On 08 Jul 2013, at 23:22, Johnathan Corgan wrote:
On 07/08/2013 02:16 PM, Jason Resch wrote:
This one is very interesting, but the fact that Pi was a poor
choice for
the constant makes the equation considerably more ugly than it should
be. There is a growing movement to usurp the number Pi with the much
more important constant "2*Pi"
(see: http://www.math.utah.edu/~palais/pi.html ). If we call that
new
number tau (t). Then Euler's identity becomes:
e^(t * i) = 1
I think part of the appeal of the original formulation is realizing
that
the result of an exponentiation of a positive number can be a negative
number. While this is unremarkable with complex exponents, many
people
are only used to seeing real (or even just integer) exponents.
I like and often give the following exercise: compute i^i. Is it real
or imaginary?
Since sometimes my most amazing result in math is the Turing
universality of the diophantine polynomials.
My favorite simple result is the irrationality of sqr(2). A good
exercise, using the fundamental theorem of arithmetic (existence and
uniqueness of decomposition of numbers in prime factors) generalizes
this for sqr(n) for any n not being a square. A result often
attributed to Theaetetus.
Of course the existence of universal numbers is also an amazing,
stunning results, especially if you know how weak are the pretense of
"universality" for mathematical notions. This needs Church thesis,
which I consider as the most amazing thesis in cognitive science.
Bruno
Johnathan
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