Re: Still is maybe offtopic maybe definitely
--- andy_cranston at LWSYS.FSNET.CO.UK wrote:
>
> Ok *decimal* fractional numbers are difficult to represent in binary two's
> complement (integer and mantissia?) with anywhere near 100% accuracy. 0.25
> is fine but a third is no go.
>
> Now PI *can't* be represented in a finite series of decimal (base 10)
> numbers. I suspect that it *can't* be represented in a finite series of
> binary (base 2) numbers either. My theorem:
>
> Can PI be presented in a finite series of fixed
> limit based integers?
>
> As an example can PI be represented in base 36 where 0 is 0, 9 is 9, A is
> 10, b is 11, Y is 35 and Z is 36?
>
> Don't limit your self to small base numbering systems. Base a million and
> one is perfectly fine in math (or as high as you need to go!).
>
> As with all my ideas I'm sure there must be stuff out there that already
> covers or has covered it. Links to it would be most welcome.
Nope. Pi is irrational and transcendental. Irrationality means that there's
no fraction where numerator and denomenator are integers. If there were a way
to do this in one base, it could be done in any base. But, of course, that
doesn't address writing out pi using a decimal point. I guess the real problem
with something like pi, it that there are no repeating patterns (that we've
found so far).
Perhaps even more interesting is the transcendental nature of pi. First, I'll
explain what algebraic numbers are:
Take an equation such as:
a0+a1*x1+a2*x2^2+...+an*xn^n = 0, where ai is a rational number. Any number
which is a root of such an equation is an algebraic number. This includes many
rational and irrational numbers:
-2+x^2=0 ==> x = sqrt(2) ==> sqrt(2) is algebraic.
No such equation exists with pi (or e, to name another famous number) as a
root, and so pi is transcendental. To prove this stuff you need to get into
field theory and abstract algebra, which is a lot of fun, but a bit outside the
scope of this list.
Matt Lewis
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