# What is the definition of $\infty$

I saw in a note that say $\infty$ is not a real number, and there is no interval of the form $(a, \infty]$? So what is the definintion of $\infty$?

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There is no standalone definition of $\infty$. But there are definitions of various expressions that involve the symbol $\infty$. In particular, there is a definition of $(a,\infty)$. Note that there is nothing peculiar about this. There is no standalone definition of $a$ either, or of $($. –  André Nicolas Oct 15 '11 at 15:41
@AndréNicolas Yeah, you're right, I would mention that $a$ maybe a real number. –  Fan Zhang Oct 15 '11 at 15:53

The symbol "$\infty$" is part of the ordinary mathematical alphabet. Like many other symbols in the mathematical alphabet, it has various uses. There is no standalone definition of $\infty$. There are, however, precise definitions of various expressions that use the letter $\infty$.

In particular, in analysis, there is a definition of what is meant by the interval $(a,\infty)$. Note that there is nothing peculiar about the fact that there is no standalone definition of $\infty$, while some expressions that use $\infty$ have a clear conventional meaning. There is no standalone definition of $a$ either, or of $($.

Comment: Perhaps the following analogy will be useful. Suppose that we have written a program that finds, for any $n$, the $n$-th decimal digit of $\pi$. Then the command "Print $[17, 42]$" might mean print all the digits from the $17$-th to the $42$-th, inclusive. The command "Print $[17, 42)$" could be used to mean print from the $17$-th (inclusive) to the $42$-th (exclusive). And if we wanted to print all the digits from the $17$-th on, we might issue the command "Print $[17,\infty)$". But the command "Print $[17,\infty]$ would make no sense, no reasonable meaning can be assigned to the phrase "the $\infty$-th digit of $\pi$."

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You might want to add the two lines giving the definition for $(a,\infty)$ in the real analysis context. –  Asaf Karagila Oct 15 '11 at 16:41
Since we're suggesting edits, how about a quick comment about how $(a, \infty)$ can be defined in the real analysis context? –  The Chaz 2.0 Oct 15 '11 at 17:48
Sorry, I meant " ] " above! –  The Chaz 2.0 Oct 15 '11 at 18:11
You could also use $\infty$ as the point that compactifies $\mathbb{C}.$ Thus, it is convenient to use the point at infinity when constructing the Riemann sphere. So, in topology, it is used to compactify certain topological spaces.
There are a variety of different concepts of infinity in mathematics. When used in interval notation of this kind, something like $(5,\infty)$ means the set of all real numbers greater than $5$. Similarly $[5,\infty)$ means the set of all real numbers greater than or equal to $5$.