# Floor, Ceiling and Sawtooth Function

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

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,

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