Abacus

Friar,

You quoted my equations before I had a chance to correct my mistakes. Every other term in the expansions for cos(x) and sin(x) should have a minus sign.

Abacus

Also, whenever you hear talk of exponential growth or decay, you can be pretty sure that "e" is involved. For example, take the following function:

y(t) = y(0)*e^(k*t)

k is a constant, t is time (or something else), and y(0) is the value of our function at t=0.

If k is positive, this function is an equation for exponential growth. Some real world examples of exponential growth are population growth or interest accumulating on a savings account (for the latter you would like as large a k value as possible).

If k is negative, then the function models exponential decay. An example of this would be the half-life decay of radioactive atoms.

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

Scrambles

I vote for:

1 + 2^(-s) + 3^(-s) + ... = (1-2^(-s))(1-3^(-s))(1-5^(-s))(1-7^(-s))...

for Re(s) &gt; 1. Euler kicked ass on that one.

Ronald Begg

wade-w

On further reflection, I want to change my entries...

1. d(e^x)/dx = e^x

2. ~(A & B) = ~A or ~B

3. The fundamental Theorem of Calculus

and e^(i*pi) + 1 = 0 ties

[ September 25, 2002: Message edited by: wade-w ]</p>

trientalis

This may be an odd question but in what way is e^iPi + 1 = 0 a "beautiful" equation? I've come across that equation before in "The Art of Mathematics" and the author's explanation didn't help. If any of the mathematically inclined out there could have a stab at explaining why the above equation is "beautiful" and not just interesting, I'd be ever so grateful.
And another thing: is the equation useful in any way, much as F = ma is useful? Is there a application for that equation? Or is it just a "beautiful" result?

Sorry to be overly, well, pathetically earnest. I'm not a mathematician but I am a lover of beauty. The beauty of mathematics is something I have yet to experience, much less understand. If you are a mathematician, I envy you for the beauty you see that I do not.

&lt;End of whine&gt;

wade-w

Its basically a matter if elegance, trientalis. in this case, e^(i*pi) + 1 = 0 shows a very simple relationship between several fundamentally important numbers. And the proof shows that the exponential function is closely related to the circular functions.

Bloop

Friar Bellows:
And a, b and c are natural numbers.

Me:

Absolutely. My bad. Wouldn't be much of a challenge otherwise

Answerer

The dirac delta function is also another mathematical function with special properties. At a single point in the graph(most often x=0), the delta function, given by sigma, will tend towards infinity but however, when intergrating the delta function over the entire plane, the solution will definitely be the value of 1. Therefore, I feel that dirac delta can be included in the top three equations.

Bloop

For me it is the simplicty of the whole thing. It is short and contains these five immensely important numbers and nothing else. Elegance.

Maybe the equation isn't the most useful. But no one can deny the usefulness of complex numbers (i), e, Pi , 1 and 0.


Maybe you have to be into maths to see the beauty. But that may be a little like saying that you must be a christian to really appreciate the (religious) music of Bach. Which I do not agree with...

Also if you personally fail to see any beauty in mathematical/physical equations that doesn't mean anything more than just that. After all we are all different. I have a really hard time liking the modern "classical" music that is being written today. To me it sounds like collections of random sounds. I'm certain Bach would laugh his ass off

Bialar Crais

Shouldn't that be: i^2 = -1?

I vaguely remember my math teacher trying to bite my head of for suggesting that (-1)^0.5 = i...