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I am no great mathematician but I have a question which I can't seem to find a answer for. How can one divide a finite by a infinite number? For example if you have a circle with a circumference of exactly 40 to get the diameter of that circle you divide 40 by π. However if pi is infinite how are we able to obtain a value for the diameter?

Aloizio Macedo
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Dan
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    The value is just $\frac{40}{\pi},$ and that is the exact value. – 3x89g2 May 14 '15 at 18:13
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    $;\pi;$ isn't, and can't be, infinite. It is a very definite, finite number, just like $;1,2,0.56;$ or $;1000;$ . Now, do you have any idea how would you want to define "an infinite number"? I mean, what is that? – Timbuc May 14 '15 at 18:21
  • Would I be correct in calling π an irrational number instead? @Timbuc – Dan May 14 '15 at 18:24
  • @Dan $\pi$ is irrational, and finite. –  May 14 '15 at 18:27

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First of all, you seem to be confused with your concepts. There is no infinite real number. What you are calling "infinite" is actually "having infinite decimal expansion".

Also, note that you may be comfortable with division by other numbers which have infinite decimal expansion: take $\frac{4}{3}$ for instance. Even simpler is division by $\frac{1}{3}$: which is simply multipying by $3$.

If you have a problem with operating with irrational numbers, there is no way I could (decently) tell you what are you exactly doing when you operate them without getting "technical". I can give you some intuitive/geometric description like: $\frac{a}{b}$ is the number $x$ for which the rectangle with sides $x$ and $b$ has area $a$... but that is sort of cheating. To understand operating with these numbers, you need to know how real numbers are defined (search for Dedekind Cuts, for instance. It is a way of constructing them.)

Aloizio Macedo
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However if $\pi$ is infinite, how are we able to obtain a value for the diameter?

That really depends on what do you mean by "a value". Is $329$ a value? What about $3.9$? What about $\sqrt2$? What about $2\sqrt 2$? If you are convinced that $\sqrt 2$ is a "value" (whatever that means), then $\pi$ should be a value. It might seem exotic since we are using a greek letter to represent that thing, but it's still a value. If you want you can use $\star$ to represent pi.

I think what really confuses you is this: how do I get the numerical value of $\pi$ or $\frac{40}{\pi}$? That would be a different subject. See Approximations of $\pi$

3x89g2
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