StringMash.com

Babylonian numerals converter

Base 60, written in wedges pressed into clay.

Conversion
4 characters
Updates as you type
Sexagesimal2,23,3
Cuneiform๐’น๐’น ๐’Œ‹๐’Œ‹๐’น๐’น๐’น ๐’น๐’น๐’น
Drawn below

Show the steps
2233
  1. Base 60: divide by 60 again and again; the remainders are the digits, read from the last one up.
  2. 8,583 รท 60 = 143 remainder 3
  3. 143 รท 60 = 2 remainder 23
  4. 2 รท 60 = 0 remainder 2
  5. Each digit is written with ten-wedges for its tens and unit-wedges for its ones. The Babylonians wrote no zero at first, only a gap, and never a mark for the fractional point: readers worked it out from context.

Using it

Type a number, with a decimal point if you like. The output gives it in base 60 with commas between the digits, such as 2,23,3 for 8583, and a semicolon before any fractional places. The wedges are drawn below, and the rows give the cuneiform characters to copy.

Swap the boxes to read base 60 back: type the digits with commas, or paste the cuneiform signs with a space between digits.

How it works

The Babylonians counted in sixties, and built each digit from just two signs: a vertical wedge for 1 and a corner wedge for 10. 23 is two corner wedges and three vertical ones. The digits are then placed side by side, each worth sixty times the one to its right, so 2,23,3 is 2 ร— 3,600 + 23 ร— 60 + 3 = 8,583.

There was no zero at first, only a gap, and later a placeholder sign used in the middle of a number but never at the end. Nor was there a mark for the fractional point, so the same wedges could mean 23, 23 ร— 60 or 23โ„60; the reader worked it out from context. This page uses the modern scholars' notation, a semicolon, to show where the fraction starts.

Where base 60 survives

Sixty divides evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, which makes fractions easy. Its traces are still on every clock and compass: 60 minutes in an hour, 60 seconds in a minute, and 360 degrees in a circle.

The most famous Babylonian number is on the tablet YBC 7289, at Yale, which gives the square root of 2 as 1;24,51,10. Type that in with the boxes swapped: it's 1.41421296โ€ฆ, correct to about six decimal places.

Questions

How do you write 60 in Babylonian numerals?

As a single unit wedge in the sixties place, followed by an empty place: the same wedge as 1, which is why context mattered.

Did the Babylonians have zero?

Not as a number. Later texts used a placeholder sign inside numbers, but never at the end, and never as a value on its own.

Which characters are these?

Unicode's cuneiform sign DISH (U+12079) for the unit wedge and U (U+1230B) for the ten, the same signs Wikipedia uses.

Sources

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