Quotient of Modulo Operation with Modulus

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Theorem

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

Let $x \bmod y$ denote the modulo operation:

$x \bmod y := \begin{cases}

x - y \floor {\dfrac x y} & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\floor {\dfrac x y}$ denotes the floor of $\dfrac x y$.


Let $y \ne 0$.


Then:

$0 \le \dfrac x y - \floor {\dfrac x y} = \dfrac {x \bmod y} y < 1$


Proof

From Real Number minus Floor:

$\dfrac x y - \floor {\dfrac x y} \in \hointr 0 1$

Thus by definition of half-open real interval:

$0 \le \dfrac x y - \floor {\dfrac x y} < 1$


Then:

\(\ds x \bmod y\) \(=\) \(\ds x - y \floor {\frac x y}\) Definition of Modulo Operation
\(\ds \leadsto \ \ \) \(\ds \frac {x \bmod y} y\) \(=\) \(\ds \frac {x - y \floor {\dfrac x y} } y\)
\(\ds \) \(=\) \(\ds \frac x y - \frac y y \floor {\dfrac x y}\)
\(\ds \) \(=\) \(\ds \frac x y - \floor {\dfrac x y}\)

Hence the result.

$\blacksquare$


Sources

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