Definition:Fraction/Improper
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Definition
An improper fraction is a fraction representing a rational number whose absolute value is greater than $1$.
Specifically, when expressed in the form $r = \dfrac p q$, where $p$ and $q$ are integers such that (the absolute value of) the numerator is greater than (the absolute value of) the denominator: $\size p > \size q$.
Examples
Example: $\frac 4 3$
- $\dfrac 4 3$ is an improper fraction.
Example: $\frac 8 7$
- $\dfrac 8 7$ is an improper fraction.
Example: $\frac {-5} 4$
- $\dfrac {-5} 4$ is an improper fraction.
Example: $\frac {16} {10}$
- $\dfrac {16} {10}$ is an improper fraction, although not in canonical form.
Example: $1 \frac 1 2$
- $1 \frac 1 2$ is not an improper fraction: it is in fact a mixed fraction.
Also known as
An improper fraction is also known informally as a top-heavy fraction.
Also see
- Definition:Vulgar Fraction
- Definition:Proper Fraction
- Definition:Mixed Fraction
- Definition:Complex Fraction
- Results about improper fractions can be found here.
Linguistic Note
The word improper has evolved to mean rude or offensive, or even invasive (of privacy, personal space or sexual integrity).
Stand-up comedians may wish to take the opportunity to riff on why language hates fractions so much, that they are either improper or merely vulgar.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): fraction
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): improper fraction
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): proper fraction
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): fraction
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): improper fraction
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): proper fraction
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): fraction