# S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Thu, 20 Jun 2013 03:22:27 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 2 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Concise and Precise Storage of (any) NumberPosted: Thu, 22 Oct 2009 17:55:12 UTC
 Member

Joined: Wed, 23 Aug 2006 18:17:18 UTC
Posts: 34

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).

//Donar

Top

 Post subject: Posted: Thu, 29 Oct 2009 14:25:54 UTC
 Senior Member

Joined: Wed, 22 Dec 2004 01:59:38 UTC
Posts: 74

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.

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 2 posts ]

 All times are UTC [ DST ]

#### Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous