# Definition:Fraction/Improper

## 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

• 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.