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
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory