Floor Function/Examples/Floor of Minus One Half

Theorem

$\floor {-\dfrac 1 2} = -1$

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

Proof

We have that:

$-1 = -\dfrac 2 2 \le -\dfrac 1 2$

and:

$0 > -\dfrac 1 2$

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

$\blacksquare$

Sources

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