Autonemesis

1 + 1 = 3

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(for very large values of 1)

Bloop

e^(i*pi)=-1

and move the 1 to the left hand side to get:

(e^(i*Pi))+1 = 0

and you have , explicitly, five numbers of grave importance mixed into the same equation 0,1,e,Pi and i

Abacus:

(replace theta with Pi)

So that is , by far, the most beautiful equation in all of Maths if you ask me. (not to say that in any way have seen them all, some in this thread are unfamiliar to me)

2. a^n+b^n != c^n for n > 2 where n is a natural number. (Fermats last theorem solved by Andrew Wiles not so long ago, mid 90's I think)

If n = 2 you get the Pythagorean theorem we all recognize.

3. Fundamental theorem of Calculus (which comes in the form of an equation)

[ September 25, 2002: Message edited by: Bloop ]</p>

Christopher Lord

What is the 'e' in e^(i*pi)=-1? is it in the same league as 0, 1, i and pi?

Jesse

Christopher Lord:
What is the 'e' in e^(i*pi)=-1? is it in the same league as 0, 1, i and pi?

Yup, it's one of the "biggies"--like pi it shows up in a lot of places, but it's probably most well known for being the basis of the <a href="http://mathworld.wolfram.com/NaturalLogarithm.html" target="_blank">natural logarithm</a>, which has a simpler derivative than logarithms in other bases. Here's some <a href="http://mathworld.wolfram.com/e.html" target="_blank">more info</a> on e, and here's an <a href="http://www.math.toronto.edu/mathnet/questionCorner/epii.html" target="_blank">explanation</a> of the equation e^(pi*i) = -1.

Answerer

According to Euler's equation, the exponential function, e^(pi*i) = -1 = cos(pi) + isin(pi). Anyway, I, myself, prefer Laplace and Poisson Equations.

RufusAtticus

<ol type="1">[*]p' = p^2+p(1-p)=p[*]p' =(w11*p^2+w12*p*(1-p))/(w11*p^2+w12*2p(1-p)+w22*(1-p)^2)[*]q* = (u/s)^0.5[/list=a]

wade-w

The "e" in these equtions is Euler's number. It is the base of what is called the exponential function, e^x, which has the fascinating property that the derivitive of e^x is e^x.

The inverse function of e^x, of course, is the natural logarithm, which is the integral of 1/x, and has other neat properties, as Jesse points out. So yes, it is one of the biggies. Oh, and like pi, its an irrational number.

Abacus

Some others that really fascinate me are the Taylor series expansions of e^x, cos(x), and sin(x):

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

I wish I knew how to display them with the sigma summation notation.

Edited to fix sign errors.

[ September 25, 2002: Message edited by: Abacus ]</p>

Feather

My favorite three:

1. Newton's formulation of gravity (in my opinion the single most important equation in all physics--it, or rather the process of determining it, started the revolution that continues today).

2. Maxwell's Equations (okay, okay--there are four of them here, but they're all "one" if you know what I mean), because they were the first step down the path of unification.

3. Euler-Lagrange's calculus of variations (perhaps the single most important mathematical contribution to physics in history, for its application ranges from the mundane "ball in a bowl" to the complex nuances of electrons passing through slits).

Friar Bellows

That's not really an equation; it's the definition of i.

Damned engineer.

And a, b and c are natural numbers.

Looking at these expansions you can figure out how the equation e^(i * x) = cos(x) + i * sin(x) comes about. It was by looking at Taylor expansions that Euler figured out the answer to my 3rd equation.