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### The Constants pi, e, i, 0, and 1.

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Date: 28 Jun 1995 18:40:28 -0400
From: Thomas D. Jacob
Subject: e^(pi*i)

A friend told me that e^(pi*i) = -1
(That is, e to the power of pi times i equals negative one.)
irrational numbers can together produce such an ordinary result.)

Thanks.

Thomas D. Jacob
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```
Date: 28 Jun 1995 20:35:56 -0400
From: Dr. Ken
Subject: Re: e^(pi*i)

Hello there!

Yes, this is one of the coolest things in Math.  In fact, there's
an even better way to write it:

i Pi
e     + 1 = 0

This way, you combine the five big constants in mathematics:
0,1,i,Pi, and e.  You also get the three main operations: +, *,
and exponentiation.  And you get the notion of equality,
which is so key in mathematics.

To understand how this works, you have to know the fact that
e^(i*t) = Cos(t) + iSin(t).  This in itself is a really cool thing.
What it means is that if you let t go from zero to 2 Pi, the
function e^(i*t) will trace out the unit circle in the complex plane.
So to rotate any complex figure by t radians, multiply it by e^(i*t).

Anyway, once you have that, plug in Pi for t, and you've got
your formula! Thanks for the question.

-K
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Associated Topics:
College Exponents

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