Continued Fractions
In one of my searches, I stumbled across Continued Fractions. I have always known they existed, but had very little or no use for them. The ones that I will discuss here are Continued Fraction representations for the Square Root of Integers. Another powerful statement for all of those Transcendental Cantor Infinity lovers is:
A decimal number must be able to have a Continued Fraction representation or it is not a legitimate decimal number.
In other words, one cannot just place a decimal point in front of a bunch of integers and say its a decimal number it must have a Continued Fraction representation.
There are two types of simple Continued Fractions, finite and infinite. A simple Continued Fraction has all of its partial denominators as integers. They take the form of:
The Finite Continued Fraction represents a rational number and can be calculated using the Euclidean Algorithm. The Infinite Continued Fraction represents an irrational number. The representation is as follows: [a0;a1,a2,a3] This is a representation of a finite Continued Fraction. For an Infinite Continued Fraction a bar would be over the repeating portion of the fraction or in the case of web pages ,etc. a set of three dots can be used to show that everything from the semicolon on repeats: [a0;a1,a2,a3...]
The Square Root representation of an integer is always infinite and follows some rules. alast is always 2*a0. and the repeating part of the Continued Fraction minus the last term is symmetrical: [a0;a1,a2,a3...a3,a2,a1,2*a0...]
The Square Root of 2 equals [1;2...]
The Square Root of 7 equals [2;1,1,1,4...]
The Square Root of 94 equals [9;1,2,3,1,1,5,1,8,1,5,1,1,3,2,1,18..]
more to come...
Last updated: 8/18/2001
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