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 $1$:
- $x \bmod 1 = x - \floor x$
Thus:
| \(\ds 1 \cdotp 1 \bmod 1\) | \(=\) | \(\ds 1 \cdotp 1 - \floor {1 \cdotp 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \cdotp 1 - 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 0 \cdotp 1\) |
$\blacksquare$
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: Exercise $10$