The fractional part function

Every real num­ber can be split into an inte­ger part and a left­over part between $0$ and $1$. The great­est inte­ger func­tion gives the inte­ger part; this les­son is about the left­over, called the frac­tional part. Its graph has a strik­ing saw­tooth shape, and it is the first exam­ple you meet of a func­tion that repeats itself reg­u­larly, an idea that becomes cen­tral when you study trigono­met­ric func­tions.

Inte­gral part and frac­tional part#

For a pos­i­tive num­ber the split is obvi­ous:

$$1.7 = 1 + 0.7$$

Here $1$ is the inte­gral part and $0.7$ is the frac­tional part.

For a neg­a­tive num­ber we insist that the frac­tional part is not neg­a­tive: it must lie in $[0, 1)$. So $-1.7$ is not split as $-1 - 0.7$. Instead, bor­row one whole unit:

$$-1.7 = -1 - 0.7 = -1 - 1 + 1 - 0.7 = -2 + 0.3$$

The inte­gral part is $-2$ and the frac­tional part is $0.3$. Notice that the inte­gral part $-2$ is exactly $[-1.7]$.

An inte­ger has frac­tional part $0$: $2 = 2 + 0$.

The frac­tional part func­tion#

Def­i­n­i­tion. For $x \in \mathbb{R}$, the func­tion

$$f(x) = x - [x]$$

is called the frac­tional part of $x$. It is writ­ten $\{x\}$ (some books write $]x[$). So

$$\{x\} = x - [x], \qquad x = [x] + \{x\}$$

Exam­ple 1#

Find $\{1.7\}$, $\{-1.7\}$, $\{2.3\}$, $\{-2.3\}$ and $\{2\}$.

$$\{1.7\} = 1.7 - [1.7] = 1.7 - 1 = 0.7$$

$$\{-1.7\} = -1.7 - [-1.7] = -1.7 - (-2) = -1.7 + 2 = 0.3$$

$$\{2.3\} = 2.3 - 2 = 0.3$$

$$\{-2.3\} = -2.3 - (-3) = 0.7, \quad \text{since } -2.3 = -3 + 0.7$$

$$\{2\} = 2 - 2 = 0$$

A use­ful short­cut: for a neg­a­tive non-inte­ger, $\{-x\} = 1 - \{x\}$. For exam­ple, $\{-2.3\} = 1 - 0.3 = 0.7$.

Domain, range and graph#

Every real num­ber has a frac­tional part, and by its def­i­n­i­tion the frac­tional part always lies in $[0, 1)$:

$$\text{Domain of } x - [x] = \mathbb{R}, \qquad \text{Range of } x - [x] = [0, 1)$$

To draw the graph, make a table.

Between $0$ and $1$ the graph is the line $y = x$, climb­ing from $(0, 0)$ towards $(1, 1)$. Just before $x = 1$ the value is almost $1$, but at $x = 1$ it drops back to $0$ and starts climb­ing again. So the graph is a row of par­al­lel line seg­ments, each ris­ing from a point $(n, 0)$ on the $x$-axis towards height $1$, with a filled dot at the bot­tom and an open cir­cle at the top. This shape is called a saw­tooth.

The pat­tern repeats every $1$ unit: $\{x + 1\} = \{x\}$ for every $x$. We say the frac­tional part func­tion has period $1$.

Exam­ple 2#

Find the range of $[x] - x$.

Start from the known range and mul­ti­ply by $-1$, revers­ing the inequal­i­ties:

$$0 \leq x - [x] < 1$$

$$0 \geq [x] - x > -1$$

$$-1 < [x] - x \leq 0$$

So the range of $[x] - x$ is $(-1, 0]$.

Exam­ple 3#

Find $\{5.25\}$ and $\{-5.25\}$.

$\{5.25\} = 5.25 - 5 = 0.25$.

$-5.25$ lies between $-6$ and $-5$, so $[-5.25] = -6$ and $\{-5.25\} = -5.25 + 6 = 0.75$.

Exam­ple 4#

Solve $2[x] = x + \{x\}$.

Replace $x$ by $[x] + \{x\}$:

$$2[x] = [x] + \{x\} + \{x\} \implies [x] = 2\{x\}$$

Since $0 \leq \{x\} < 1$, we have $0 \leq 2\{x\} < 2$, so the inte­ger $[x]$ can only be $0$ or $1$.

• If $[x] = 0$, then $\{x\} = 0$, so $x = 0 + 0 = 0$.
• If $[x] = 1$, then $\displaystyle \{x\} = \frac{1}{2}$, so $\displaystyle x = 1 + \frac{1}{2} = \frac{3}{2}$.

So $\displaystyle x \in \left\{0, \frac{3}{2}\right\}$. Check $\displaystyle x = \frac{3}{2}$: $2[1.5] = 2$ and $1.5 + 0.5 = 2$.

Exam­ple 5#

For which $x$ is $\{x\} = 0$?

$\{x\} = 0$ means $x - [x] = 0$, that is, $[x] = x$, which hap­pens exactly when $x$ is an inte­ger. So $x \in \mathbb{Z}$.

Prac­tice#

1. Find $\{4.6\}$.
2. Find $\{-4.6\}$.
3. Find $\{7\}$.
4. Find $\{-0.25\}$.
5. Find $\{x\} + \{-x\}$ when $x = 2.3$, and when $x = 3$.
6. Find the range of $2 - \{x\}$.
7. Find the range of $\{x\} + 3$.
8. Find all $x$ with $2 \leq x < 4$ and $\{x\} = 0.4$.
9. Solve $[x] = 3\{x\}$.
10. For $x = -6.3$, find $[x]$, $\{x\}$ and $[x] + \{x\}$.

Answers#