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Numeral system

Sexagesimal, also known as 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, the largest sexagesimal digit is "59".

According to Otto Neugebauer, the origins of sexagesimal are not as simple, consistent, or singular in time as they are often portrayed. 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 sexagesimal digits are written. 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. 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 sexagesimal is used most uniformly and consistently in mathematical tables of data.

In the Chinese calendar, a system 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. 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 of a degree in base 60, and was practically equivalent to a modern day table of values of the sine function. Medieval astronomers also used sexagesimal numbers to note time. Al Biruni first subdivided the hour sexagesimally into minutes, seconds, thirds and fourths in 1000 while discussing Jewish months. Around 1235 John of Sacrobosco continued this tradition, although Nothaft thought Sacrobosco was the first to do so. The Parisian version of the Alfonsine tables (ca. 1320) used the day as the basic unit of time, recording multiples and fractions of a day in base 60 notation. The sexagesimal number system continued to be frequently used by European astronomers for performing calculations as late as 1671. For instance, Jost Bürgi in Fundamentum Astronomiae (presented to Emperor Rudolf II in 1592), his colleague Ursus in Fundamentum Astronomicum, and possibly also Henry Briggs, used multiplication tables based on the sexagesimal system in the late 16th century, to calculate sines. In the late 18th and early 19th centuries, Tamil astronomers were found to make astronomical calculations, reckoning with shells using a mixture of decimal and sexagesimal notations developed by Hellenistic astronomers. 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.

Modern uses for the sexagesimal system include measuring angles, geographic coordinates, electronic navigation, 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 (3 hours, 23 minutes, and 17 seconds) can be interpreted as a whole sexagesimal number (no sexagesimal point), meaning 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 version 1.1 of the YAML data storage format, sexagesimals are supported for plain scalars, and formally specified both for integers and floating point numbers. This has led to confusion, as e. g. some MAC addresses would be recognised as sexagesimals and loaded as integers, where others were not and loaded as strings. In YAML 1.2 support for sexagesimals was dropped.

In Hellenistic Greek astronomical texts, such as the writings of Ptolemy, sexagesimal numbers were written using Greek alphabetic numerals, with each sexagesimal digit being treated as a distinct number. Hellenistic astronomers adopted a new symbol for zero, , which morphed over the centuries into other forms, including the Greek letter omicron, ο, normally meaning 70, but permissible in a sexagesimal system where the maximum value in any position is 59. The Greeks limited their use of sexagesimal numbers to the fractional part of a number. In medieval Latin texts, sexagesimal numbers were written using Arabic numerals; the different levels of fractions were denoted minuta (i. e., fraction), minuta secunda, minuta tertia, etc. By the 17th 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′2″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. 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 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, of a second was called a "tierce" or "third". In the 1930s, Otto Neugebauer introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon (;) to separate the integer and fractional portions of the number and using a comma (,) to separate the positions within each portion. For example, the mean synodic month used by both Babylonian and Hellenistic astronomers and still used in the Hebrew calendar is 29;31,50,8,20 days. This notation is used in this article.