How does e to the pi i = -1?
Date: 6 Jan 1995 11:03:48 -0500 From: Winnie So Subject: Question Just out of curiosity, I want to know how "e to the pi i" = -1 Can you show me how that may be evaluated and the type of formulas that are involved? Please e-mail back to me at email@example.com Thanks!!! Winnie
Date: 7 Jan 1995 22:16:45 -0500 From: Dr. Sydney Subject: Re: Question Dear Winnie, Hello again! Thanks for writing Dr. Math. You asked a great question. In fact, I was wondering the same thing earlier this year as well. Here's how it was explained to me: Do you know that e^(ix) = cosx + isinx? You might wonder why this is true. Are you familiar with Taylor polynomials? I'm not sure how much calculus you have had, but it is possible to approximate many functions with a polynomial. It turns out that we can, in fact, approximate the exponential function and the trig functions as follows: e^x = 1 + x + x^2/2! + x^3/3! + ... sinx = x - x^3/3! + x^5/5! - ... cosx = 1 - x^2/2! + x^4/4! - ... If you play around with these Taylor polynomials, I think you'll see why e^ix = cosx + isinx. If you are having trouble seeing why this is true, or if you want to know more about Taylor polynomials, please write back. Anyway, once you know e^ix = cosx + isinx, figuring out what e^i(pi) becomes pretty straightforward. e^i(pi) = cos(pi) + i sin(pi) = -1 + 0 = -1. I hope this helps. Write back if you have any more questions. --Sydney
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