If we subtract from a real number $x$ the greatest integer not exceeding $x$ , we obtain a number $y$ between 0 and 1, which can equal 0 if $x$ is an integer. In other words, $$y \;=\; x\!-\!\lfloor{x}\rfloor,$$ where $\lfloor{x}\rfloor$ is the floor of $x$ . Such a number $y$ is called the mantissa of $x$ . So we have for example

$2.7-2 \;=\; 0.7$ ,
$1.7-1 \;=\; 0.7$ ,
$0.7-0 \;=\; 0.7$ ,
$-0.3\!-\!(-1) = 0.7$ ,
$-1.3\!-\!(-2) = 0.7,$

i.e. these numbers 2.7, 1.7, 0.7, $-0.3$ , $-1.3$ at mutual distances an integer have the same mantissa (0.7). This is apparently always true -- thus the mantissa function $$x \mapsto x\!-\!\lfloor{x}\rfloor$$ is periodic: its least period is 1.

The mantissa is identic with the mantissa used in the Briggsian logarithm calculations.

When $x$ increases from an integer $n$ towards the next integer $n\!+\!1$ , its mantissa $x\!-\!\lfloor{x}\rfloor$ increases with the same speed from 0 tending to 1, but at $n\!+\!1$ it falls back to 0.

Being a periodic function, the Fourier expansion of the function is easy to form: $$x\!-\!\lfloor{x}\rfloor \;=\; \frac{1}{2}-\sum_{n=1}^\infty\frac{\sin 2n\pi{x}}{n\pi}$$ This is valid for $x \not\in \mathbb{Z}$ , since the series gives in the jump discontinuity points the arithmetic means ($= \frac{1}{2}$ ) of left and right limits.