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}

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Floor, Ceiling and Sawtooth Function

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