Definition:Modulo Operation

Definition
Let $$x, y \in \R$$ be real numbers.

Then the operation $$\mod$$ is defined as:
 * $$x \, \bmod \, y \ \stackrel {\mathbf {def}} {=\!=} \ \begin{cases}

x - y \left \lfloor {\frac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$$

From the definition of the floor function, we see that, when $$y \ne 0$$:


 * $$0 \le \frac x y - \left \lfloor {\frac x y}\right \rfloor = \frac {x \, \bmod \, y} y < 1$$.

Hence:
 * $$y > 0 \implies 0 \le x \, \bmod \, y < y$$;
 * $$y < 0 \implies 0 \ge x \, \bmod \, y > y$$;
 * $$x - \left({x \, \bmod \, y}\right)$$ is an integral multiple of $$y$$.

The operation is most usually defined when $$x$$ and $$y$$ are both integers.

From the Quotient-Remainder Theorem it follows that the quantity $$x \, \bmod \, y$$ is called the remainder when $$x$$ is divided by $$y$$. This still holds when $$x$$ and $$y$$ are not actually integers.

The value $$\left \lfloor {\frac x y}\right \rfloor$$ is called the quotient.

Modulo Zero
We see that, from the definition:
 * $$x \, \bmod \, 0 = x$$

This can be considered as a "special case", but it is interesting to note that most of the results concerning the modulo operation still hold.

Modulo 1
Note also that from the definition:
 * $$x \, \bmod \, 1 = x - \left \lfloor {x}\right \rfloor$$

from which it follows directly that:
 * $$x = \left \lfloor {x}\right \rfloor + \left({x \, \bmod \, 1}\right)$$

The value $$x \, \bmod \, 1$$ is called the fractional part of $$x$$, and sometimes denoted $$\left\{{x}\right\}$$.

From Real Number Minus Floor we confirm that $$0 \le x \, \bmod \, 1 < 1$$.

Also see
Compare with congruence modulo $z$ which approaches the subject from a slightly different direction.