# Mathematical Wizardry **URL:** \ **Category:** Factual Questions\ **Created:** [November 15, 2002, 7:49am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838 "2002-11-15T07:49:53Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 16, 2002, 4:42pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/21 "2002-11-16T16:42:13Z")
> [@](#): > > \*Originally posted by Joe Mahma \* > \*\*How about a “circle” with zero circumference and zero diameter. That would give pi=0. \*\* No, that would allow pi to be any number. 0/0 ain’t 0, or anything else. ---
**Author:** ![Joe\_Mahma](https://avatars.discourse-cdn.com/v4/letter/j/5daacb/32.png) [@Joe\_Mahma](https://boards.straightdope.com/u/Joe_Mahma)\ **Post date:** [November 16, 2002, 4:47pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/22 "2002-11-16T16:47:05Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\* > > No, that would allow pi to be any number. 0/0 ain’t 0, or anything else. \*\* oops. ---
**Author:** ![DarrenS](https://avatars.discourse-cdn.com/v4/letter/d/977dab/32.png) [@DarrenS](https://boards.straightdope.com/u/DarrenS)\ **Post date:** [November 16, 2002, 6:53pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/23 "2002-11-16T18:53:10Z")
I’d never thought of the degenerate case - does that mean that the rule should be, “the ratio of the circumference to the diameter of any circle is equal to _pi_, _providing the radius is greater than zero_” ? Or should we not consider a point a circle of radius zero? Or is this a proof that in fact by letting radius -\> 0, 0/0 = _pi_ ?? (Juuust kidding on the last part!) ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [November 16, 2002, 6:55pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/24 "2002-11-16T18:55:33Z")
Here’s [Dr. Math’s take](http://mathforum.org/library/drmath/view/51629.html) on an attempt to use Euler’s formula to prove that pi = 0. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 16, 2002, 7:15pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/25 "2002-11-16T19:15:46Z")
> [@](#): > > \*Originally posted by DarrenS \* > \*\*I’d never thought of the degenerate case - does that mean that the rule should be, “the ratio of the circumference to the diameter of any circle is equal to _pi_, _providing the radius is greater than zero_” ? Or should we not consider a point a circle of radius zero? Or is this a proof that in fact by letting radius -\> 0, 0/0 = _pi_ ?? (Juuust kidding on the last part!) \*\* Probably best to do it as the limit as r goes to 0. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [November 16, 2002, 9:28pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/26 "2002-11-16T21:28:26Z")
**ultrafilter** , your proof that 0 is not equal to -1 is an impressive bit of mathematical tomfoolery. I’m still not sold on your implicit assumption that if something is an element of a set, then it is in that set (kidding!), but I’ll let it slide for now. 😉 I think we need a definition of pi that does not depend on Calculus. How is pi defined nowadays? ---
**Author:** ![rowrrbazzle](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rowrrbazzle/32/414_2.png) [@rowrrbazzle](https://boards.straightdope.com/u/rowrrbazzle)\ **Post date:** [November 16, 2002, 11:38pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/27 "2002-11-16T23:38:00Z")
The “proof” beginning with e[sup]2pi[/sup] = 1 is equivalent to this “proof”: sin(2p) = 0 sin(0) = 0 sin(2p) = sin(0) Take the arcsin of both sides: 2p = 0 Divide by 2: p = 0 ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 17, 2002, 6:19am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/28 "2002-11-17T06:19:20Z")
> [@](#): > > \*Originally posted by Achernar \* > \*\*I think we need a definition of pi that does not depend on Calculus. How is pi defined nowadays? \*\* It’s the ratio of the circumference of a circle to its diameter, same as it always has been. Not sure I understand why you don’t like the calculus of it. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [November 17, 2002, 6:32am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/29 "2002-11-17T06:32:29Z")
Because now, just about everything is defined in terms of set theory. I would think that something as straightforward as pi would have gotten there too, but of course you’re right, it hasn’t. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 17, 2002, 6:34am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/30 "2002-11-17T06:34:55Z")
Well, keep in mind that real numbers are defined in terms of set theory. Once you’ve got a set-theoretic entity that has all the same properties as the reals, every theorem that applies to the reals applies to these entities as well. IOW, [symbol]p[/symbol] is a Dedekind cut, but it’s the one that corresponds to a particular notion from calculus. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [November 17, 2002, 6:39am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/31 "2002-11-17T06:39:21Z")
Well yes of course it all goes back to sets eventually. But I was hoping you knew what I meant. Something more direct. ---
**Author:** ![Tyrrell\_McAllister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tyrrell_mcallister/32/16772_2.png) [@Tyrrell\_McAllister](https://boards.straightdope.com/u/Tyrrell_McAllister)\ **Post date:** [November 17, 2002, 10:36am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/32 "2002-11-17T10:36:25Z")
To define pi in the modern formal spirit, you can define sin in terms of power series sin(x) = x - x^3/3! + x^5/5! - . . . and then define pi to be the smallest root of sin in the positive real numbers. ---
**Author:** ![Tyrrell\_McAllister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tyrrell_mcallister/32/16772_2.png) [@Tyrrell\_McAllister](https://boards.straightdope.com/u/Tyrrell_McAllister)\ **Post date:** [November 17, 2002, 10:39am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/33 "2002-11-17T10:39:20Z")
… but be aware that even the smallest root of sin can make baby Jesus cry. ---
**Author:** ![DarrenS](https://avatars.discourse-cdn.com/v4/letter/d/977dab/32.png) [@DarrenS](https://boards.straightdope.com/u/DarrenS)\ **Post date:** [November 18, 2002, 2:38am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/34 "2002-11-18T02:38:23Z")
\If [symbol]p[/symbol] is defined as the ratio of the circumference of a circle to its diameter, is there any simple geometrical interpretation of _e_ ? Why do I think this has something to do with the Golden Section or logarithmic spirals? \ ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [November 18, 2002, 2:49am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/35 "2002-11-18T02:49:00Z")
A simple interpretation? I doubt it. Maybe you could come up with a contrived one. The fact is that pi is _the_ number of geometry, and e is _the_ number of calculus. ---
**Author:** ![Shalmanese](https://avatars.discourse-cdn.com/v4/letter/s/45deac/32.png) [@Shalmanese](https://boards.straightdope.com/u/Shalmanese)\ **Post date:** [November 18, 2002, 6:20am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/36 "2002-11-18T06:20:04Z")
> [@](#): > > \*Originally posted by Tyrrell McAllister \* > \*\*… but be aware that even the smallest root of sin can make baby Jesus cry. \*\* bwhahaha ---
**Author:** ![LifeWillFall](https://avatars.discourse-cdn.com/v4/letter/l/46a35a/32.png) [@LifeWillFall](https://boards.straightdope.com/u/LifeWillFall)\ **Post date:** [November 18, 2002, 7:47am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/37 "2002-11-18T07:47:25Z")
I think e is defined as: the limit of (1+1/n)^n but what the hell do I know ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 18, 2002, 4:39pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/38 "2002-11-18T16:39:48Z")
> [@](#): > > \*Originally posted by LifeWillFall \* > \*\*I think e is defined as: > > the limit of (1+1/n)^n > > but what the hell do I know \*\* That is one way of getting e, but that’s not the definition. I think there are a couple ways you could define e: 1. The value of x such that the integral of 1/t from 1 to x is equal to 1. OR 1. The sum over n from zero to infinity of x[sup]n[/sup]/n! evaluated at x = 1. I think the second definition would be preferred these days. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [November 18, 2002, 6:39pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/39 "2002-11-18T18:39:59Z")
Is there some reason you don’t say it like this: 1. The sum over n from zero to infinity of 1/n! The whole bit with x seems like an unnecessary complication. LifeWillFall’s definition of e is the only one I’ve ever seen (for instance, equation 1 on [this MathWorld page](http://mathworld.wolfram.com/e.html)). However, it’s probably better to define the function exp(x) differently, and then show that it’s equivalent to e[sup]x[/sup]. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 18, 2002, 6:41pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/40 "2002-11-18T18:41:43Z")
I wasn’t thinking clearly. Let’s use the simpler form instead. [Previous page](https://boards.straightdope.com/t/mathematical-wizardry/136838.md?page=1) [Next page](https://boards.straightdope.com/t/mathematical-wizardry/136838.md?page=3)