modular arithmetic

Discrete mathematics is the study of mathematics in systems that are countable, such as the set of integers.

Modular arithmetic is a topic of discrete mathematics that is defined as integer arithmetic. This means that there are no decimals. Most operations are very similar to "regular" arithmetic:

• 5 + 3 = 8
• 5 - 3 = 2
• 5 * 3 = 15

It actually gets tricky with division:

• 5 / 3 = 1

5 ÷ 3 actually equals 1.66666, but since there are no decimals, the result is truncated to the integer 1. The remaining .66666 is calculated using a new operator called modulo.

• 5 % 3 = 2

The result of the modulo operation is the remainder of the division. So, in our example:

```   5    ÷    3    =
(5 / 3) + (5 % 3) =
1    +  2 / 3  =
1.66666
```

Note that the result of a modulo operation will always be an integer between 0 and the modulus minus 1. This special property is what makes modular arithmetic very useful in programming.

Math For Artists Who Now Need to Program

Elie Zananiri
ITP DriveBy
1 Oct 2008