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Computer Representation of Real Numbers

IP.com Disclosure Number: IPCOM000079362D
Original Publication Date: 1973-Jun-01
Included in the Prior Art Database: 2005-Feb-26
Document File: 1 page(s) / 12K

Publishing Venue

IBM

Related People

Hwang, PB: AUTHOR

Abstract

This technique allows rational and irrational numbers to be expressed in exact values by combinations of integers, permitting improved accuracy in digital computer calculations.

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Computer Representation of Real Numbers

This technique allows rational and irrational numbers to be expressed in exact values by combinations of integers, permitting improved accuracy in digital computer calculations.

A software package utilizing the system of real-number representation can be implemented using the following concepts.

Rational numbers, such as X/Y, where X and Y are integers, define a field which is closed under addition, subtraction, multiplication, and division (provided Y is not equal to 0). Therefore, performing these operations with rational numbers by utilizing only integers proceeds as follows: (a/b)+/-(c/d) = (ad +/- bc)/bd, (a/b)x(c/d) = (ac/bd), and (a/b)/(c/d) = (ad)/(bc).

Where a, b, c, and d are all integers. The result of such operations produces a number also of the same field, i.e., having the form X/Y and can be expressed as an ordered pair, i.e., -- (X,Y). For example, the rational number 1/3 is expressed as (1,3) as opposed to 0.33333... conventionally used.

Irrational numbers, also forming closed field may also be used.

An example is, the pythagorean type, which is defined as the ratio
(r) of the length of the diagonal (d) of a square and the length of its side (s), i.e., r=d/s, must satisfy the equation d/2/= (rs)/2/= s/2/+s/2/. Therefore, the operation of addition, subtraction, multiplication, and division may be performed as follows:

Let A=(a,b) and B=(c,d); where A and B are pythagorean type irrational numbers and a, b, c,...