On 2/17/2013 7:56 AM, Bruno Marchal wrote:


On 13 Feb 2013, at 04:29, socra...@bezeqint.net wrote:

After proving Euler's identity during a lecture, Benjamin Peirce,
a noted American 19th-century philosopher, mathematician,
and professor at Harvard University, stated that
"it is absolutely paradoxical; we cannot understand it,
and we don't know what it means, but we have proved it,
and therefore we know it must be the truth."
#
Stanford University mathematics professor Keith Devlin said,
"Like a Shakespearean sonnet that captures the very essence
of love, or a painting that brings out the beauty of the human
form that is far more than just skin deep, Euler's Equation reaches
down into the very depths of existence."

http://en.wikipedia.org/wiki/Euler's_identity
=====..

"it is absolutely paradoxical; we cannot understand it,
and we don't know what it means, .  . . . .’
. . .  but . . .
‘ Euler's Equation reaches down into the very depths of existence."
===..

Yes. Euler identity is wonderful.

It amazes me also that it makes the square of any complex number into a (non normalized) gaussian:
(e^ix)^2 = e^(-x^2)

?? (e^ix)^2 = e^(2ix)

Brent


I love also Euler even deeper identity relating the square of the integers and the prime numbers:

Sum from n = 1 to infinity of 1/n^s = Product on all primes p of (1/(1- 1/p^s). This led Riemann to the deeper of all open problem in math
(Riemann hypothesis).

Ramanujan found quite amazing number relations. Some are so deep that they link gravitation, quantum computing, prime numbers, string theory and the arithmetic of the integers, all given a key role to the number 24.

Jacobi found amazing relations too, involving 24.

Math is full of surprising relations. That's a reason, I think, to believe in their objectivity or 3p-independence.

Bruno

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