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Educators and Professors: when you teach first year calculus students that infinity is not a number, how would you logically present to them $-\infty < x < +\infty$, where $x$ is a real number?

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In the sense of extended real number, this 'inequality' represents $x$ is real number. –  tetori Jul 18 '13 at 9:12
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I would present it as a notational convention, which we interpret to mean that $x$ is unbounded. –  Gerry Myerson Jul 18 '13 at 9:15
    
Infinity is not a number –  zerosofthezeta Aug 15 '13 at 18:50

5 Answers 5

up vote 1 down vote accepted

I'd make a couple points:

The word "number," on its own, doesn't really mean anything, aside from "something that somehow resembles something else we're already calling 'numbers.'" Lots of people have called lots of different things "numbers." As a few examples, the modern notion of an ideal in a ring gets its name from what Kummer called "ideal numbers;" we call the two other real algebras commutative algebras the "dual numbers" and the "hyperbolic numbers," by analogy with the "complex numbers;" and of course there are things like the extended real numbers or the surreal numbers which are things we call numbers which do include infinite elements.

Of course I probably wouldn't say it that way to students, but the point is that "infinity isn't a number" is either a false or meaningless statement. The points you want to get across are this:

  • The word "number," on its own, is just sort of a vague idea, not one specific concept.
  • The phrase "real number," on the other hand, indicates a very specific thing.
  • In calculus, we're working over the real numbers.
  • There is no real number which is infinity.
  • Sometimes, however, we use it as a notational convention.
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The symbols $+\infty,-\infty$ (and $\infty$) simply denote a formal symbol which means "larger/small than any real number".

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No real number can be larger than any real number. So $\infty$ is not any number? –  skullpatrol Aug 8 '13 at 13:05
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That is correct. –  Asaf Karagila Aug 8 '13 at 13:08
    
$\infty$ is not any number and so is $-\infty$. not any number = not any number, therefore $\infty=-\infty$ ? –  skullpatrol Aug 8 '13 at 13:21
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$3\neq 2, 4\neq 2\implies 3=4!$ –  Asaf Karagila Aug 8 '13 at 13:25
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I'm a student, and I best understood it by seeing ∞ as a symbol that tells you that whatever you think of, there is something larger (smaller if -∞), always! –  Fiire Aug 15 '13 at 19:13

To make sense out of $-\infty < 1000 < \infty$, for example, you do not need $\infty$ and $-\infty$ to be "numbers," it is enough to extend the definition of "$<$" so that the truth value of $x < y$ was defined for all $x,y\in\mathbb R\sqcup\{\pm\infty\}$. This is easy to do, in an obvious way.

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To accept $\infty$ and $-\infty$ as numbers, they would have to satisfy the usual rules of arithmetic of numbers. In particular, for all numbers $a$, $a-a=0$, so if infinity $\infty$ is a number we shoild have $\infty - \infty=0$. But that is not necessarily true, as we can let $\infty$ be represented by any sequence growing beyoind all limits, such as $\lim_{n\rightarrow \infty} 2n$ or $\lim_{n \rightarrow \infty} n$. But the difference between these two representatives of $\infty$ is itself $\infty$, indicating, with this representatives, that $\infty - \infty=\infty$. With other representatives you can get any result you want! So the arithmetic of real numbers cannot be extended to the symbols $\infty$ and $-\infty$ in any consistent way.

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Once upon a time, the "usual rules of arithmetic" included things like "every number is greater than or equal to zero", or even "every number is a ratio of integers". –  Hurkyl Jan 1 at 1:41

Just think of $\infty$ and $-\infty$ as beyond our number system; beyond the scope of our knowledge of numbers. Remember that $\infty$ and $-\infty$ are simply IDEAS, not actual numbers. If there is a number line that goes on forever and ever (technically ALL number lines go on forever actually), then $\infty$ would be on the right "end" and $-\infty$ would be on the left. Because $\infty$ and $-\infty$ are the ends of a never ending number line, ALL the real numbers are in the region $[-\infty, \infty]$. So you can logically say that $-\infty < x < \infty$.

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