Freethought & Rationalism Archive

ohwilleke · 09-26-2002, 03:12 PM

Quote:

Originally posted by 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.

<End of whine>

It is beautiful because there is no good reason for e, i, and Pi to be related in such a simple way. Each of these constants arise independently in mathematics. Pi flows from the ratio of the circumfrence of a cirlce and its diameter. The number i, flows from the square root function and a gap in the number system. The number e is defined as a rate of growth which is growing at the rate of growth. Why should these numbers have a simple relationship? Yet, this equation indicates that the nubmers are in fact deeply connected in a profoundly simple way. The exact number Pi can be used to derive the exact number e and visa versa.

Beauty in mathematics is fundamentally about the existence of simple, but non-obvious relationships.

ohwilleke · 09-26-2002, 03:20 PM

Quote:

Originally posted by 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.

Ooh! I forgot those. My signature quirk taking first year physics at a local college while I was in high school, was that I did all my calculation on a simple +, -, *, divide calculator and figured out all of the sines, cosines and exponentials in the class using a Taylor's series approximation and still managed to finish before my peers with scientific calculators. I wrote the Taylor's series on a sticker on the back of the calculator. I carried a one page photocopy of a log table to do logs.

Secular Elation · 09-26-2002, 03:21 PM

1)The Fundamental Theorem of Calculus (okay, it's a theorem, but still neat)

2)The quadratic formula. [(-b +or- sqrt(b^2-4ac)]/2a

3)C/d = pi

4)(1+x)^(1/x)= e (Euler's number) <when x approaches zero)

[ September 26, 2002: Message edited by: Secular Elation ]

wade-w · 09-26-2002, 07:14 PM

Quote:

Originally posted by Secular Elation:
1)The Fundamental Theorem of Calculus (okay, it's a theorem, but still neat)

Its also an equation, so it definitely qualifies, imho.

Abacus · 09-26-2002, 07:19 PM

Quote:

Originally posted by ohwilleke:


Ooh! I forgot those. My signature quirk taking first year physics at a local college while I was in high school, was that I did all my calculation on a simple +, -, *, divide calculator and figured out all of the sines, cosines and exponentials in the class using a Taylor's series approximation and still managed to finish before my peers with scientific calculators. I wrote the Taylor's series on a sticker on the back of the calculator. I carried a one page photocopy of a log table to do logs.

Awesome dude! You don't know how glad I am to hear that. When I was in college, I actually had classmates in engineering that had to use their $100+ graphing calculators to do trivial arithmetic such as 22+13.

I kid you not! Bleh!

Abacus · 09-26-2002, 07:24 PM

Another good one is the wave equation. You can see it <a href="http://mathworld.wolfram.com/WaveEquation.html" target="_blank">here.</a>

Trying to type it out here won't do it justice. The second derivatives and the formatting just don't mix.

09-26-2002, 03:21 PM	#43
Secular Elation Veteran Member Join Date: Apr 2001 Location: California Posts: 6,196	1)The Fundamental Theorem of Calculus (okay, it's a theorem, but still neat) 2)The quadratic formula. [(-b +or- sqrt(b^2-4ac)]/2a 3)C/d = pi 4)(1+x)^(1/x)= e (Euler's number) <when x approaches zero) [ September 26, 2002: Message edited by: Secular Elation ]</p>

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09-26-2002, 07:24 PM	#46
Abacus Veteran Member Join Date: Mar 2002 Location: South Dakota Posts: 2,214	Another good one is the wave equation. You can see it <a href="http://mathworld.wolfram.com/WaveEquation.html" target="_blank">here.</a> Trying to type it out here won't do it justice. The second derivatives and the formatting just don't mix.

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