Sexagesimal

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

The number 60, a highly composite number, has twelve factors, namely { 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 } of which two, three, and five 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, six minutes, five minutes, etc. Sixty is the smallest number that is divisible by every number from one to six. This is because 60 = 2 × 2 × 3 × 5 = 4 × 3 × 5.

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

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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. Later Babylonian texts used a dot 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 123,000.