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### How does e to the pi i = -1?

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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 wso@walrus.mvhs.edu  Thanks!!!

Winnie
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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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Associated Topics:
High School Polynomials

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