Floor is between Number and One Less
Jump to navigation
Jump to search
Theorem
- $x - 1 < \floor x \le x$
where $\floor x$ denotes the floor of $x$.
Proof
By definition of floor function:
- $\floor x \le x < \floor x + 1$
Thus by subtracting $1$:
- $x - 1 < \paren {\floor x + 1} - 1 = \floor x$
So:
- $\floor x \le x$
and:
- $x - 1 < \floor x$
as required.
$\blacksquare$
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