Definition:Remainder/Real

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Definition

Let $x, y \in \R$ be real numbers such that $y \ne 0$.

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 see


Sources