Definition:Fractional Part

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Definition

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.

Also see

• 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 $\fractpart x \in \hointr 0 1$.

Generalizations

• Definition:Modulo Operation
• Compare with the definition of Definition:Modulo 1:

$x \bmod 1 = \fractpart x$

• Results about fractional parts can be found here.

Sources

• 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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): integer part
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): integer part
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): fractional part