George Sassoon ([EMAIL PROTECTED]) wrote (regarding base-e arithmetic):
>Maybe pi could be expressed exactly in such a
>system. After all, e^(pi*i) = -1.
Indeed, it follows that pi=-i*log(-1). But now one has the problem that
one has defined one transcendental number in terms of another, and is
thus no closer to writing pi in closed form in terms of whole numbers.
I prefer to use base i, i.e. to no longer restrict oneself to the reals.
In the complex (x,y) plane (i.e. every complex number x + i*y is viewed
as the point (x,y) in a Cartesian-style coordinate system), one has that
i = (0,1), which involves just integers. One also has the nifty identity
i^i = -pi/2
i.e. the imaginary number i, raised to itself, gives - voila!- a real
result involving pi in a simple way.
(Proof: log(i^i) = i*log(i). But in complex polar form, i = e^(i*pi/2),
so log(i) = i*pi/2, from which the result quickly follows.)
Thus, pi=-2*i^i.
>This led to a discussion as to whether or not it is possible to have a number
>system based on a non-integer base.
Your example with base e shows that indeed, one can. On the other hand,
i (as in my example) is not an eligible base for a number system, since
it has complex modulus one (i.e. one can raise i to any power one likes
but never get off the unit circle.) But one can simply use, say, b = 2*i
as the base, and then one has the identity pi = -2*(b/2)^(b/2).
And now, at this point of our seemingly off-topic foray into the wonders
of pi, e and the complex numbers, we suddenly are back on-topic. Here's
why: Richard Crandall, who devised DWT algorithm at the heart of the
Mersenne testing codes all of use in one form or another, refers to it,
in words, as the "irrational-base discrete weighted transform" (IBDWT).
I've quibbled with him about his choice of wording here, since it seems
to cause much confusion among folks who have not actually implemented
such an algorithm computationally. The source of the confusion is that,
even though representing a prime-exponent Mersenne number 2^p-1 using
N binary words is in some sense mathematically equivalent to using an
irrational-base number system, in practice one uses a mixture of base
2^s and 2*2^s, where s = floor(p/N), i.e. is an integer. Thus, one
doesn't actually use irrational bases anywhere in the actual algorithm.
-Ernst
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