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


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.)
Can you tell me anything about this?  (Such as why two 
irrational numbers can together produce such an ordinary result.)

Thanks.

Thomas D. Jacob   


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
    
Associated Topics:
College Exponents

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