Why is e^(pi i) = -1?

Imagine you're a complex number, which is just a type of 2-vector.<p>Exponentiation is to do with growth at a speed which is a multiple of how big you are already.<p>i is the multiplication which turns you through ninety degrees.<p>If you grow in a direction which is at right angles to yourself, you turn rather than increasing in magnitude.<p>Pi is how long it takes you to turn through a half circle.<p>So if you grow at right angles to yourself for time pi, you are pointing the opposite way.

Pi is Wrong: <a href="http://www.math.utah.edu/~palais/pi.pdf" rel="nofollow">http://www.math.utah.edu/~palais/pi.pdf</a><p>It should be e^(pi i) = 1, but pi was unfortunately defined at half the appropriate value in the 17th century.

The article didn't have any mention of Euler, but this is typically called Euler's Formula and is considered one of the most beautiful formulas to Mathematicians.<p>Here's some more info if you want to find out some info, applications and such: <a href="http://en.wikipedia.org/wiki/Eulers_formula" rel="nofollow">http://en.wikipedia.org/wiki/Eulers_formula</a>

Write this as e^(iπ) + 1 = 0 and you have my favourite equation.<p>It relates all the fundamental mathematical numbers: e, i, π, 1 and 0.

You remember those half-angle formulae<p>sin 2A = 2 sin A cos A<p>cos 2A = (cos A)^2 - (sin A)^2<p>If you are happy using sine of small x is x and cosine of small x is 1 as your base cases, you can write sine and cosine as mutually recursive functions:<p><pre><code> (defun sin (z) (if (small z) z (* 2 (sin (half z)) (cos (half z))))) (defun cos (z) (if (small z) 1 (- (square (cos (half z))) (square (sin (half z)))))) </code></pre> (sin pi) =&#62; 6.167817939221069d-7 quite close to zero (cos pi) =&#62; -1.0012055113842453d0 you can get this much closer to -1 by using 1-x^2/2 as your base case.<p>It had never occurred to me to do exponential the same way, as<p><pre><code> (defun exp (z) (if (small z) (+ 1 z) (square (exp (half z))))) </code></pre> So it is a bit of a shock to try it and see it work just fine for complex numbers<p>(exp (complex 0 pi)) =&#62; #C(-1.0012055113842453d0 6.167817939221069d-7)

The explanations mapping complex exponentiation to rotations are basically right. In this context, it's worth noting that the conventional choice of circle constant is off by two. We should be using tau = <i>τ = C/r</i> as the circle constant, rather than pi = <i>π = C/D</i>. Read <i>τ</i> as "turn" and all those radian angle measures suddenly make sense. Ninety degrees? Instead of the confusing <i>π</i>/2 we have 90° = <i>τ</i>/4 = one quarter turn. And so on: 60° = <i>τ</i>/6 = one sixth of a turn, 180° = <i>τ</i>/2 = one half turn, etc.<p>In these terms, Euler's formula would be recast as<p><i>e</i>^(<i>i</i> <i>τ</i>) = 1<p>That is, the exponential of the imaginary unit <i>i</i> times the circle constant <i>τ</i> is unity: one full rotation.

I enjoyed Lakoff and Núñez's section on this in their book <i>Where Mathematics Comes From</i>[1]. The relevent pages are available at Google Books[2].<p>1- <a href="http://en.wikipedia.org/wiki/Where_Mathematics_Comes_From" rel="nofollow">http://en.wikipedia.org/wiki/Where_Mathematics_Comes_From</a><p>2- <a href="http://books.google.com/books?id=YXv6SEjTNKsC&#38;pg=PA432&#38;lpg=432" rel="nofollow">http://books.google.com/books?id=YXv6SEjTNKsC&#38;pg=PA432&#...</a>

For those of you interested in a longer look at this, check out "E, the Story of a Number" by Eli Maor.<p>Yes, I read when I should probably be hacking.

I like Feynman's description, where he actually used 10^x first, just noting that 10^x for small real x was 1 + ln(10)*x (approximating from the derivative), assuming that this worked for small complex values too, and then extended to larger values by squaring.

So to broadcast my math ignorance to all:<p>Why does cos(pi) + i sin(pi) == -1 ??<p>Why does i disappear in this?<p>Edit: Thanks to the replies below.

<a href="http://xkcd.com/179/" rel="nofollow">http://xkcd.com/179/</a><p>I'm sure almost everyone has seen this anyway, but why not post.

So many texts on complex analysis simply define e^{i \theta} = \cos \theta + i \sin \theta without ever explaining how. This provides a good introduction to the reason behind with only minimal recourse to Calculus.<p>Another good description is discussed in "Visual complex Analysis" by Needham.

I have known it as the celebrities formula, because if you write it this way: e^(pi i) + 1 = 0<p>It contains all the "celebrities" of the math world.<p>It's quite cool actually.

Pages like this will be so much better when we have MathML in the browser and can stop using fuzzy images for mathematical formulas.

A better question is how can you:<p>(pi/2)^2e

umm i m no smart ass.. but i think e^(pi i) = -1 cuz LHS = cos PI + isin PI = -1 .... :O