# Modulo Operation/Examples/1.1 mod 1

## 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:

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

$\blacksquare$