Sexagesimal

[2] 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.

[2] Another practical factor that helped expand the use of sexagesimal in the past, even if less consistently than 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.

In the late 3rd millennium BC, Sumerian/Akkadian units of weight included the kakkaru (talent, approximately 30 kg) divided into 60 manû (mina), which was further subdivided into 60 šiqlu (shekel); the descendants of these units persisted for millennia, though the Greeks later coerced this relationship into the more base-10–compatible ratio of a shekel being one 50th of a mina.

The details 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.

[5] Ptolemy's Almagest, a treatise on mathematical astronomy written in the second century AD, uses base 60 to express the fractional parts of numbers.

Al-Biruni first subdivided the hour sexagesimally into minutes, seconds, thirds and fourths in 1000 while discussing Jewish months.

[9] 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.

[12][13] Modern uses for the sexagesimal system include measuring angles, geographic coordinates, electronic navigation, and time.

In version 1.1[15] of the YAML data storage format, sexagesimals are supported for plain scalars, and formally specified both for integers[16] and floating point numbers.

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.

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.

[23][24] 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.

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 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... (OEIS: A070197) The value of π as used by the Greek mathematician and scientist Ptolemy was 3;8,30 = 3 + ⁠8/60⁠ + ⁠30/602⁠ = ⁠377/120⁠ ≈ 3.141666....[28] Jamshīd al-Kāshī, a 15th-century Persian mathematician, calculated 2π as a sexagesimal expression to its correct value when rounded to nine subdigits (thus to ⁠1/609⁠); his value for 2π was 6;16,59,28,1,34,51,46,14,50.