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Sexagesimal

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The sexagesimal system is a numeral system that uses sixty as its base. Most people today use a decimal system, which is based on ten. However, sixty is a highly composite number. This means it has many divisors. The number sixty can be divided evenly by twelve different numbers: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Because it has so many factors, it is very useful for calculating fractions.

Babylonian numerals
Babylonian numerals

To understand how this works, think about how we divide an hour. One hour can be split into many equal parts. You can have 30, 20, 15, 12, 10, 6, 5, 4, 3, 2, or even 1-minute sections. This mathematical advantage makes the system perfect for writing and calculating fractions. In ancient times, this was a huge benefit for merchants. They used sexagesimal math to divide large quantities of goods fairly during trades.

Babylonian 1
Babylonian 1

The system was not a pure base-60 system in the way we think of modern math. Ancient Babylonians used a sub-base of ten to write their digits. They used cuneiform, which are wedge-shaped marks made on clay. A single sexagesimal digit was made by adding small marks for units and wider marks for tens. For example, a digit could represent any value up to 59.

Babylonian 10
Babylonian 10

Writing these numbers was sometimes complicated because they lacked a consistent zero. For a long time, there was no symbol for zero at all. This meant that the symbol for 1 and the symbol for 60 looked identical. Later Babylonian texts eventually used a placeholder to represent zero. However, they only used this placeholder in the middle of numbers, not at the end.

Babylonian digit 0
Babylonian digit 0

History shows that the sexagesimal system has deep roots. The Sumerians originated the system in the 3rd millennium BC. The Babylonians later adopted and used it. The system has also appeared in other cultures. For example, the Chinese zodiac uses a sexagenary cycle. This cycle repeats every 60 steps through a sequence of ten stems and twelve branches.

sexagenary cycle years
sexagenary cycle years

Great scientists have relied on this system for centuries. In the 2nd century AD, Ptolemy used base 60 in his work, the *Almagest*. He used it to express the fractional parts of degrees in astronomy. In the year 1000, Al-Biruni subdivided the hour into minutes, seconds, thirds, and fourths. Later, in the 17th century, European astronomers still used sexagesimal calculations. Even the mathematician Jost Bürgi used sexagesimal multiplication tables to calculate sines.

Ybc7289-bw
Ybc7289-bw

We still use modified versions of this system every single day. We use it to measure time, where one hour is 60 minutes and one minute is 60 seconds. We also use it for angles and geographic coordinates. A full circle contains 360 degrees, which is six groups of sixty. Each degree contains 60 minutes of arc, and each minute contains 60 arcseconds. This ancient math remains a vital part of modern navigation and science.

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🖼️ Images & Media (15)
File:Proto-cuneiform sexagesimal type Sa.svg
Proto-cuneiform sexagesimal type Sa.svg
File:Babylonian 1.svg
Babylonian 1.svg
File:Babylonian 2.svg
Babylonian 2.svg
File:Babylonian 3.svg
Babylonian 3.svg
File:Babylonian 4.svg
Babylonian 4.svg
File:Babylonian 9.svg
Babylonian 9.svg
File:Babylonian 10.svg
Babylonian 10.svg
File:Babylonian 20.svg
Babylonian 20.svg
File:Babylonian 30.svg
Babylonian 30.svg
File:Babylonian 40.svg
Babylonian 40.svg
File:Babylonian 50.svg
Babylonian 50.svg
File:Babylonian numerals.svg
Babylonian numerals.svg

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