1. Re: Still is maybe offtopic maybe definitely
- Posted by MichaelANelson at WORLDNET.ATT.NET
Feb 24, 2001
-
Last edited Feb 25, 2001
Andy,
You would need to check a math text for a formal proof, but in answer to
your question: A rational number, that is a number representable as a ratio
of integers (2/3, 7/5, 355/113,10002/343434, 4 (=4/1), etc.) when expressed
as a decimal fraction will take one of two forms:
A. Terminating decimal
example: 1/4 = 0.25 (exact)
B. Repeating decimal
example: 1/3 =0.33333333333 . . . (3's repeat infinitely)
(the repeating group may have one or more digits and need not start
right after the decimal point.
In binary (or any other) base, the same pattern holds, although a
terminating decimal may be a repeating binary. (Because 10 is evenly
divisible by 2, a terminating binary cannot be a repeating decimal.)
Terminating binary: 1/4 = 0.01
Repeating binary: 1/3 = 0.010101010101 . . . (01's repeat infinitely)
Pi, however, is an irrational number, which by definiton can neither repeat
nor terminate in any base. A given number must be either rational or
irrational regardless of base.
-- Mike Nelson
Andy Cranston wrote:
<snip>>
> 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?
>
