Rational number
Elementary
A rational number is a number that can be written as a fraction of whole numbers, or the negative of one.
High school
A rational number is a number that can be written as $p/q$, where $p$ and $q$ are integers and $q \neq 0$.
Undergraduate
A rational number is an element of $\mathbb{Q}$, constructed as equivalence classes of pairs $(p, q)$ of integers such that $q \neq 0$, where $(p, q) \sim (r, s)$ if $p s = q r$.
Examples
Examples
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$\frac{1}{2}$ and $\frac{2}{4}$
Elementary
$\frac{1}{2}$ and $\frac{2}{4}$ represent the same rational number.
Non-examples
Things that may be mistaken for examples.
-
$\pi$
Elementary
$\pi$ is not a rational number, because it is not a ratio of two integers.
- $\sqrt{2}$ High school
Properties
- A real number is rational if and only if its decimal expansion either ends or repeats.
- Between every two real numbers there is a rational number: $\mathbb{Q}$ is dense in $\reals$.