Free Floor Function Calculator

What Is the Floor Function?

The floor function is a basic math operation that takes any real number and returns the greatest integer that is less than or equal to it. It is often called the greatest integer function, the integer part function, or simply the round‑down function. When you want the floor of a number, you are looking for the integer immediately below (or equal to) that number, ignoring any fractional part. An online floor(x) calculator performs this operation instantly, handling any positive or negative input without manual enumeration.

Definition and Notation

Formally, for a real number $x$, the floor of $x$ — written as $\lfloor x \rfloor$ or in programming languages as floor(x) — is defined as the largest integer $m$ satisfying $m \le x$:

$$\lfloor x \rfloor = \max\{\, m \in \mathbb{Z} \mid m \le x \,\}.$$

Thus the floor function maps the entire real line onto the set of integers: $\text{floor} : \mathbb{R} \to \mathbb{Z}$.

Examples of the Floor Function

The table below shows how the floor function behaves with different values.

$x$	Integers $\le x$	Greatest among them	$\lfloor x \rfloor$
21.3	21, 20, 19, …	21	21
7	7, 6, 5, …	7	7
–1.3	–2, –3, –4, …	–2	–2
$\pi \approx 3.14159$	3, 2, 1, …	3	3
–2	–2, –3, –4, …	–2	–2

Several key observations can be made:

• For a positive non‑integer (e.g., 21.3), the floor is simply the integer part.
• When the input is already an integer, the floor equals that integer.
• For negative numbers, the floor is not the same as dropping the sign; it moves to the next lower integer (e.g., $\lfloor -1.3 \rfloor = -2$, not $-1$).
• The floor function always returns an integer, never a fraction.

Important Properties

The floor function satisfies a set of useful algebraic and analytical properties:

• Bounds: Every real number lies between its floor and the floor plus one:
  $x - 1 < \lfloor x \rfloor \le x$.
• Integer translation: Adding an integer $n$ shifts the floor accordingly:
  $\lfloor x + n \rfloor = \lfloor x \rfloor + n$ for any integer $n$.
• Idempotence: Applying the floor twice has no further effect:
  $\lfloor \lfloor x \rfloor \rfloor = \lfloor x \rfloor$.
• Monotonicity: If $x \le y$, then $\lfloor x \rfloor \le \lfloor y \rfloor$ — the function never decreases.
• Relation to the ceiling function: The ceiling, written $\lceil x \rceil$, gives the smallest integer $\ge x$ and is linked to the floor by $\lceil x \rceil = -\lfloor -x \rfloor$.

The floor function is not continuous; it has a jump discontinuity at every integer. It is also not one‑to‑one because an entire interval $[n, n+1)$ maps to the single integer $n$.

Graphical Appearance

When plotted, the floor function produces a staircase‑shaped graph characteristic of step functions. For each integer $n$, the function equals $n$ on the interval $[n, n+1)$ and then jumps up by one at the next integer. On the graph, the points $(n, n)$ are shown as filled circles (the function takes that value), while points $(n, n-1)$ are open circles (the function does not take that value). For example, at $x = 1$ the value is $1$, not $0$.

Typing the Floor Function in LaTeX

In LaTeX documents, produce the floor symbols using \lfloor (left) and \rfloor (right). For example, \lfloor x \rfloor renders as $\lfloor x \rfloor$. To size brackets automatically around a tall expression, use \left\lfloor and `\right\rfloor$.

Using the Floor Function Calculator

This floor(x) calculator acts as a greatest integer function calculator and a round down calculator in one. Simply enter any real number — positive, negative, integer, or decimal — and the tool instantly returns the floor. It eliminates the need to mentally list integers and check inequalities, which is especially helpful for negative inputs or large numbers. Whether you need the integer part for a programming problem, a math assignment, or a quick check, this online calculator provides a fast, error‑free result.