y = x - \lfloor x \rfloor

Graph of the Fractional Part $y = x - \lfloor x \rfloor$ (Sawtooth Wave)

$y = x - \lfloor x \rfloor$ extracts the fractional part of $x$ , and is often written $\{x\}$ ¹. Subtracting the floor function $\lfloor x \rfloor$ , the largest integer not exceeding $x$ , removes the integer part and leaves the fraction behind.

$x$	$\lfloor x \rfloor$	$x - \lfloor x \rfloor$
$2.3$	$2$	$0.3$
$5$	$5$	$0$
$-1.2$	$-2$	$0.8$

For a negative number the floor function returns the integer below, so the result is always non-negative. That the fractional part of $-1.2$ is $0.8$ sits a little oddly with intuition, so it is worth noting.

Domain and range

The domain is all real numbers and the range is $0 \leq y < 1$ . The value never reaches $1$ .

Periodicity and shape

The function has period $1$ , with $\{x + 1\} = \{x\}$ . Over each interval between consecutive integers it is a straight line of slope $1$ .

y = x - n \quad (n \leq x < n+1)

It starts at $0$ when $x = n$ , climbs straight up, and approaches $1$ just before the next integer. That shape repeating is why the graph is called a sawtooth wave.

Discontinuities and jumps

The function is discontinuous at every integer. Approaching an integer $n$ from the left the value tends to $1$ , but at $x = n$ it is reset to $0$ . The jump is therefore $-1$ : the curve falls back to the floor just before it would reach the ceiling. Each step includes its left end, where the value is $0$ , and excludes its right end.

That the floor function is right-continuous carries over directly to this one.

Relations to other functions

The fractional part is built from the floor function and agrees with $x \bmod 1$
The value is $0$ exactly when $x$ is an integer
Writing $\lfloor x \rfloor = x - \{x\}$ recovers the floor function from the fractional part

Fourier series

At non-integer points the function is given by the following series.

x - \lfloor x \rfloor = \frac{1}{2} - \frac{1}{\pi}\sum_{n=1}^{\infty}\frac{\sin(2\pi n x)}{n}

The constant term $\dfrac{1}{2}$ is the average height over one period. Unlike a square wave this one contains even harmonics as well, and the coefficients decay as $\dfrac{1}{n}$ . Since the waveform has jumps the convergence is slow, and partial sums show the Gibbs phenomenon.

Applications

The oscillator waveform of a synthesizer
The phase accumulator that advances a phase repeatedly from $0$ to $1$
Computations with remainders in general

Containing every harmonic gives it a bright timbre, which makes it a favorite starting point for subtractive synthesis, where a filter carves the sound out.

Sawtooth wave, Wikipedia

Graph of the Fractional Part y=x−⌊x⌋y = x - \lfloor x \rfloory=x−⌊x⌋ (Sawtooth Wave)