# Magic Equation **URL:** \ **Category:** Factual Questions\ **Created:** [April 20, 2000, 7:30pm UTC](https://boards.straightdope.com/t/magic-equation/11053 "2000-04-20T19:30:00Z")\ **Posts on this page:** 18\ **Page:** 1
**Author:** ![FrankGrimes](https://avatars.discourse-cdn.com/v4/letter/f/82dd89/32.png) [@FrankGrimes](https://boards.straightdope.com/u/FrankGrimes)\ **Post date:** [April 20, 2000, 7:30pm UTC](https://boards.straightdope.com/t/magic-equation/11053/1 "2000-04-20T19:30:00Z")
I read something about an equation that links the four most important numbers in math: pi, i, 1 and 0. Does anyone know it? ---
**Author:** ![whitetho](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whitetho/32/1059_2.png) [@whitetho](https://boards.straightdope.com/u/whitetho)\ **Post date:** [April 20, 2000, 7:40pm UTC](https://boards.straightdope.com/t/magic-equation/11053/2 "2000-04-20T19:40:00Z")
Well, Grimey, I you are referring to the famous [Euler Formula: e^(pi\*i) = -1](http://forum.swarthmore.edu/dr.math/problems/drizzt.6.5.96.html), which is explained by the linked page. * * * “I live in an apartment over a bowling alley, which is under _another_ bowling alley” ---
**Author:** ![Strainger](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/strainger/32/244_2.png) [@Strainger](https://boards.straightdope.com/u/Strainger)\ **Post date:** [April 20, 2000, 7:42pm UTC](https://boards.straightdope.com/t/magic-equation/11053/3 "2000-04-20T19:42:00Z")
IIRC, it’s something like e^(-i\*pi)=1. I’ll see if I can dig up the proof. ---
**Author:** ![Strainger](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/strainger/32/244_2.png) [@Strainger](https://boards.straightdope.com/u/Strainger)\ **Post date:** [April 20, 2000, 7:43pm UTC](https://boards.straightdope.com/t/magic-equation/11053/4 "2000-04-20T19:43:00Z")
Dammit, whitho beat me while I was trying to see if there was an ASCII code for “pi.” ---
**Author:** ![KarlGauss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/karlgauss/32/3713_2.png) [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)\ **Post date:** [April 20, 2000, 7:47pm UTC](https://boards.straightdope.com/t/magic-equation/11053/5 "2000-04-20T19:47:00Z")
Indeed, it is truly a _magical_ equation. Probably the single most important factor contributing to what has become a lifelong love. If only I had the talent…sigh. ---
**Author:** ![FrankGrimes](https://avatars.discourse-cdn.com/v4/letter/f/82dd89/32.png) [@FrankGrimes](https://boards.straightdope.com/u/FrankGrimes)\ **Post date:** [April 20, 2000, 7:50pm UTC](https://boards.straightdope.com/t/magic-equation/11053/6 "2000-04-20T19:50:00Z")
Thanks Strainger and whitetho. ---
**Author:** ![Boris\_B](https://avatars.discourse-cdn.com/v4/letter/b/fbc32d/32.png) [@Boris\_B](https://boards.straightdope.com/u/Boris_B)\ **Post date:** [April 20, 2000, 7:50pm UTC](https://boards.straightdope.com/t/magic-equation/11053/7 "2000-04-20T19:50:00Z")
I’ve never understood this one, and my advanced algebra skills have gotten rusted shut. Tell me some simpler ones: What is e^i? I should probably be able to do this one, but I’ve totally forgotten how to raise to imaginary powers. What is 1^(i x pi)? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 20, 2000, 7:59pm UTC](https://boards.straightdope.com/t/magic-equation/11053/8 "2000-04-20T19:59:00Z")
Boris: To take exponential powers, you have to use the power series for e^x: e^x = x^0/0! + x^1/1! + x^2/2! + x^3/3! + … Or, you can use the shortcut a’la’ Euler (pronounced “Oiler”, btw): e^ix = cos(x) + i\*sin(x) For complex exponents (not real or pure imaginary), you can break it up into its real and imaginary parts: e^(a + bi) = (e^a)\*(e^bi). For bases other than e, use the formula a^b = e^(ln(a)\*b) Does this help? * * * “There are only two things that are infinite: The Universe, and human stupidity-- and I’m not sure about the Universe” –A. Einstein ---
**Author:** ![Rasputin](https://avatars.discourse-cdn.com/v4/letter/r/dec6dc/32.png) [@Rasputin](https://boards.straightdope.com/u/Rasputin)\ **Post date:** [April 21, 2000, 7:39am UTC](https://boards.straightdope.com/t/magic-equation/11053/9 "2000-04-21T07:39:00Z")
e^i: 0.540302305868 + 0.841470984808i 1^(Pi\*i): 1 to any power is always 1…but we know you knew that. * * * “I don’t want to achieve immortality through my work… I want to achieve it through not dying.” – Woody Allen ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 21, 2000, 6:54pm UTC](https://boards.straightdope.com/t/magic-equation/11053/10 "2000-04-21T18:54:00Z")
Another interesting one: i^i = .2078796 . That’s right, it’s pure real. Fun, eh? * * * “There are only two things that are infinite: The Universe, and human stupidity-- and I’m not sure about the Universe” –A. Einstein ---
**Author:** ![Boris\_B](https://avatars.discourse-cdn.com/v4/letter/b/fbc32d/32.png) [@Boris\_B](https://boards.straightdope.com/u/Boris_B)\ **Post date:** [April 21, 2000, 7:00pm UTC](https://boards.straightdope.com/t/magic-equation/11053/11 "2000-04-21T19:00:00Z")
> [@](#): > > 1^(Pi\*i): > 1 to any power is always 1…but we know you knew that. Well … I did kinda know that, but I didn’t know imaginaries were included with “any power”. And Chronos, I _never_ woulda guessed that i^i was a real number. Thanks for blasting my world-view into nasty little splinters. You guys are starting to make my brains wiggle. I’m going to head over to a nice pleasant gun control thread. ---
**Author:** ![Arnold\_Winkelried](https://avatars.discourse-cdn.com/v4/letter/a/3d9bf3/32.png) [@Arnold\_Winkelried](https://boards.straightdope.com/u/Arnold_Winkelried)\ **Post date:** [April 21, 2000, 7:06pm UTC](https://boards.straightdope.com/t/magic-equation/11053/12 "2000-04-21T19:06:00Z")
Of course, the form of the equation that answers the original question is e^(i \* pi) + 1 = 0 ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [April 22, 2000, 10:09pm UTC](https://boards.straightdope.com/t/magic-equation/11053/13 "2000-04-22T22:09:00Z")
Arnold Finally! And, of course, that’s the _five_ most important numbers, not four… ---
**Author:** ![Konrad](https://avatars.discourse-cdn.com/v4/letter/k/f475e1/32.png) [@Konrad](https://boards.straightdope.com/u/Konrad)\ **Post date:** [April 23, 2000, 2:31am UTC](https://boards.straightdope.com/t/magic-equation/11053/14 "2000-04-23T02:31:00Z")
It’s not really that magical. Remember that the pi stands for pi radians, which is just a human convention. You can also write e^(i\*180 degrees) = -1. Nothing but convention and hype. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 24, 2000, 5:43am UTC](https://boards.straightdope.com/t/magic-equation/11053/15 "2000-04-24T05:43:00Z")
Yeah, Konrad, but remember, it’s just convention that 1 stands for the unit, and that e stands for lim(x-\>inf) (1+1/x)^x, and that = stands for “is equal to”. Yes, it’s a human convention to refer to angles in radians, but it’s a very logical convention: were it not for that convention, all the trig differentials/integrals would be more complicated, the simple analogy wouldn’t exist between exponentials and trig functions, etc. For that matter, even if we DIDN’T use radian notation, e^(i_pi) would STILL equal -1, since sum(n=0 to inf) (i_pi)^n/n! = -1 . It’s only because of this that the pi in that equation is interpreted to mean pi radians in the first place. * * * “There are only two things that are infinite: The Universe, and human stupidity-- and I’m not sure about the Universe” –A. Einstein ---
**Author:** ![Konrad](https://avatars.discourse-cdn.com/v4/letter/k/f475e1/32.png) [@Konrad](https://boards.straightdope.com/u/Konrad)\ **Post date:** [April 24, 2000, 11:12pm UTC](https://boards.straightdope.com/t/magic-equation/11053/16 "2000-04-24T23:12:00Z")
Chronos: That’s true. I guess that does make the equation pretty interesting. ---
**Author:** ![emarkp](https://avatars.discourse-cdn.com/v4/letter/e/3be4f8/32.png) [@emarkp](https://boards.straightdope.com/u/emarkp)\ **Post date:** [April 24, 2000, 11:38pm UTC](https://boards.straightdope.com/t/magic-equation/11053/17 "2000-04-24T23:38:00Z")
Also remember that radians are defined as arclength divided by radius (is the angle in radians that subtends the given arc). That means that radians have no human-defined calibration (like degrees, etc.), but are a pure ratio. ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [April 25, 2000, 4:47pm UTC](https://boards.straightdope.com/t/magic-equation/11053/18 "2000-04-25T16:47:00Z")
> [@](#): > > emarkp wrote: > That means that radians have no human-defined calibration (like degrees, etc.), but are a pure ratio. I think you’re saying that radians are unitless, but degrees are unitless also. There’s just 360 of them in a circle instead of two pi of them. The reason radians are used is because they’re more “natural” - equations become much simpler with them. Same thing with using “e” for logarithms. By the way, no one has pointed out that the easy way to find e^(i\*x) is this way: e^(i\*x) = cos(x) + i \* sin(x). cos(pi) is -1, and sin(pi) is zero, so e^(i\*pi) is -1.