Floor function

Definition

Let $x$ be a real number. The floor of $x$, written $\lfloor x \rfloor$, is the greatest integer $n$ with $n \le x$. The floor function is the function $\mathbb{R} \to \mathbb{Z}$, $x \mapsto \lfloor x \rfloor$.

Remarks

The greatest such integer exists because the reals are Archimedean and because of the well-ordering principle. For each integer $n$, $\lfloor n \rfloor = n$. For example, $\lfloor \pi \rfloor = 3$ and $\lfloor -0.5 \rfloor = -1$. The characterization of the floor gives an equivalent definition.

Explanations

L1 human

The floor of a number rounds it down to a whole number. On the number line, it is the nearest integer at the number or to its left. For negative numbers, take care: the floor of $-1.2$ is $-2$, not $-1$.