# Definition:Babylonian Number System

## Contents

## Definition

The number system as used in the Old Babylonian empire was a positional numeral system where the number base was a combination of decimal (base $10$) and sexagesimal (base $60$).

The characters were written in cuneiform by a combination of:

- a thin vertical wedge shape, to indicate the digit $1$
- a fat horizontal wedge shape, to indicate the digit $10$

arranged in groups to indicate the digits $2$ to $9$ and $20$ to $50$.

At $59$ the pattern stops, and the number $60$ is represented by the digit $1$ once again.

Thus these groupings were placed side by side:

- the rightmost grouping would indicate a number from $1$ to $59$
- the one to the left of that would indicate a number from $60 \times 1$ to $60 \times 59$

and so on, each grouping further to the left indicating another multiplication by $60$

For fractional numbers there was no actual radix point. Instead, the distinction was inferred by context.

The fact that they had no symbol to indicate the zero digit means that this was not a true positional numeral system as such.

For informal everyday arithmetic, they used a decimal system which was the decimal part of the full sexagesimal system.

## Also denoted as

When representing numbers using the Babylonian number system, it is laborious to represent the actual cuneiform symbols themselves.

Hence it is commonplace to use their decimal counterparts, separated by commas, so that the number represented, for example, in cuneiform as:

would be represented as:

- $7, 7, 7$

In such a system, the radix point is represented by a semicolon.

## Examples

### Example: $25 \, 267$

The number $25 \, 267$ is represented in the Babylonian number system as:

That is:

- $7 \times 60 \times 60 + 7 \times 60 + 7 = 25 \, 200 + 420 + 7 = 25 \, 267$

### Example: $12,59;57,17$

The number represented in the Babylonian number system as:

- $12,59;57,17$

is represented in decimal as:

- $12 \times 60 + 29 + \dfrac {57} {60} + \dfrac {17} {60 \times 60} \approx 779 \cdotp 955$

## Sources

- 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $1$: Tokens, Tallies and Tablets: The first numerals