Extended real number line

In mathematics, the extended real number system^[a] is obtained from the real number system ${\displaystyle \mathbb {R} }$ by adding two infinity elements: ${\displaystyle +\infty }$ and ${\displaystyle -\infty ,}$^[b] where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration.^[1] The extended real number system is denoted ${\displaystyle {\overline {\mathbb {R} }}}$ or ${\displaystyle [-\infty ,+\infty ]}$ or ${\displaystyle \mathbb {R} \cup \left\{-\infty ,+\infty \right\}.}$^[2] It is the Dedekind–MacNeille completion of the real numbers.

When the meaning is clear from context, the symbol ${\displaystyle +\infty }$ is often written simply as ${\displaystyle \infty .}$^[2]

There is also the projectively extended real line where ${\displaystyle +\infty }$ and ${\displaystyle -\infty }$ are not distinguished so the infinity is denoted by only ${\displaystyle \infty }$.

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