Everything You Need to Know about the Sexagesimal Numeral System

Everyone knows there are 60 seconds in a minute, 60 minutes in an hour, and 60 hours in a... Oops! 24 hours in a day, not 60 anymore. But anyway, the sexagesimal numeral system is used by all of us on a day-to-day basis. In this blog I will talk about its origin and historical uses, as well as how to convert between decimal and sexagesimal.

This blog uses the notational system developed by Otto Neugebauer in the 1930s, in which:

- each position in the sexagesimal number is represented by a decimal number between 0 and 59 inclusive;

- positions within the integral or fractional portion are separated by a comma (,);

- the sexagesimal equivalent of the "decimal point", i.e., the symbol that separates the integral portion and the fractional portion of the number, is denoted by a semicolon (;).

The exact origin of the sexagesimal numeral system is disputed. Many people argue that it comes from the fact that a year has 365 days. The number 365 is so close to 360, which is a multiple of 60, that some ancient civilizations decided to use 60 as the base of their numeral systems.

Some other people suggest that some ancient civilizations used a different method of finger counting. Instead of counting using the fingers themselves, they used the thumb to point at each of the three finger bones on the other four fingers, enabling finger counting from 1 to 12. In addition, the other hand may be used to indicate the number of times 12 has been reached on their first hand. This allowed people to count from 1 to 60, which could be how the sexagesimal numeral system originated.

People in ancient times used the sexagesimal numeral system far more often than we do today. In the Babylonian civilization, for example, people used it for basic counting as well as measurement of time, the latter having been preserved all the way till today.

The sexagesimal number system was also used extensively by many civilizations in the field of mathematics, with countless examples:

- The value of the square root of 2 (1.4142135623...) was approximated by the ancient Babylonians as 1;24,51,10 (1.41421296...);

- The value of π (3.1415926535...) used by the Greek mathematician and scientist Ptolemy was 3;8,30 (3.141666...);

- Ptolemy's Almagest, a treatise on mathematical astronomy written in the 2nd century AD, uses base 60 to express the fractional parts of numbers;

- and many more...

Now let's get into the conversion.

First we will talk about how to convert a sexagesimal number into its decimal equivalent. The sexagesimal system is a positional numeral system just like decimal, which means every position has its own place value. For example, the units digit of a base-60 number indicates the number of "ones", the "sixties" digit (one place higher than the units digit) denotes how many 60s there are in the number, and the "sixtieths" digit (one place lower than the units digit) means how many 60ths are present in the number.

This makes conversion from base 60 to base 10 easy.

Example: Convert the sexagesimal number 4,18;23,17 to decimal.

Assigning place values to each sexagesimal digit:

(4,18;23,17)₆₀ = 4 × 60¹ + 18 × 60⁰ + 23 × 60^-1 + 17 × 60^-2 = 240 + 18 + (23/60) + (17/3600) = 258 + 1397/3600 = 258.388₁₀ (to 3 d.p.)

Now it's time to discuss how to do the exact opposite: converting from base 10 to base 60. This is slightly more complicated, but it's not difficult once you learn it.

For integers:

- First, write down the number to be converted and then divide it by 60. Write the quotient below the original number and the remainder next to it.

- Next, repeat this process for the quotient obtained each time, until you get a quotient of zero.

- Finally, write down all the remainders, each separated by a comma, from bottom to top.

Example: Convert the decimal number 8993 to sexagesimal.

8993 ÷ 60 ... 53

149 ÷ 60 ... 29

2 ÷ 60 ... 2

0

∴ 8993₁₀ = (2,29,53)₆₀

For decimals:

- First, write down the number to be converted and convert the integral portion first according to the method above.

- Next, write down the fractional portion of the number and multiply it by 60.

- Next, repeat this process on only the fractional portion of the number you get each time, until the fractional portion is cleared (that is, every single digit after the decimal point is zero).

- Finally, put a semicolon after the integral portion of the sexagesimal number, and then write down all the integral portions obtained in the last step from top to bottom, each separated by a comma.

Example: Convert the base-10 decimal 16.26 to sexagesimal.

The integral portion is 16

.26 × 60 = 15.6

.6 × 60 = 36.0

∴ 16.26₁₀ = (16;15,36)₆₀

Hopefully this blog helped you achieve a better understanding of the origin, usage, and conversion of sexagesimal numbers. If you need any clarifications or want to know more about sexagesimal, leave a comment below. You are also welcome to report any errors in this blog.