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When learning mathematics we are told that infinity is undefined. (*)

Recently I read about the infinitesimal version of Calculus and how we can in fact treat $dy/dx$ as a fraction under this approach (something we can't do with limits). This is achieved by constructing the Hyperreal numbers $^*\mathbb{R}$, which contain the real numbers and the infinitesimal numbers (all positive numbers greater than zero, yet less than any real number).

Thinking back to (*) I wonder, could sense of infinity be made somehow by including the reciprocal infinitesimal numbers? Presumably these numbers would be defined since the infinitesimal numbers are never zero.

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In fact, infinite numbers are a part of the Hyperreal Numbers. Just check this out on Wikipedia.

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    $\begingroup$ and it should be stressed that $\infty $ is still an entity that does not exist. For every non-zero infinitesimal in the hyperreal system, its reciprocal is an infinitely large number. Conversely, for every infinitely large number, its reciprocal is a non-zero infinitesimal. This correspondence is bijective. $\endgroup$ – Ittay Weiss Dec 20 '14 at 8:28
  • $\begingroup$ I find the term "transfinite number" better. $\endgroup$ – Asaf Karagila Dec 21 '14 at 22:12
  • $\begingroup$ The correct technical term is "unlimited", see Goldblatt page 27. $\endgroup$ – Mikhail Katz Dec 22 '14 at 11:20
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The symbol $\infty$ was introduced by John Wallis in the 17th century as a sign for an infinite number that can be used in calculations such as computation of areas, and he also used infinitesimals of the form $\frac{1}{\infty}$ for this purpose. The procedures relied upon by Wallis are similar to those found in modern books on the hyperreals. It may therefore be tempting to use the symbol $\infty$ as Wallis did to denote an infinite number, but this may be confusing because the symbol is often used as a part of notation such as sums of series $\sum_{i=0}^\infty u_i$ where it has no meaning other than as a reminder that a limit was taken in evaluating the series. In this sense the claim that "$\infty$ is an entity that does not exist" is a matter of convention.

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No. The symbol $\infty$ as used in analysis usually means either positive infinite element of affinely extended real line (in this case you also would have $-\infty$ as another element) or (more rarely) the infinite element of the projectively extended real line.

In the later case it is the multiplicative inverse of zero ($\frac 10=\infty$).

So in fact you can use the symbol as part of a numerical system, but such system is called differently (affine real line or projective real line).

If you consider the hyperreal numbers, no hyperreal number has the properties usually ascribed to symbol $\infty$. For instance, for any hyperreal number $a$, $a+1\ne a$, while $\infty$ is usually defined as having the property of invariancy against addition: $\infty+1=\infty$.

That said, you of course can extend the hyperreal numbers in affine or projective manner just as we do with real numbers by adding $\infty$ as an element but this is not different from how reals can be extended.

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