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 are widely applied in number theory—the floor function, ceiling function, sawtooth function.

For a real number $x$,

• the floor function returns the largest integer less than or equal to $x$, denoted as $\lfloor x \rfloor ;$
• the ceiling function returns the smallest integer larger than or equal to $x$, denoted as $\lceil x \rceil ;$
• the sawtooth function returns the fractional part of $x$, denoted as $\{x\}$.

Note: The variable $n$ in this section is assumed to be an integer.

The floor function has the following properties:

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. $\lfloor xy\rfloor \geq \lfloor x \rfloor \lfloor y\rfloor \quad \mathrm{for} \quad x, y \ge 0$
7. $\displaystyle \lfloor \sqrt[n]{x} \rfloor ^n \leq \lfloor x \rfloor$
8. $\displaystyle \bigg \lfloor \frac{nx}{y} \bigg \rfloor = n\bigg \lfloor \frac{x}{y} \bigg \rfloor$

Solve the following equation for a non-zero solution:

$$x+2\{x\}=3\lfloor x \rfloor .$$

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We have

$$\begin{aligned} x+2\{x\}&=3\lfloor x \rfloor \\ \lfloor x \rfloor + \{x\} + 2\{x\} &= 3\lfloor x \rfloor \\ 3\{x\} &= 2\lfloor x \rfloor \\ \{x\} &= \frac{2}{3}\lfloor x \rfloor. \end{aligned}$$

Since $0 \leq \{x\} < 1$, we have $0 \leq \lfloor x \rfloor < 1\frac{1}{2}$. Substituting $\lfloor x \rfloor = 1$, we have $\{x\} = \frac{2}{3}$. Then we finally have

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