Extended Real Number¶

Definition¶

We can extend the real number system to include $+\infty$ and $-\infty$, resulting in the extended real number system denoted as $\overline{\mathbb{R}}\text{:}$ it consists of all real numbers together with the two symbols $-\infty$ and $+\infty\text{,}$ ordered so that $-\infty<x<+\infty$ for every real $x\text{.}$

Interpretation¶

The extended real number system augments the real numbers by symbols for positive and negative infinity, placed as a least and a greatest element. This extension allows limits, divergence and unbounded behaviour to be described within a single framework.

The figure draws $\overline{\mathbb{R}}$ as the real line with two endpoints attached. Every real $x$ lies strictly between them, $-\infty<x<+\infty$$+\infty$, and the order of $\mathbb{R}$ is kept. With these endpoints every nonempty set of real numbers has a supremum and an infimum in $\overline{\mathbb{R}}$: a set without an upper bound, such as $\{1,2,3,\ldots\}$ in the Examples, gets the supremum $+\infty$.

The gain lies in the order, not in arithmetic. A common slip is to treat $+\infty$ as a very large real number and compute with it. Some operations have a natural value, such as $x+(+\infty)=+\infty$ for real $x$, but $(+\infty)-(+\infty)$ has none: $a_n=n+1$ and $b_n=n$ both tend to $+\infty$ with $a_n-b_n=1$ for every $n$, while $a_n=2n$ and $b_n=n$ give $a_n-b_n=n$$n$. Likewise, a sequence with limit $+\infty$ is still called divergent, because convergence asks for a real limit $L$.

Key Results¶

• Infinities are symbols, rather than big numbers.
• Many limit statements naturally take values in the extended real numbers.

Examples¶

The sequence $a_n=n$ has extended-real limit $+\infty$, and the set $\{1,2,3,\ldots\}$ has supremum $+\infty$ in $\overline{\mathbb{R}}$. Neither statement makes infinity an ordinary real number: expressions such as $+\infty-(+\infty)$ are not defined by real arithmetic.

Prerequisites¶

• Real Number

Related Concepts¶

• Divergence to Infinity
• Limit at Infinity
• Infinity

Sources¶

• Math I: Calculus · Lecture 3

Aliases: extended real line Symbols: infinity, real numbers

Last updated: 20 October 2026 03:02