Floor Function Calculator

What Is the Floor Function?

The floor function, denoted by the mathematical notation $\lfloor x \rfloor$ or often written as $\text{floor}(x)$ or $[x]$, maps any real number $x$ to the greatest integer less than or equal to $x$.

Informally, the floor function "rounds down" a number to the nearest integer on the number line. For positive real numbers, this corresponds to removing the fractional part (e.g., $\lfloor 3.75 \rfloor = 3$). For negative numbers, however, rounding down moves further left on the number line away from zero (e.g., $\lfloor -3.25 \rfloor = -4$).

Formal Mathematical Definition

For any real number $x \in \mathbb{R}$, the floor of $x$ is defined as:

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

Equivalently, $\lfloor x \rfloor$ is the unique integer $k$ satisfying the inequality:

$$k \le x < k + 1 \quad \text{where } k \in \mathbb{Z}$$

Floor Function vs. Ceiling, Rounding, and Truncation

Understanding how the floor function differs from other rounding operations is essential in computer science, mathematics, and discrete probability:

Input $x$	Floor $\lfloor x \rfloor$	Ceiling $\lceil x \rceil$	Round $[x]$	Truncate $\text{trunc}(x)$	Fractional $\{x\}$
3.75	3	4	4	3	0.75
3.00	3	3	3	3	0.00
-3.25	-4	-3	-3	-3	0.75
-4.00	-4	-4	-4	-4	0.00
0.50	0	1	1	0	0.50

Key Properties of the Floor Function

Idempotence: $\lfloor \lfloor x \rfloor \rfloor = \lfloor x \rfloor$. Applying floor multiple times produces the same integer.
Integer Addition: For any integer $n \in \mathbb{Z}$, $\lfloor x + n \rfloor = \lfloor x \rfloor + n$.
Duality with Ceiling: $\lfloor -x \rfloor = -\lceil x \rceil$. You can compare results using our Ceiling Function Calculator.
Fractional Part: The fractional part $\{x\}$ is defined as $\{x\} = x - \lfloor x \rfloor$, which always satisfies $0 \le \{x\} < 1$.
Relation to Floor Division: In integer arithmetic, $a // b = \lfloor a / b \rfloor$. Explore detailed remainders with the Floor Division Calculator.

Step-by-Step Calculation Examples

Example 1: Positive Decimal

Compute $\lfloor 8.92 \rfloor$:

Identify integers surrounding $8.92$: $8 < 8.92 < 9$.
Select the greatest integer that does not exceed $8.92$, which is $8$.
Result: $\lfloor 8.92 \rfloor = 8$, with fractional part $\{8.92\} = 8.92 - 8 = 0.92$.

Example 2: Negative Fraction

Compute $\lfloor -11/4 \rfloor$:

Convert fraction to decimal: $-11/4 = -2.75$.
Locate integers on the real line: $-3 < -2.75 < -2$.
The greatest integer less than or equal to $-2.75$ is $-3$.
Result: $\lfloor -11/4 \rfloor = -3$, with fractional part $\{-2.75\} = -2.75 - (-3) = 0.25$.

Frequently Asked Questions

Is the floor function continuous?

No, the floor function is piecewise constant and has jump discontinuities at every integer $x \in \mathbb{Z}$. It is right-continuous everywhere because $\lim_{x \to a^+} \lfloor x \rfloor = \lfloor a \rfloor$, but it is not left-continuous at integer points.

What is the floor of negative numbers?

For negative non-integer numbers, the floor function rounds down towards $-\infty$. For example, $\lfloor -1.2 \rfloor = -2$, and $\lfloor -7.89 \rfloor = -8$. If the number is an exact integer, such as $-5$, then $\lfloor -5 \rfloor = -5$.

How do I write the floor function in LaTeX?

In LaTeX, the floor brackets are generated with \lfloor and \rfloor. For scalable height, use \left\lfloor x \right\rfloor.

What is the difference between floor and truncation?

Truncation discards the fractional part and moves towards zero ($\text{trunc}(-2.7) = -2$), whereas floor always rounds towards negative infinity ($\lfloor -2.7 \rfloor = -3$). For non-negative numbers, floor and truncation produce identical outputs.

Can I evaluate mathematical constants like pi and e?

Yes, enter constants such as pi ($\pi \approx 3.14159 \implies \lfloor \pi \rfloor = 3$) or e ($e \approx 2.71828 \implies \lfloor e \rfloor = 2$) or radical expressions like sqrt(2) directly into the calculator.

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