Floor Function/Examples/Floor of Minus 5 over 2

Theorem

$\floor {-\dfrac 5 2} = -3$

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

Proof

We have that:

\(\ds -\dfrac 5 2\)

\(=\)

\(\ds -3 + \dfrac 1 2\)

\(\ds \leadsto \ \ \)

\(\ds -3\)

\(\le\)

\(\ds -\dfrac 5 2\)

\(\ds \)

\(<\)

\(\ds -2\)

Hence $-3$ is the floor of $-\dfrac 5 2$ by definition.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 10$: The well-ordering principle