Date: Apr 9, 2013 12:16 PM
Subject: Fractional Digits in Number theory
I was going to post this in Wikipedia, but they don't take original research. So I'm posting it here since I don't know where else to go:
A fractional digit number system is similar to a binary digit number system in that it is used to represent numbers in forms different from common use. With a fractional digit number system, digits normally written as whole numbers can be written as fractions. Parenthesis are often used to distinguish a fractional digit within a number. Fractional digits involve both a base and a step (sometimes called a degree). The step is the finite quantity a number must increase for a digit to change to another digit, and the base is the quantity of steps required for an additional digit to be added to the number. The higher a base is, the more unique characters are required to represent a digit in a number. The base must always be a larger quantity than the step and the step must divide the base evenly. Numbers in a fractional digit number system can be represented by a base that's between zero and one (a fractional base) provided that the step divides the base evenly.
Counting in the Fractional Digit Number System
The first thing to do in any kind of counting is to choose what amount to count by. Counting can be done by twos, threes, hundreds, or even halves. The amount chosen to count by would be the step. Next a base must be chosen. When the number of steps in a digit increases enough to equal the quantity of the base, an additional digit must be added for counting to continue. For fractional digits to occur, the step must be less than one. There are limitless ways to count using fractional digits. Below is one example.
Counting with base 3/2 step 1/2 gives the following sequence, where each number is separated by a comma, and certain digits within a number are separated by parenthesis:
0, 1/2, 1, (1/2)0, (1/2)(1/2), (1/2)1, 10, 1(1/2), 11, (1/2)00, (1/2)0(1/2), (1/2)01, (1/2)(1/2)0, (1/2)(1/2)(1/2), (1/2)(1/2)1, (1/2)10, (1/2)1(1/2), (1/2)11, 100...
This sequence translates to the more familiar quantities:
0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9....
So the number two in the familiar base ten step one system is written in the base 3/2 step 1/2 system as (1/2)(1/2). And the number four would translate to 11. Four and a half (4.5) would be represented as (1/2)00, and nine would be 100.
Conversion to Base 10
A number with base n and step m that has digits A,B,C,D,E, etc. where the digits progress from right to left with A being the rightmost digit, can be converted to a familiar base ten number by this formula:
A*(n/m)^0 + B*(n/m)^1+ C*(n/m)^2+ D*(n/m)^3......
Distinctions Between Mixed Fractions, Improper Fractions, and Fractional Digits in a Fractional Digit Number System
Fractional digits in a fractional digit number system should not be confused with traditional fractional notations such as mixed and improper fractions. Unlike traditional fractional notations, numbers in a fractional digit number system can have multiple digits with a consistent base and step throughout the entire number. Mixed fractions have a whole number digit that has a different base and step than the fractional digit, and improper fractions cannot go beyond a single fractional digit.
To date, there are no known applications for fractional digits, as they are an exercise in pure mathematics. It has been suggested that they have potential application in the field of encryption.