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

$y = x - \lfloor x \rfloor$ extracts the fractional part of $x$, often written $\{x\}$. Subtracting the floor $\lfloor x \rfloor$ (the greatest integer not exceeding $x$) removes the integer part and leaves only the fraction.

• $2.3 - \lfloor 2.3 \rfloor = 2.3 - 2 = 0.3$
• $5 - \lfloor 5 \rfloor = 5 - 5 = 0$
• $-1.2 - \lfloor -1.2 \rfloor = -1.2 - (-2) = 0.8$

For negative numbers the floor returns the integer below, so the result is always non-negative. Note that the fractional part of $-1.2$ is $0.8$, which is slightly counter-intuitive.

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

Periodicity and shape. The function has period $1$, with $\{x + 1\} = \{x\}$. On each integer interval $[n,\ n+1)$ it is the line $y = x - n$ of slope $1$: it starts at $0$ when $x = n$, rises straight up, and approaches $1$ just before the next integer. Because this shape repeats, 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 resets to $0$. Each jump therefore has size $-1$: the value climbs almost to the ceiling, then drops to the floor, over and over. Each step includes its left end (the integer point, value $0$) but not its right end, so it is right-continuous.

Relation to other functions. The fractional part is built from the floor function, and $x - \lfloor x \rfloor$ equals the remainder $x \bmod 1$. Only at integers do $x$ and its floor coincide, making the value $0$.

Applications. The sawtooth wave is widely used in signal processing: as the oscillator of an audio synthesizer, and as a phase accumulator that repeatedly advances a phase from $0$ to $1$ (or $0$ to $2\pi$). It is also a classic example for Fourier series.