a friend told me that the equation (e^(-i * pi)) + 1 = 0 is evidence
that the universe has intelligent design.  i understand the equation
but would like a few scientific articles and summaries throughly
explaining and discussing it, and possibly how it was derived.

the equation is correct:
://www.google.com/search?hl=en&lr=&safe=off&q=e%5E%28-i*pi%29%2B1&btnG=Search

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Request for Question Clarification by mathtalk-ga on 13 Jan 2005 07:50 PST

Hi, drkay4414-ga:

I don't grasp what Euler's formula, beautiful as it is, has to do with
the intelligent design hypothesis, but there is an ongoing discussion
of whether mathematical "truths" are discovered or invented.

In any case if it would be satisfactory, I could give a bit of
historical background to Euler and how he discovered his formula.  As
ticbol-ga comments, the particular equation you ask about is a special
case of the relationship between the exponential function (for complex
values) and the trigonometric functions sine and cosine.

regards, mathtalk-ga

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Clarification of Question by drkay4414-ga on 14 Jan 2005 18:01 PST

yes, that would be fine, thanks

(e^(-i * pi)) + 1 = 0

Let us play with that and convert it into a more familiar equation.

(e^(-i * pi)) + 1 = 0
1/[e^(i*pi) +1 = 0
Multiply both sides by e^(i*pi),
1 +e^(i*pi) = 0
Or,
e^(i*pi) +1 = 0  ----***

That is the special form of the Euler's Formula. 
It shows the relationship of 0, 1, pi, e, and i in one equation.

The general form of the euler's formula is
e^(i*x) = cos(x) +i*sin(x)

So if x = pi,
e^(i*pi) = cos(pi) +i*sin(pi)
e^(i*pi) = -1 +0
e^(i*pi) +1 = 0  ----***

For the story, intelligent design, whatever, behind that formula, go
search Google for euler's formula.

If we look at the Taylor Series Expansion for the exp function, we see it is:

exp (x) = 1 + x/1! + x^2/2! + x^3/3! + x^4/4! + ...

If we look at the expansion for the sin and cos functions, we see it is:

sin (x) = x - x^3/3! + x^5/5! - x^7/7! + ...

cos (x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...

Now, add the cos and sin terms

You get...

cos (x) + sin(x) = 1 + x - x^2/2! - x^3/3! ...

Substituting x = (i*pi), you get...

cos (i*pi) + sin(i*pi) = 1 + i*pi - (i*pi)^2/2! - (i*pi)^3/3! ...

But wait!  i^1 = i
           i^2 = -1
           i^3 = -i
           i^4 = +1 and so on...

So, cos (i*pi) + sin (i*pi) = 1 + i (pi) + pi^2/2 + i (pi^3)/3! + ...

Note all the sin terms from the Taylor expansion have a factor of i in them.

Now, note that this is almost exactly identical to the above expansion
of the exp term.  The only difference is those pesky i terms in the
odd powers, which are in the sin terms.

Well, we can just factor those out, and get...

exp (i*x) = cos (x) + i * sin(x), which was already pointed out by ticobal.

Doing the actual math, we find that

cos (pi) = -1
i* sin (pi) = 0

cos (pi) + i* sin(pi) = -1
exp (i*pi) = -1

Note that cos (-pi) = cos (pi) = -1, so it doesn't matter if you have
exp (i*pi) or exp (-i*pi) -- you get the same answer.

Hope that helped.

-touf

In response to whether this is Divine or defined (haha), in my
opinion, probably a bit of both.  Most of it has to do with
definitions and the like, but I do believe that math is Divinely
inspired, as its perfection cannot of human design.