Extended real number line

Real numbers with + and - infinity added

In mathematics, the extended real number system is obtained from the real number system R {\displaystyle \mathbb {R} } by adding two elements denoted + ∞ {\displaystyle +\infty } and − ∞ {\displaystyle -\infty } that are respectively greater and lower than every real number. This allows for treating the potential infinities of infinitely increasing sequences and infinitely decreasing series as actual infinities. For example, the infinite sequence ( 1 , 2 , …) {\displaystyle (1,2,\ldots)} of the natural numbers increases infinitively and has no upper bound in the real number system (a potential infinity); in the extended real number line, the sequence has + ∞ {\displaystyle +\infty } as its least upper bound and as its limit (an actual infinity).

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