Floor Function/Examples/Floor of Root 5

Theorem

$\floor {\sqrt 5} = 2$

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

Proof

The decimal expansion of $\sqrt 5$ is:

$\sqrt 5 \approx 2 \cdotp 23606 \, 79774 \, 99789 \, 6964 \ldots$

Thus:

$2 \le \sqrt 5 < 3$

Hence $2$ is the floor of $\sqrt 5$ by definition.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $2$. Equivalence Relations: Exercise $3$