# Integer Function - Floor, Ceiling, Sawtooth

An integer function maps a real number to an integer value. In this wiki, we're going to discuss three integer functions that is widely applied in Number Theory — Floor Function, Ceiling Function, Sawtooth Function.

#### Contents

## Summary

For a real number \(x\),

Floor Function returns the largest integer less than or equal to \(x\), denote as \(\lfloor x \rfloor\).

Ceiling Function returns the smallest integer larger than or equal to \(x\), denote as \(\lceil x \rceil\).

Sawtooth Function returns the fractional part of \(x\), denote as \(\{x\}\).

## Properties

Note: The variable \(n\) in this section is assumed to be an integer.

\(1\). \(\lfloor x \rfloor + \{x\} = x\)

\(2\). \(\lfloor x+y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor\)

\(3\). \(\{x\} + \{y\} \geq \{x+y\}\)

\(4\). \(\lfloor x + n \rfloor = \lfloor x \rfloor + n\)

\(5\). \(\{x+n\}=\{x\}\)

\(6\). \(\lfloor xy\rfloor \geq \lfloor x \rfloor \lfloor y\rfloor\)

\(7\). \(\displaystyle \lfloor \sqrt[n]{x} \rfloor ^n \leq \lfloor x \rfloor\)

\(8\). \(\displaystyle \bigg \lfloor \frac{nx}{y} \bigg \rfloor = n\bigg \lfloor \frac{x}{y} \bigg \rfloor\)

## Problem Solving

## Solve the equation for non-zero solution.

\[x+2\{x\}=3\lfloor x \rfloor\]

\[\begin{aligned} x+2\{x\}&=3\lfloor x \rfloor \\ \lfloor x \rfloor + \{x\} + 2\{x\} &= 3\lfloor x \rfloor \\ 3\{x\} &= 2\lfloor x \rfloor \\ \{x\} &= \frac{2}{3}\lfloor x \rfloor \end{aligned}\]

Since \(0 \leq \{x\} < 1 \), then \(0 \leq \lfloor x \rfloor < 1\frac{1}{2}\). Substitute \(\lfloor x \rfloor = 1\), we have \(\{x\} = \frac{2}{3}\). Finally,

\[x = \lfloor x \rfloor + \{x\} = 1 + \frac{2}{3} = 1\frac{2}{3}\]

**Cite as:**Integer Function - Floor, Ceiling, Sawtooth.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/integer-function-floor-ceiling-sawtooth/