The matematical constant e (2.718...)
The definitions of PI and e are simple, it''s why they show up all over the place instead of the other infinity of similar numbers I was questioning.
in you e base, or pi base, how do you reckon the digits will look like. will there be a finite number of digits? Can you even talk about a base? I don''t think so.
quote:Original post by d000hg
The definitions of PI and e are simple, it's why they show up all over the place instead of the other infinity of similar numbers I was questioning.
One could ask why we have any universal constants, like e, pi, G, h_bar, etc. It comes down to the fundamental nature of mathematics/physics in our universe. e is simply the number that has the properties listed above: e.g. d/du(eu) = eu. It is possible to formulate formal systems with other wonderful tricks like this...
One thing to be wary of... e isn't that spectacular really... for example, any logarithmic equation can be expressed as a different logarithmic equation in a new base. E.g:
x = loga(b)can be written as logc(b)x = ------- logc(a) Which means that any special formula in which loge appears could be written with say, log10, or how about logpi.
If you want to consider something really funky, I learned this the other day... given the Halting probability of a self-delimiting Universal Turing machine, Omega, which in base two has the form 0.w1w2w3... (where the wi are binary digits, any binary string of length N appears with probability 1/N in the Halting probability. That's FUNKY!!!
That means that if I use 8 bits for each character in my name (Timkin), then my name appears in any halting probability with probability 1/48. WOOT! I'm famous... at least I expect to be for every 48th Turning machine!!! (rofl)
Cheers,
Timkin
[edited by - Timkin on June 9, 2004 11:15:34 AM]
uber_n00b: You misunderstood me. I was trying to demonstrate a possible line of proof that pi can''t be the root of a polynomial with integer coefficients.
Any rational number can be written in this form, as well as many others, numbers that cannot(eg pi) are transcendental.
http://mathworld.wolfram.com/TranscendentalNumber.html
Any rational number can be written in this form, as well as many others, numbers that cannot(eg pi) are transcendental.
http://mathworld.wolfram.com/TranscendentalNumber.html
quote:Original post by janos
in you e base, or pi base, how do you reckon the digits will look like. will there be a finite number of digits? Can you even talk about a base? I don''t think so.
Take a positive number. Substract the largest power of e that you can from it. Rinse. Repeat.
That gives you a representation in base e, although I don''t know how interesting it is.
For example, 3 is e+2*e^-2+e^-5+..., so you can write it as 1.02001... in base e.
A more interesting base to use is (1+sqrt(5))/2. You get
3 = 100.01
4 = 101.01
7 = 1010.101
A number is an integer if and only if its representation in base (1+sqrt(5))/2 is symmetrical around the digit that represents "1".
I think someone should mention phi right about now just to stir things up ;-). The number of god
->alvaro,
ok, you can have a good integer decompostions of a number considering a non integer base as you just said. Maybe I talked too fast when I said it was totally not a base, but some strong properties do not hold in the sense that such things as "mechanical" addition can require an infinite amount of time. In base e, 1+2 gives :
10.2001 and continues for long.
btw, you golden number propery is rather cool.
ok, you can have a good integer decompostions of a number considering a non integer base as you just said. Maybe I talked too fast when I said it was totally not a base, but some strong properties do not hold in the sense that such things as "mechanical" addition can require an infinite amount of time. In base e, 1+2 gives :
10.2001 and continues for long.
btw, you golden number propery is rather cool.
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