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I have read that infinity is not an element of R, the set of all real numbers and infinity is not a number. So can we say that infinity and minus infinity does not belongs to R ? Or can we say that plus infinity and minus infinity belongs to the set of all real numbers ? Please help .

Ami
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3 Answers3

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$\infty$ or $-\infty$ are not elements of $\mathbb R$. However, we have the extended real number system $\mathbb R\cup\{-\infty,\infty\}$ (see here for more details) which contains $\infty$ and $-\infty$ as its elements.

Cm7F7Bb
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Infinity isn’t a member of the set of real numbers. One of the axioms of the real number set is that it is closed under addition and multiplication. That is if you add two real numbers together you will always get a real number. However there is no good definition for $\infty + (-\infty)$ And $\infty \times 0$ which breaks the closure rule.

However there extended real number set that includes positive and negative infinity with the trade off that some operatorions have indeterminate values.

Q the Platypus
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  • I understand what you meant, but "it has indeterminate values" is a little confusing to read. The extended reals just have regular real values, and $\infty$ and $-\infty$, and no other values in it. Instead, there are just more arithmetic operations than division by $0$ that are undefined, and they're undefined because they correspond to indeterminate expressions representing limits that are not determined by the limits of the parts. – Mark S. Jun 08 '18 at 10:47
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I'm reminded of this video when I see this sort of question. Essentially, $\infty$ is just a concept. It doesn't behave in the same way that the common real number does. To see this, consider the two equations: $$x=x+1$$ $$x=2x$$ Notice that apart from the second being satisfied by $0$, there exists no number which satisfies either of these. However, due to the arbitrary nature of the concept $\infty$, it solves both, as explained in the video.

Rhys Hughes
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