2π was 6;16,59,28,1,34,51,46,14,50.[27][28]

## References

1. Ifrah, Georges (2000), The Universal History of Numbers: From prehistory to the invention of the computer., John Wiley and Sons, ISBN 0-471-39340-1. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk.
2. Macey, Samuel L. (1989). The Dynamics of Progress: Time, Method, and Measure. Atlanta, Georgia: University of Georgia Press. p. 92. ISBN 978-0-8203-3796-8.
3. Neugebauer, O. (1969). The Exact Sciences In Antiquity. Dover. ISBN 0-486-22332-9.
4. Neugebauer, O. (1969). The Exact Sciences in Antiquity. Dover. pp. 17, para. 2. ISBN 0-486-22332-9. The example of our present system of numeration for degrees, hours, measures and ordinary numbers should suffice totally to discredit the popular idea that a number system was “invented” at a certain moment. Yet innumerable “reasons” have been advanced why the Babylonians used the basis 60 for their number system. I shall not make any attempt to discuss here the history of the sexagesimal system in any detail, but a few points must be mentioned because they are of importance for the historical approach to the development of number systems as a whole.
5. Neugebauer, O. (1969). The Exact Sciences in Antiquity. Dover. pp. 19, para. 2 (middle). ISBN 0-486-22332-9. Variations of these systems, both decimal and more or less sexagesimal, can be established at different localities. The main facts, however, are common to all of them, namely, the existence of a decimal substratum and the use of bigger symbols to represent higher units. This latter fact is obviously the root for the development of the place value notation.
6. Neugebauer, O. (1969). The Exact Sciences in Antiquity. Dover. pp. 18, para. 3. ISBN 0-486-22332-9. First of all, there exists a common misconception as to the generality of the use of the sexagesimal system. The very same tablet which contains hundreds of sexagesimal numbers, column beside column, to compute the dates of the new moons for a given year, might end with a “colophon” containing the name of the owner of the tablet, the name of the scribe, and the date of writing of the text, the year being expressed in the form 2 me 25 “2 hundred 25” where the main text would express the very same date sexagesimally as 3,45. In other words, it is only in strictly mathematical or astronomical contexts that the sexagesimal system is consistently applied. In all other matters (dates, measures of weight, areas, etc.), use was made of mixed systems which have their exact parallel in the chaos of 80-division, 24-division, 12-division, 10-division, 2-division which characterizes the units of our own civilization. The question of the origin of the sexagesimal system is therefore inextricably related to the much more complex problem of the history of many concurrent numerical notations and their innumerable local and chronological variations.
7. Neugebauer, O. (1969). The Exact Sciences in Antiquity. Dover. pp. 17, para. 3 (middle). ISBN 0-486-22332-9. In other words: it is only in strictly mathematical or astronomical contexts that the sexagesimal system is consistently applied. In all other matters (dates, measures of weight, areas, etc.), use was made of mixed systems which have their exact parallel in the chaos of 60-division, 24-division, 12-division, 10-division, 2-division which characterizes the units of our own civilization.
8. Neugebauer, O. (1969). The Exact Sciences in Antiquity. Dover. pp. 19, para. 3. ISBN 0-486-22332-9. Combined with this, another process was taking place. In economic texts units of weight, measuring silver, were of primary importance. These units seem to have been arranged from early times in a ratio 60 to 1 for the main units “mana” (the Greek μνα ̃ “mina”) and shekel. Though the details of this process cannot be described accurately, it is not surprising to see this same ratio applied to other units and then to numbers in general. In other words, any sixtieth could have been called a shekel because of the familiar meaning of this concept in all financial transactions. Thus the “sexagesimal” order eventually became the main numerical system and with it the place value writing derived from the use of bigger and smaller signs. The decimal substratum, however, always remained visible for all numbers up to 60.
9. Neugebauer, O. (1969). The Exact Sciences in Antiquity. Dover. pp. 19, para. 2. ISBN 0-486-22332-9. Beside these basic elements, many modifications of number symbols were in use for different classes of objects, such as capacity measures, weights, areas, etc. Among these a clear decimal system has been recognized with signs for 1, 10, and 100. The numbers 1 and 10 we have already described. The 100 was written as a circular impression which looks like 10, but is made much bigger. Thus 100 is simply “big 10”. Another system proceeds sexagesimally, at least partially. Distinct units are 1 and 10 as before. A big 1 represents 60. Two big units written in opposing directions are combined into one sign to form 120. A 10-sign added in the middle gives 1200. A very big 10 sign stands for 3600. Variations of these systems, both decimal and more or less sexagesimal, can be established at different localities. The main facts, however, are common to all of them, namely, the existence of a decimal substratum and the use of bigger symbols to represent higher units.
10. Neugebauer, O. (1969). The Exact Sciences in Antiquity. Dover. pp. 17, para. 1. ISBN 0-486-22332-9. The other inconsistency of the modern astronomical notation, namely, to continue beyond the seconds with decimal fractions, is a recent innovation. It is interesting to see that it took about 2000 years of migration of astronomical knowledge from Mesopotamia via Greeks, Hindus, and Arabs to arrive at a truly absurd numerical system.
11. Bello, Ignacio; Britton, Jack R.; Kaul, Anton (2009), Topics in Contemporary Mathematics (9th ed.), Cengage Learning, p. 182, ISBN 9780538737791.
12. Lamb, Evelyn (August 31, 2014), "Look, Ma, No Zero!", Scientific American, Roots of Unity
13. Barton, George A. (1908), "On the Babylonian origin of Plato's nuptial number", Journal of the American Oriental Society, 29: 210–219, doi:10.2307/592627, JSTOR 592627. McClain, Ernest G.; Plato, (1974), "Musical "Marriages" in Plato's "Republic"", Journal of Music Theory, 18 (2): 242–272, doi:10.2307/843638, JSTOR 843638
14. Newton, Isaac (1671). The Method of Fluxions and Infinite Series: With Its Application to the Geometry of Curve-lines. London: Henry Woodfall (published 1736). p. 146. The most remarkable of these is the Sexagenary or Sexagesimal Scale of Arithmetick, of frequent use among Astronomers, which expresses all possible Numbers, Integers or Fractions, Rational or Surd, by the Powers of Sixty, and certain numeral Coefficients not exceeding fifty-nine.
15. Neugebauer, Otto (1952), "Tamil Astronomy: A Study in the History of Astronomy in India", Osiris, 10: 252–276, doi:10.1086/368555; reprinted in Neugebauer, Otto (1983), Astronomy and History: Selected Essays, New York: Springer-Verlag, ISBN 0-387-90844-7
16. Bowers, Nancy (1977), "Kapauku numeration: Reckoning, racism, scholarship, and Melanesian counting systems" (PDF), Journal of the Polynesian Society, 86 (1): 105–116.
17. Lean, Glendon Angove (1992), Counting Systems of Papua New Guinea and Oceania, Ph.D. thesis, Papua New Guinea University of Technology. See especially chapter 4.
18. Aaboe, Asger (1964), Episodes from the Early History of Mathematics, New Mathematical Library, 13, New York: Random House, pp. 103–104
19. Cajori, Florian (2007) [1928]. A History of Mathematical Notations. 1. New York: Cosimo, Inc. p. 216. ISBN 9781602066854.
20. Neugebauer, Otto; Sachs, Abraham Joseph; Götze, Albrecht (1945), Mathematical Cuneiform Texts, American Oriental Series, 29, New Haven: American Oriental Society and the American Schools of Oriental Research, p. 2
21. Aaboe (1964), pp. 15–16, 25
22. Wade, Nicholas (1998), A natural history of vision, MIT Press, p. 193, ISBN 978-0-262-73129-4
23. Lewis, Robert E. (1952), Middle English Dictionary, University of Michigan Press, p. 231, ISBN 978-0-472-01212-1
24. Neugebauer, Otto E. (1955), Astronomical Cuneiform Texts, London: Lund Humphries
25. YBC 7289 clay tablet
26. Toomer, G. J., ed. (1984), Ptolemy's Almagest, New York: Springer Verlag, p. 302, ISBN 0-387-91220-7
27. Youschkevitch, Adolf P., "Al-Kashi", in Rosenfeld, Boris A., Dictionary of Scientific Biography, p. 256.
28. Aaboe (1964), p. 125