Real number
High school
A real number is any number on the number line. Every real number can be written as a (possibly infinite) decimal.
Undergraduate
A real number is an element of $\reals$, the unique complete ordered field.
Examples
Examples
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$\pi$
Elementary
$\pi$ is a real number, because it is the ratio of a circle’s circumference to its diameter, a point on the number line.
- $e$ High school
- $\sqrt{2}$ High school
Properties
- Every nonnegative real number $x$ can be written as a decimal $x = n.d_1 d_2 d_3 \ldots$, where $n$ is a natural number and each digit $d_k$ is one of $0, 1, \dots, 9$, which means that \[x = n + \sum_{k = 1}^{\infty} \frac{d_k}{10^k}.\] The digits are found one at a time: $n$ is the largest whole number at most $x$, and each digit $d_k$ is the largest digit such that the decimal $n.d_1 \ldots d_k$ is at most $x$. The decimals $n.d_1 \ldots d_k$ then lie within $10^{-k}$ of $x$, so the decimals converge to $x$. A negative real number is written as the negative of a decimal.
- A real number is rational exactly when its decimal expansion either ends or repeats, such as $\frac{1}{4} = 0.25$ and $\frac{1}{3} = 0.333\ldots$, so the decimal expansion of an irrational number, such as $\sqrt{2} = 1.41421\ldots$, neither ends nor repeats.
- Some real numbers have two decimal expansions: a decimal that ends, such as $1.0$, equals the decimal that instead ends with repeating $9$s, such as $0.999\ldots$, because $\sum_{k = 1}^{\infty} \frac{9}{10^k} = 1$.
- $\reals$ can be constructed from $\mathbb{Q}$ using Dedekind cuts or Cauchy sequences.
- Every nonempty set of real numbers that is bounded above has a least upper bound (the completeness of $\reals$).