Non-Integer Bases for Number Systems

We use base 10, a place value system, to denote our numbers. It's a brilliant system: a single number used as a base, integers used as exponents, a place location to denote the exponent. It makes writing numbers, comparing numbers, and doing arithmetic, all very systematic and easy.

But recognizing what makes it work, can lead us to look for alternatives. What would we get if we didn't use a single counting number base? What would happen if place meant something other than exponent value? How could we represent numbers without using place value?

Here we will present other possibilities for numbering systems. Your challenge is to take these ideas further. Can you complete the analysis? How can these ideas be put to real use?

February 2010

 

Part 1: Non-Integers as Bases (The Golden Mean)

It is easy to organize numbers using place value based on whole numbers. But what happens if we don't limit ourselves to whole numbers? Can we create a consistent numbering system using an irrational number as the base? Here's the start for using the golden mean as a number system base.

Introduction: The Golden Mean & Fibonacci & Lucas Series

The golden mean (Phi = 1.680339....) is well known for its beauty, as well as its relationship to the Fibonacci series and the Lucas series. As used in the Lucas series the golden mean has a fascinating quality. Adding or subtracting any power of the golden mean to its inverse will make the irrational tail drop off leaving just a whole number. This chart shows how that characteristic relates to the Lucas series.

Exponent, X Phi^X Phi^(-X) Phi^X + Phi^(-X) Phi^X - Phi^(-X) Lucas Series

1

1.618033989
0.618033989
1
1
2
2.618033989
0.381966011
3
3
3
4.236067977
0.236067977
4
4
4
6.854101966
0.145898034
7
7
5
11.09016994
0.090169944
11
11

Wouldn't this simple property suggest that Phi can be used as a base for a numbering system? Here's what you get when you try.

Base 10 Base Phi
6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6
1 1.  
2 1 0. 0 1
3 1 0 0. 0 1
4 1 0 1. 0 1
5 1 0 0 0. 1 0 0 1
6 1 0 1 0. 0 0 0 1
7 1 0 0 0 0. 0 0 0 1
8 1 0 0 0 1. 0 0 0 1
9 1 0 0 1 0. 0 1 0 1
10 1 0 1 0 0. 0 1 0 1
11 1 0 1 0 1. 0 1 0 1
12 1 0 0 0 0 0. 1 0 1 0 0 1
13 1 0 0 0 1 0. 0 0 1 0 0 1
14 1 0 0 1 0 0. 0 0 1 0 0 1
15 1 0 0 1 0 1. 0 0 1 0 0 1
16 1 0 1 0 0 0. 1 0 0 0 0 1
17 1 0 1 0 1 0. 0 0 0 0 0 1
18 1 0 0 0 0 0 0. 0 0 0 0 0 1

To the left, we show 1 through 18 written in base phi. Notice that every number requires a decimal. However notice that integers can be written with a terminating decimal. The number of digits after the decimal point is about the same as the number of digits before the decimal point. (Technically that should be called the golden point, since decimal means ten.) Also notice that no number contains the sequence 11. That's because 11 results in a carry. 11 = 100. This is analogous to 0.999 repeating = 1 in base 10.

Now the challenge is can you determine how to do arithmetic in base phi? Can you determine whether this, or any other irrational base can have practical uses? How would you write pi or e in base phi? Is there an irrational base which makes special numbers like pi, e, and phi easy to write?

Note for those who have studied functions and calculus:

We can represent sin(x) and cos(x) using Eulers formulas. We can extend this idea to make the Lucas series fill complex space:

  • Sin(x) = (e^ix + e^-ix )/2
  • Cos(x) = (e^ix - e^-ix)/2i
  • Lucas(x) = phi^ix + phi^-ix

This model might add a variant for representing complex numbers using the golden number as a base.

Part 2: Base is Not a Real Number

The footnote above leads us to the amusing problem of using i = sqrt(-1) as our base.

  • i^0 = 1
  • i^1 = i
  • i^2 = -1
  • i^3 = -1
  • i^n = i^(n-4)

This pattern would seem to limit our place value system to 4 places. However, it would add the possibility of representing negative numbers and complex numbers simply using place value. Technically, the form for complex numbers, a+bi , does work as this place value system.

 

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Part 3: Bases Not Using Powers as Positional Notation (Factorial)

With decimal, binary, and hexidecimal, place value represents the exponent of the base. But this is not a necessity. Place can represent other distinct patterns. For example, place could represent the a factorial, or primorial value. Remember factorial 5! = 5*4*3*2*1 = 120. Less well know primorial represents the product of primes, eg: primorial(4) = 2*3*5*7 = 210.

Index Factorial Primorial
1 1 1 *redefined for this purpose
2 2 2
3 6 6
4 24 30
5 120 210

Since both factorial and primorial create a distinct well structured pattern, both may be used as a place value numbering system. The charts below demonstrate how roll over will occur for the first four positions in both factorial base and primorial base.

Factorial Base 10
120 24 6 2 1
1 1
1 0 2
2 1 5
1 0 0 6
3 2 1 23
1 0 0 0 24
4 3 2 1 119
1 0 0 0 0 120
Primorial Base 10
210 30 6 2 1
1 1
1 0 2
2 1 5
1 0 0 6
4 2 1 29
1 0 0 0 30
6 4 2 1 209
1 0 0 0 0 210

The patterns are simple and easy to follow. But are they useful? Will any operations of arithmetic, or number theory work any more efficiently when representing numbers using either of these base systems?

 
 

Part 4: Numbers represented where value of place changes

The possibility of changing the value of place is already found in the representation of numbers. Scientific and Engineering notation each effectively change the value of place by changing the exponent. This same idea is used as the basis of floating point notation used in computer applications.

Here we will add a variant. All numbers may be represented in a simple form: ax^2 + bx + c. Students who have had algebra will recognize this as the quadratic form, and know how to factor it. With this we may represent every number with just 4 places: a, b, c, x. We may reduce this to just three places by requiring a to be 1. Since a is always 1, we don't have to show it.

Examples
Base 10 3 Digit Base: x, b, c meaning
49 7, 0, 0 49 is 7^2
74 8, 1, 2 74 = 8^2 + 8 + 2
165 11, 4, 0 165 = 11^2 + 4*11 + 0

A value to this system is that many composites will be easy to recognize by their factorable form. eg: any sequence ending in 00, 21, or 44 will be a perfect square. It will be possible to represent every composite according to its factorable form. However, it will not prove efficient to do so.

Can you find a way to make this base useful for a specific mathematical purpose?

Summation:

Above we have demonstrated that there are many ways to represent numbers by modifying our place value system. We can change the value of the base, or the value of the place. We are not limited to using counting numbers or integers. It is interesting to know, but can it be made useful? This is a challenge for another generations of mathematical hobbyists.

 

 

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