Babylonian Numbers Converter

This Babylonian numbers converter turns an everyday decimal whole number into the base-60 digits that the ancient Babylonians used. You enter a number and the tool repeatedly divides it by 60, collecting the remainders, to produce the sexagesimal digits, each a value from 0 to 59, written from the largest place down to the units. It shows the base-60 digit string, the number of sexagesimal places, and an expanded form so you can see how each place is multiplied by a power of 60 and added together. The Babylonians wrote these digits with cuneiform wedges pressed into clay, a vertical wedge standing for one and a corner wedge for ten, grouping up to nine ones and five tens to build each digit up to 59. Their base-60 system, inherited from the earlier Sumerians, is the reason a full circle still has 360 degrees, an hour has 60 minutes and a minute has 60 seconds. Students of the history of maths, teachers preparing lessons, and anyone curious about ancient counting can use this converter to see how a familiar number looks in the world's oldest positional system. Note that early Babylonians had no true zero, so an empty middle place was ambiguous until a placeholder mark was later added. Enter a whole number below and the base-60 breakdown updates as you type.

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1, 1, 1
Babylonian base-60 digits (largest place first)
Decimal value3,661
Sexagesimal places3
Expanded form1×3,600 + 1×60 + 1

Each base-60 digit runs from 0 to 59. Whole numbers only; any decimal part is dropped. Early Babylonian writing had no symbol for zero. Educational tool.

How it works

Base-60, or sexagesimal, is a positional system just like our base-10, except each place is worth 60 times the one to its right: units, then sixties, then 3,600s, and so on. To convert, divide the decimal number by 60 and record the remainder as the units digit, divide the quotient by 60 for the next digit, and repeat until nothing is left. Reading the remainders from last to first gives the base-60 digits, each between 0 and 59. The expanded form multiplies each digit by its place value and adds the parts back to the original number.

Worked example

Take 3,661. Dividing by 60 gives 61 remainder 1, so the units digit is 1. Dividing 61 by 60 gives 1 remainder 1, so the sixties digit is 1. Dividing 1 by 60 gives 0 remainder 1, so the 3,600s digit is 1. Reading top down, the base-60 digits are 1, 1, 1 across three places. The expanded form is 1×3,600 + 1×60 + 1, which adds back to 3,600 + 60 + 1, or 3,661.

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Frequently asked questions

What base did the Babylonians use?

The Babylonians used a base-60 or sexagesimal system, with a sub-base of 10 for writing each digit. It is why we still divide an hour into 60 minutes, a minute into 60 seconds, and a full circle into 360 degrees.

How do you convert a number to base-60?

Divide the number by 60 and keep the remainder as the last digit, then divide the quotient by 60 again for the next digit, and repeat until the quotient reaches zero. Reading the remainders from last to first gives the base-60 digits, each between 0 and 59.

Did the Babylonians have a zero?

Early Babylonian writing had no symbol for zero, which made an empty place ambiguous. Later scribes added a placeholder mark to show a missing position, but it was never used on its own as the number zero the way we use it today.