Modulo Operation/Examples/1.1 mod 1

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Theorem

$1 \cdotp 1 \bmod 1 = 0 \cdotp 1$

where $\bmod$ denotes the modulo operation.


Proof

By definition of modulo $1$:

$x \bmod 1 = x - \left \lfloor {x}\right \rfloor$


Thus:

\(\displaystyle 1 \cdotp 1 \bmod 1\) \(=\) \(\displaystyle 1 \cdotp 1 - \left\lfloor{1 \cdotp 1}\right\rfloor\)
\(\displaystyle \) \(=\) \(\displaystyle 1 \cdotp 1 - 1\)
\(\displaystyle \) \(=\) \(\displaystyle 0 \cdotp 1\)

$\blacksquare$


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