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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: $\size p > \size q$.


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

  • 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.