Let $x \in \R$ be a real number.
Let $\floor x$ be the floor function of $x$.
The fractional part of $x$ is the difference:
- $\fractpart x := x - \floor x$
Beware, of course, not to get $\fractpart x$ confused with the singleton set containing $x$.
Also known as
Some sources, particularly those aimed for the grade-school and muggle market, refer to this as the decimal part, or (even worse) just the decimal.
This misnomer arises from the fact that it is the part of the number after the decimal point.
Burn it with fire.
- Definition:Floor Function
- Definition:Ceiling Function
- Definition:Nearest Integer Function
- Definition:Distance to Nearest Integer Function
- Real Number minus Floor: we have that $0 \le \fractpart x < 1$, or $\fractpart x \in \hointr 0 1$.
- $x \bmod 1 = \fractpart x$
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 1.4$: Decomposition of a set into classes. Equivalence relations: Example $3$ (footnote $4$)
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory