# e^i(pi) = -1; proof? **URL:** \ **Category:** Factual Questions\ **Created:** [April 24, 2003, 6:39pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919 "2003-04-24T18:39:02Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![aubries](https://avatars.discourse-cdn.com/v4/letter/a/53a042/32.png) [@aubries](https://boards.straightdope.com/u/aubries)\ **Post date:** [April 24, 2003, 6:39pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/1 "2003-04-24T18:39:02Z")
This is a special case of Euler’s Theorem, right? I remember learning this in Number Theory (like ten years ago), but not how it was derived. A little help? ---
**Author:** ![aubries](https://avatars.discourse-cdn.com/v4/letter/a/53a042/32.png) [@aubries](https://boards.straightdope.com/u/aubries)\ **Post date:** [April 24, 2003, 6:42pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/2 "2003-04-24T18:42:16Z")
By which of course I meant -1.:smack: ---
**Author:** ![Saxman](https://avatars.discourse-cdn.com/v4/letter/s/f6c823/32.png) [@Saxman](https://boards.straightdope.com/u/Saxman)\ **Post date:** [April 24, 2003, 6:48pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/3 "2003-04-24T18:48:12Z")
Man, after reading John Allen Paulos’s “Beyond Numeracy” when I was in junior high and coming across this amazingly sublime equation, I was the bane of math teachers’ existences…until I took AP Math senior year of high school and one glorious, glorious day, the teacher show us how to derive this. But I can’t remember exactly how, except that it used polar coordinates and infinite series. Sorry I can’t offer more – just loved the memory you evoked. 🙂 ---
**Author:** ![panamajack](https://avatars.discourse-cdn.com/v4/letter/p/47e85d/32.png) [@panamajack](https://boards.straightdope.com/u/panamajack)\ **Post date:** [April 24, 2003, 7:04pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/4 "2003-04-24T19:04:46Z")
It’s just a case of the [Euler Formula](http://mathworld.wolfram.com/EulerFormula.html), which is e[sup]_i_x[/sup] = cos x + _i_sin x. You can see how plugging pi into that gives you the result. The Formula can be derived from the series expression of e[sup]x[/sup], and remembering that sine is the odd terms and cosine is the even terms. See the link, or maybe other posts in this thread, for that. [Euler’s Theorem](http://mathworld.wolfram.com/EulersTheorem.html), refers to any of a number of theorems of Euler, as mentioned in the link. ---
**Author:** ![Dr\_Paprika](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr_paprika/32/3042_2.png) [@Dr\_Paprika](https://boards.straightdope.com/u/Dr_Paprika)\ **Post date:** [April 24, 2003, 7:20pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/5 "2003-04-24T19:20:56Z")
Can’t you kids do your own homework? Since e^iX = cos X + i sin X (Euler’s formula) e^ipi = cos pi + i sin pi = -1 ---
**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\ **Post date:** [April 24, 2003, 7:24pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/6 "2003-04-24T19:24:41Z")
e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! . . . . so e^ix = 1 + i + i^2 \* x^2 /2 + i^3 \* x^3/3! + i^4 \* x^4/4! . . . . = 1 + i - (x^2/2!) -i (x^3/3!) + (x^4/4!) + i(x^5/5!) . . . = (1 - x^2/2! + x^4/4! . . . ) + i (1 - x^3/3! + x^5/5! . . . . ) = cosx +isinx. Thus, e^(pi \*i) = cos (pi) + i \* sin(pi) = -1. ---
**Author:** ![Dr\_Paprika](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr_paprika/32/3042_2.png) [@Dr\_Paprika](https://boards.straightdope.com/u/Dr_Paprika)\ **Post date:** [April 24, 2003, 7:29pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/7 "2003-04-24T19:29:15Z")
You could also play with the Taylor series expansion, show that e^x = sum (n=0 to infinity) X^n/n! (this can be derived by remembering if f(x)=e^x, so is the (n+1)th derivative of f(x)=e^x, then using the remainder and squeeze theorems). Hence, e^x = 1 + x + x^2/2 + x^3/6 + x^4/24 + … Also recall i^2=-1, thus i^3=-i and i^4=1, hence i=i^5=i^9…; i^2=i^6=-1… Ah forget it. 🙂 ---
**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\ **Post date:** [April 24, 2003, 10:57pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/8 "2003-04-24T22:57:54Z")
> [@](#): > > \*Originally posted by \*\*\*aubries \*\* > By which of course I meant -1.:smack: I fixed the title for you. **DrMatrix** - General Questions Moderator ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [April 24, 2003, 11:09pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/9 "2003-04-24T23:09:01Z")
In higher mathematics it is a matter of definition not of proof that e[sup]ix[/sup] = cos x + i sin x (_) ( that is assuming the elementary theory of infinite series). Nevertheless, it is possible to use the elementary theory of differential equations to make (_) plausible. Since e[sup]ix[/sup] is a complex function, write e[sup]ix[/sup] = f(x) + i g(x) Differentiating twice and assuming that e[sup]z[/sup] behaves for complex z as it does for real z ( this is why it’s plausible, not a proof) -e[sup]ix[/sup] = f’’(x) + i g’’(x) Equating real and imaginary parts, f’’(x) = -f(x) g’’(x) = -g(x) Together with the obvious initial conditions, this gives f(x) = cos x g(x) = sin x ---
**Author:** ![hajario](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hajario/32/171_2.png) [@hajario](https://boards.straightdope.com/u/hajario)\ **Post date:** [April 25, 2003, 1:32am UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/10 "2003-04-25T01:32:12Z")
My Complex Analysis prof in college wrote it out like this: e^i(pi)=0-1. He said that you now have the five most important numbers in mathematics in one equation: 0, 1, e, i and pi! Haj ---
**Author:** ![Race\_Bannon](https://avatars.discourse-cdn.com/v4/letter/r/5f8ce5/32.png) [@Race\_Bannon](https://boards.straightdope.com/u/Race_Bannon)\ **Post date:** [April 25, 2003, 2:05am UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/11 "2003-04-25T02:05:48Z")
You guys are full of baloney. Everybody knows that e^(i pi) = -0.99999… ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 25, 2003, 2:57am UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/12 "2003-04-25T02:57:48Z")
> [@](#): > > \*Originally posted by hajario \* > \*\*My Complex Analysis prof in college wrote it out like this: > > e^i(pi)=0-1. He said that you now have the five most important numbers in mathematics in one equation: 0, 1, e, i and pi! > > Haj \*\* I always liked it better as e[sup][symbol]ip[/symbol][/sup] + 1 = 0. ---
**Author:** ![TGWATY](https://avatars.discourse-cdn.com/v4/letter/t/b5e925/32.png) [@TGWATY](https://boards.straightdope.com/u/TGWATY)\ **Post date:** [April 25, 2003, 5:51am UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/13 "2003-04-25T05:51:03Z")
> [@](#): > > \*Originally posted by Jabba \* > \*\*In higher mathematics it is a matter of definition not of proof that > e[sup]ix[/sup] = cos x + i sin x \*\* No. The Taylor expansions of both sides are identically equal. See a few posts back. ---
**Author:** ![hajario](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hajario/32/171_2.png) [@hajario](https://boards.straightdope.com/u/hajario)\ **Post date:** [April 25, 2003, 7:57am UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/14 "2003-04-25T07:57:33Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*I always liked it better as e[sup][symbol]ip[/symbol][/sup] + 1 = 0. \*\* Now that you mention it, that must have been how he did it. Thanks and nice coding by the way. Haj ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [April 25, 2003, 8:12am UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/15 "2003-04-25T08:12:00Z")
**TGWATY** : That’s sort of what I meant by “a matter of definition”. Sin z, cos z and e[sup]z[/sup] are all defined as power series and it is obvious from these series that e[sup]z[/sup] = cos z + i sin z ---
**Author:** ![TGWATY](https://avatars.discourse-cdn.com/v4/letter/t/b5e925/32.png) [@TGWATY](https://boards.straightdope.com/u/TGWATY)\ **Post date:** [April 25, 2003, 2:13pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/16 "2003-04-25T14:13:02Z")
**Jabba** , no _sin_ and _cos_ are defined in terms of ratios of sides of right triangles to the hypotenuse. _e_ is defined as the limit as _n_ goes to infinity of (1 + 1/n)[sup]n[/sup]. Historically, I think it was first discovered to be the value taken by _x_ such that _∫[sub]1[/sub][sup]x[/sup] 1/u du = 1_ The identity of their Taylor series is apparently just a happy accident. Yes, you could equivalently define them in terms of their Taylor series. But then you still get this surprising relation between the value of certain integral and the ratio of sides of right triangles. _e_ is a very strange critter. See [here](http://mathworld.wolfram.com/e.html) for more. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 25, 2003, 2:48pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/17 "2003-04-25T14:48:21Z")
sin, cos, and exp are defined as Taylor series in higher mathematics (cf. Rudin). ln is defined by the integral. Any relation to right triangles is coincidental. ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [April 25, 2003, 3:17pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/18 "2003-04-25T15:17:36Z")
Since it’s possible to derive the Taylor series of sin and cos from the “right triangles” definition of those functions, I wouldn’t say the relationship is just a coincidence. However, I would agree that in calculus textbooks those functions are defined more often using the Taylor series than using right triangles. ---
**Author:** ![TGWATY](https://avatars.discourse-cdn.com/v4/letter/t/b5e925/32.png) [@TGWATY](https://boards.straightdope.com/u/TGWATY)\ **Post date:** [April 25, 2003, 4:00pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/19 "2003-04-25T16:00:11Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*sin, cos, and exp are defined as Taylor series in higher mathematics (cf. Rudin). ln is defined by the integral. > > Any relation to right triangles is coincidental. \*\* Sine and cosine are defined by their series in “higher” math simply out of convenience. You don’t have to go to the trouble of defining geometrical entities like _triangle_ and _angle_ when all you need is an algebraic expression. But that doesn’t make it prior. “Any relation to right triangles is coincidental” is the silliest thing I have heard today. So-called higher math texts are silent about the triangle origin because it is assumed you already know that. Nobody introduces sine/cosine to a student wholly ignorant of them by defining them as a series expansion and then adds, btw, there is also this accidental triangle relation. Try looking it up in an encyclopedia or dictionary and you’ll see what I mean. I would argue that such most-common definitions are the “real” definitions. E.g., that _π_ is the ratio of the circumference to the diameter of a circle. There may be a dozen other _equivalent_ definitions, but that is the “real” one. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 25, 2003, 4:03pm UTC](https://boards.straightdope.com/t/e-i-pi-1-proof/170919/20 "2003-04-25T16:03:47Z")
Yeah, OK, that was a bit over the top. Still, as you pointed out, it’s more convenient to do the Taylor series definition, and that’s the one that’s used. [Next page](https://boards.straightdope.com/t/e-i-pi-1-proof/170919.md?page=2)