S.O.S. Mathematics CyberBoard

Good day, Ladies and Gents

I am writing some tool to deal with precise numbers. This of course implies that I store and manipulate precise numbers. I am looking for a suitable format for storing such numbers concisely in a computer's memory.

This is how far I have come (the symbol "|" being used as "exclusive or"):

Let N be a 4-tupel {s,n,d,f}
where
s is the sign (-1 if neg|0|1 if pos)
n is the numerator
d is the denominator
f are a collection of distinct binary flags being 1 when a certain irrational mathematical factor is present and 0 otherwise, e.g.
The flags ...eipf could indicate ...e, i, pi, phi
Then the binary value ...1010 would indicate the presence of the factors e*pi

Examples of numbers written in such 4-tupels N:
3 = {1,3,1,...0000}
-pi/4 = {-1,1,4,...0010}

Taking this a step further we include roots and powers (using "^" as the exponentiation symbol, and "[...]" to indicate optionality), and N1, N2, N3... to indicate the 4-tupels):
E=N1[^N2[^N3[...[^Nn]]]]

Examples:
(pi/2)^4 = {1,1,2,...0010}^{1,4,1,...0000}
sqrt(3) = {1,3,1,...0000}^{1,1,2,...0000}
(3e)^(i*pi) = {1,3,1,...1000}^{1,1,1,...0110}


Now, if we also allow factors of such exponents E:
E1[*E2[*E3[...[*En]]]]
(needed because E alone could not handle for example 3sqrt(2)e^p )

then I think that I can store every possible number in a concise and *precise* format.

But I am not sure and am looking for your input (mainly looking for a counter-example, actually).

Thanks in advance for your time
//Donar

Some instant comments:

1. Seems "n" belong to Natural Number, "d" belong to Postive Integer. Why "n" can't simply belong to Integer? In such case "s" will be useless.

2. A number is stored as {s,n,d,f}? Seems not! sqrt(3) is stored as a pair of 4-tuples with an extra operator defined in your example. So this number system seems to use sequence of 4-tuples with some operators to represent number.

3. If the flags are eipf, that means all irrational numbers could be expressed in this four irrational numbers in such a way. You need a formal Mathematical proof!

4. Actually, I never saw anyone classify "i" as irrational number before. Any supporting information for your classification?


What I see is: it is just to store Rational Numbers and assume it can use four extra, or may be some finite set of, irrational numbers with some operation sequence to generate all Real (or may be Complex) Number.

Do you think the sequence must be finite?

To me, I can't even sure infinite sequence of such 4-tuples can generate all Real/Complex number.

If you want to create such a number system, you need to do some formal Mathematical proofs to show to people it is working good. Not to ask people to provide counter examples to disprove it for you.