FractionalPart[x] gives the fractional part of x.

Functions that are specified by simple algebraic formulas tend to be such that when their input is changed only slightly, their output also changes only slightly. But ...

Functions relating real numbers and integers. Extracting integer and fractional parts. IntegerPart[x] and FractionalPart[x] can be thought of as extracting digits to the left ...

Throughout Mathematica there is support not only for approximate real numbers, but also for exact numbers represented in algebraic or symbolic form. Functions like Floor, ...

Mathematica handles both integers and real numbers with any number of digits, automatically tagging numerical precision when appropriate. Mathematica internally uses several ...

SawtoothWave[x] gives a sawtooth wave that varies from 0 to 1 with unit period.SawtoothWave[{min, max}, x] gives a sawtooth wave that varies from min to max with unit period.

Nested logical and piecewise functions can be expanded out much like nested arithmetic functions. You can do this using LogicalExpand and PiecewiseExpand. Expanding out ...

Ceiling[x] gives the smallest integer greater than or equal to x. Ceiling[x, a] gives the smallest multiple of a greater than or equal to x.

IntegerPart[x] gives the integer part of x.

Mod[m, n] gives the remainder on division of m by n. Mod[m, n, d] uses an offset d.