Definition:Remainder

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Definition

Let $a, b \in \Z$ be integers such that $b \ne 0$.

From the Division Theorem, we have that:

$\forall a, b \in \Z, b \ne 0: \exists_1 q, r \in \Z: a = q b + r, 0 \le r < \size b$


The value $r$ is defined as the remainder of $a$ on division by $b$, or the remainder of $\dfrac a b$.


Real Arguments

When $x, y \in \R$ the remainder is still defined:


The remainder of $x$ on division by $y$ is defined as the value of $r$ in the expression:

$\forall x, y \in \R, y \ne 0: \exists! q \in \Z, r \in \R: x = q y + r, 0 \le r < \size y$


From the definition of the Modulo Operation:

$x \bmod y := x - y \floor {\dfrac x y}$

it can be seen that the remainder of $x$ on division by $y$ is defined as:

$r = x \bmod y$


Also known as

A remainder as defined here is also known as a principal remainder.


Also see


Sources