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Can someone give an explanation/proof of whether these two numbers lie on the real number line?

mathlove
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Bosnia
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    They are elements of $\overline{ \Bbb R}$. –  Aug 30 '14 at 09:37
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    $\infty$ and $-\infty$ are not considered as numbers. – Taladris Aug 30 '14 at 09:40
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    no, they are not real numbers. – Ittay Weiss Aug 30 '14 at 09:42
  • The $\pm\infty$ are not on the line. But this follows from the definition of the real numbers. So either you accept this, or plow through the definition of real numbers. Starting from Peano axioms (one of the most satisfactory routes) that is 20+ pages, so is not going to appear here. For many purposes it is useful to extend the line to contain these (for example this is a useful way of compactifying the real line). – Jyrki Lahtonen Aug 30 '14 at 10:06

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They are not real numbers as such. They are part of the extended real number line $\bar{\Bbb R}=\Bbb R \cup\{\infty,-\infty\}$. These elements do not have the usual arithmetic with other real numbers and abide strict rules however they are useful in fields such as analysis. They provide an unlimited behavior to certain concepts such as limits and integrals.

Let me add to why they are not in the real number line. The rationals are dense in $\Bbb R$ so every real number can be represented as the limit of a sequence of rational numbers. This can be done from both sides of any real number which poses obvious problems if 2 elements are the largest and the smallest etc.

Ali Caglayan
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    More accurately, they quantify unlimited behavior in the same sense that 0 quantifies the behavior of "decreasing in magnitude to eventually be smaller than any positive number". –  Aug 30 '14 at 09:57