Can a math wiz explain this?

Test Preparation
1

<p>I was bored in math one day (and devoted my time to pressing buttons on my calc) so I decided to take the natural log of negitive 1 (in imaginary mode). It's pi(i). Someone explain?</p>

2

<p>e^(pi * i) = -1</p>

<p>don't ask me to explain why</p>

3

<p>You didn't really help...</p>

4

<p>tanonev we went over that equation for 15 minutes in calculus one day, lol.</p>

5

<p>sorry...I never got that complex math thing...</p>

6

<p>neither did I lol</p>

7

<p>wait, I got it!</p>

<p>e^(x*i) = (1, x) in polar coordinates on the complex plane</p>

8

<p>again, I have no idea why...lol</p>

9

<p>Essentially, it's just a definition of what e^x means for imaginary x. e^(ix) is defined to be cosx+i*sinx. The reason for that definition is that it remains consistent with most of the important properties of e^x. In particular, the Taylor series remains valid:</p>

<p>e^x=1+x+x^2/2!+x^3/3!+...
e^(ix)=1+ix-x^2/2!-ix^3/3!+...
=(1-x^2/2!+x^4/4!+...)+i(x-x^3/3!+...)
=cosx+isinx</p>

<p>Doesn't really give insight into WHY it should be true, but it's still pretty neat, I think.</p>

<p>A couple justifications for the why include that this definition retains the fact that e^a * e^b = e^(a+b), and that e^x is the unique function that is its own derivative and has f(0)=1.</p>

10

<p>I'm guessing the Fourier expansion works out as well? That explains why it looked familiar...my dad gave me a book on Fourier when I was 9...I have no idea what he was thinking...</p>

11

<p>i think it has something to do with De Moivre's Theorem. perhaps.</p>

12

<p>The way I've been taught to think about it is due to Taylor series. The Taylor series for e^(ix) is the same as that of cos(x) + i sin(x), so </p>

<p>e^(ix)=cos(x) + i sin(x)</p>

<p>This is called Euler's theorem. </p>

<p>DeMoivre's Theorem, which is related, states that </p>

<p>(cos(x) + i sin(x))^n = cos(nx) + i sin (nx)</p>

<p>(Note that the proof is obvious when we consider Euler's theorem.</p>

13

<p>how the heck did you guys learn so much about math - i did calc BC but never touched on Fourier or DeMoivre & I never did those equations</p>

<p>any books u would recommend?</p>

14

<p>Always check the appendices of your math texts. They often have some extra stuff, such as complex numbers. Since everything on calc BC deals with real numbers (I think...it's been a while :p), complex numbers usually are taken out of the main text and shoved into the appendix.</p>