# 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:** 1
**Author:** ![KeeblerElf](https://avatars.discourse-cdn.com/v4/letter/k/47e85d/32.png) [@KeeblerElf](https://boards.straightdope.com/u/KeeblerElf)\ **Post date:** [November 15, 2002, 7:49am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/1 "2002-11-15T07:49:53Z")
I’ve been told by one of my Math Wizard friends that it’s possible to make pi equal zero, as well as any other number. I haven’t seen him actually do it yet, but the possibility intrigues me. Any of you Math Dopers out there know what he’s talking about? Care to give a demonstration? ---
**Author:** ![Pythagras](https://avatars.discourse-cdn.com/v4/letter/p/45deac/32.png) [@Pythagras](https://boards.straightdope.com/u/Pythagras)\ **Post date:** [November 15, 2002, 8:06am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/2 "2002-11-15T08:06:42Z")
Whaaa? pi is by definition not 0. Here, I can make pi 0: Define “pi” to be 0. done. Too bad in most notation pi is the ratio of a circumference to the diameter. The only way this would be 0 is if you had a circle with a 0 circumference and non-zero diameter. Show me that, Ill mail you a dollar. Though maybe there’s some weirdness in non-euclidean geometry Im not aware of… ---
**Author:** ![Joey\_P](https://avatars.discourse-cdn.com/v4/letter/j/919ad9/32.png) [@Joey\_P](https://boards.straightdope.com/u/Joey_P)\ **Post date:** [November 15, 2002, 11:25am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/3 "2002-11-15T11:25:33Z")
Multiply it by zero? You said your friends can make any other number zero. How exacty would you make 7 equal to zero (7=0)? ---
**Author:** ![NardoPolo](https://avatars.discourse-cdn.com/v4/letter/n/ecae2f/32.png) [@NardoPolo](https://boards.straightdope.com/u/NardoPolo)\ **Post date:** [November 15, 2002, 12:13pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/4 "2002-11-15T12:13:56Z")
Give him a pie and he’ll eat it. Then it’s zero. (shrug) ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [November 15, 2002, 12:19pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/5 "2002-11-15T12:19:01Z")
Possibly by solving a pair of linear equations that are inconsistent. ---
**Author:** ![muttrox](https://avatars.discourse-cdn.com/v4/letter/m/a8b319/32.png) [@muttrox](https://boards.straightdope.com/u/muttrox)\ **Post date:** [November 15, 2002, 12:48pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/6 "2002-11-15T12:48:54Z")
Or having a circle be defined in a non-euclidian space, so it effectively covers the surface of a globe. Imagine a circle that is on the 80th north parralel, covering the land north of that. Now extend the circle southwards. After you cross the equator, the circumference shrinks, but the diameter still grows. When the circle envelops the whoel globe, the circurference will be zero, diameter will by 28K miles or so. Then you can pull all kinds of stupid math tricks. Or you could use this old trick… assume a=b 1. a=b 2. ab=b^2 (mult by b) 3. ab-b^2=a^2-b^2 (subtract b^2) 4. b(a-b) = (a+b)(a-b) (factor) 5. b=a+b (divide out the common factor) 6. b=b+b (we assumed they’re equal, now substitute 7. b=2b 8. 1=2 (divide by b) This is a fun proof (find the flaw) that comes up a lot. Now go further. 9. 0=1 (subtract 1) 10. pi=pi (duh) 11. pi_1=pi_0 12. pi=0 Your friend may have similar tricks in mind. As far as I know though, there is no mathematically legitimate way to show that pi=0 through mathematics using standard definitions and procedures. ---
**Author:** ![hermn8r](https://avatars.discourse-cdn.com/v4/letter/h/53a042/32.png) [@hermn8r](https://boards.straightdope.com/u/hermn8r)\ **Post date:** [November 15, 2002, 2:44pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/7 "2002-11-15T14:44:17Z")
> [@](#): > > \*Originally posted by muttrox \* > \*\*Or you could use this old trick… assume a=b > > 1. a=b > 2. ab=b^2 (mult by b) > 3. ab-b^2=a^2-b^2 (subtract b^2) > 4. b(a-b) = (a+b)(a-b) (factor) > 5. b=a+b (divide out the common factor) > 6. b=b+b (we assumed they’re equal, now substitute > 7. b=2b > 8. 1=2 (divide by b) > This is a fun proof (find the flaw) that comes up a lot. Now go further. > 9. 0=1 (subtract 1) > 10. pi=pi (duh) > 11. pi_1=pi_0 > 12. pi=0 > \*\* One flaw in your equation is in line (3): if you subtract b^2 from both sides of (2) ab=b^2 you would get ab-b^2=b^2-b^2 (which is ab-b^2=0) NOT your line: 3) ab-b^2=a^2-b^2 (subtract b^2) I think the ‘proof’ you may be looking for is: Theorem 1: 1=2 a=b postulate a^2=ab (multiply both sides by a) a^2-b^2=ab-b^2 (subtract b2 from both sides) (a+b)(a-b)=b(a-b) (factor) a+b=b (divide by (a-b)) b+b=b (as a=b, substitute) 2b=b (b+b=2b) 2=1 (divide by b) QED Or many more of similar relation. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 15, 2002, 4:41pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/8 "2002-11-15T16:41:59Z")
> [@](#): > > \*Originally posted by muttrox \* > **Or having a circle be defined in a non-euclidian space, so it effectively covers the surface of a globe. Imagine a circle that is on the 80th north parralel, covering the land north of that. Now extend the circle southwards. After you cross the equator, the circumference shrinks, but the diameter still grows. When the circle envelops the whoel globe, the circurference will be zero, diameter will by 28K miles or so. Then you can pull all kinds of stupid math tricks.** No time for a detailed explanation, but that’s not how a circle in spherical geometry would look. Search for more information. > [@](#): > > \*\*Your friend may have similar tricks in mind. As far as I know though, there is no mathematically legitimate way to show that pi=0 through mathematics using standard definitions and procedures. \*\* This is absolutely correct. The only way to show that [symbol]p[/symbol] = 0 is to do some fakery. ---
**Author:** ![muttrox](https://avatars.discourse-cdn.com/v4/letter/m/a8b319/32.png) [@muttrox](https://boards.straightdope.com/u/muttrox)\ **Post date:** [November 15, 2002, 5:20pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/9 "2002-11-15T17:20:42Z")
Arrgh… hermn8r is absolutely right. Lousy stupid early morning posts. Should have subtracted a^2 not b^2, then proceed from there. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [November 15, 2002, 6:23pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/10 "2002-11-15T18:23:06Z")
exp(2pi×i) = 1 = exp(0) Taking the log of both sides gives… 2pi i = 0 Divide both sides by 2i. pi = 0 ---
**Author:** ![Monstre](https://avatars.discourse-cdn.com/v4/letter/m/e274bd/32.png) [@Monstre](https://boards.straightdope.com/u/Monstre)\ **Post date:** [November 15, 2002, 8:03pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/11 "2002-11-15T20:03:53Z")
Well, I have a **wonderful** proof for pi=0, but this thread is too small to contain it… 😃 ---
**Author:** ![NotMrKnowItAll](https://avatars.discourse-cdn.com/v4/letter/n/6bbea6/32.png) [@NotMrKnowItAll](https://boards.straightdope.com/u/NotMrKnowItAll)\ **Post date:** [November 15, 2002, 8:29pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/12 "2002-11-15T20:29:20Z")
**muttrox** Is the problem step 5 where you divide by zero? ---
**Author:** ![muttrox](https://avatars.discourse-cdn.com/v4/letter/m/a8b319/32.png) [@muttrox](https://boards.straightdope.com/u/muttrox)\ **Post date:** [November 15, 2002, 8:32pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/13 "2002-11-15T20:32:23Z")
Yep, that’s the fakery. Curious to hear what’s the OP’s friends trick was. ---
**Author:** ![DarrenS](https://avatars.discourse-cdn.com/v4/letter/d/977dab/32.png) [@DarrenS](https://boards.straightdope.com/u/DarrenS)\ **Post date:** [November 15, 2002, 11:46pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/14 "2002-11-15T23:46:19Z")
Achernar - where is the trickery in yours? Is it something to do with _log_ being a multi-valued function in the complex domain? ---
**Author:** ![Pencil\_Pusher](https://avatars.discourse-cdn.com/v4/letter/p/b38774/32.png) [@Pencil\_Pusher](https://boards.straightdope.com/u/Pencil_Pusher)\ **Post date:** [November 16, 2002, 1:56am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/15 "2002-11-16T01:56:38Z")
I think its simpler than that, **DarrenS**. If 2pi i = 0, and we know 2 and pi are both =/= 0, then i must = 0. Therefore, you cannot divide both sides by i (or 2i). ---
**Author:** ![Enderw24](https://avatars.discourse-cdn.com/v4/letter/e/ba9def/32.png) [@Enderw24](https://boards.straightdope.com/u/Enderw24)\ **Post date:** [November 16, 2002, 3:30am UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/16 "2002-11-16T03:30:05Z")
Well, think about it. Have you ever seen a pie that doesn’t look like O? ---
**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, 3:59pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/17 "2002-11-16T15:59:04Z")
**DarrenS** is correct. i[sup]2[/sup] = -1, and 0[sup]2[/sup] != -1, so i != 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, 4:20pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/18 "2002-11-16T16:20:12Z")
**ultrafilter** , you’re assuming that 0 is not equal to -1. I’m going to need a proof for that “minor” detail you just glossed right over. And yeah, **DarrenS** is totally right. ---
**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, 4:38pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/19 "2002-11-16T16:38:31Z")
> [@](#): > > \*Originally posted by Achernar \* > **ultrafilter, you’re assuming that 0 is not equal to -1. I’m going to need a proof for that “minor” detail you just glossed right over.** All right. I’m going to assume that you’ll give me all the properties of natural numbers that I need. Recall that an integer is defined to be an equivalence class of ordered pairs of natural numbers. In this case, (a, b) = (c, d) iff a + d = b + c. Also remember that [(a, b)] + [(c, d)] = [(a + c, b + d)]. First, you’ll note that the integer 0 is the class [(a, a)]. This is easy to see: [(b, c)] + [(a, a)] = [(a + b, a + c)]. (a + b, a + c) = (b, c) means that (a + b) + c = (a + c) + b, and due to the commutative and associative properties of addition of natural numbers, that’s true. You don’t want me to prove those. 😉 So what’s the representation of 1? Well, you may know that the idea of writing c as [(a, b)] is to make a - b = c in the informal sense. That means that [(a + 1, a)] is 1. You could show that it satisfies the multiplicative identity property if you want to, but let’s just agree that I’m right. This also means that [(a, a + 1)] is the additive inverse of [(a + 1, a)]. This is easy to show: [(a, a + 1)] + [(a + 1, a)] = [(2a + 1, 2a + 1)], which is zero! Now for the meat of it. Is [(a, a + 1)] = [(a, a)]. No, of course not! (0, 0) is in the second, but not the first, and (0, 1) is in the first but not the second. Additionally, a + a != a + a + 1 (this is using set equality on the naturals). Is that good enough? ---
**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, 4:40pm UTC](https://boards.straightdope.com/t/mathematical-wizardry/136838/20 "2002-11-16T16:40:53Z")
> [@](#): > > \*Originally posted by Pythagras \* > \*\*Whaaa? pi is by definition not 0. Here, I can make pi 0: > > Define “pi” to be 0. > > done. > > Too bad in most notation pi is the ratio of a circumference to the diameter. The only way this would be 0 is if you had a circle with a 0 circumference and non-zero diameter. Show me that, Ill mail you a dollar. > \*\* How about a “circle” with zero circumference and zero diameter. That would give pi=0. [Next page](https://boards.straightdope.com/t/mathematical-wizardry/136838.md?page=2)