Floor Function/Examples/Floor of Root 2

Theorem

$\floor {\sqrt 2} = 1$

where $\floor x$ denotes the floor of $x$.

Proof

The decimal expansion of $\sqrt 2$ is:

$\sqrt 2 \approx 1.41421 \ 35623 \ 73095 \ 0488 \ldots$

Thus:

$1 \le \sqrt 2 < 2$

Hence $1$ is the floor of $\sqrt 2$ by definition.

$\blacksquare$

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