Negative Bases

Date: 06/10/99 at 10:26:13
From: Ms. Wile's Class
Subject: Negative bases

Are there such things as negative bases?

Date: 06/10/99 at 12:31:00
From: Doctor Rob
Subject: Negative bases

Yes, there are negative bases. They aren't used much, but they are 
quite interesting. They allow you to represent both positive and 
negative numbers with only positive digits. For example, if the base 
is -2, then you have the following equalities:

   Base 10     Base -2
     -11       110101  
     -10         1010
      -9         1011
      -8         1000
      -7         1001
      -6         1110
      -5         1111
      -4         1100
      -3         1101
      -2           10
      -1           11
       0            0
       1            1
       2          110
       3          111
       4          100
       5          101
       6        11010
       7        11011
       8        11000
       9        11001
      10        11110
      11        11111
      12        11100
      13        11101
      14        10010
      15        10011
      16        10000

and so on. Negative numbers have an even number of digits, and 
positive ones an odd number of digits.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   

Date: 06/10/99 at 11:32:40
From: Doctor Steve
Subject: Re: Negative bases

Hello,

I don't know how often negative bases are used or the first 
mathematician to use them, but one can certainly use negative numbers 
as bases in an exponential expression. You can answer the question, 
what is (-3) squared? Or, what is (-3) cubed? 

I imagine you were thinking of our number system, where the one that 
most of us use on a daily basis is in base ten. The first placeholder 
or digit is the "ones" which is defined by 10 to the zero power. The 
second place is the tens, defined by ten to the first power. The third 
is the hundreds, defined by ten to the second power (or ten squared). 
Etc.

Instead of using ten to define the base for the number system, one can 
use other numbers, negative or positive. So if -3 was our base, then 
the first place would still be the "ones" place. The second would be 
the "negative threes." The third would be the "nines" (positive). Etc.

One challenge, then, is can we represent base ten numbers in base -3? 
Let's take a number like 7.

In base -3 I might write it like 111. In other words 1 (1) + 1 (-3) + 
1 (9) = 7 in base ten. Is there only one way to write each number in 
base ten in base -3? Can all of the base ten numbers be written in 
base -3? Try a few more conversions. If the answer is no to either of 
these questions, then negative bases may prove to be a problem and not 
so useful. On the other hand, maybe negative bases make it easy to do 
certain kinds of calculations. I don't know much more at this point 
but it would be fun to search around for more information.

Here's a problem on the web that could have a negative base for a 
solution:

   University of Idaho Internet Math Challenge
  http://www.uidaho.edu/LS/Math/imc/1997/p97-06.html   

Here's a calculator from the University of Arkansas Community College 
of Hope that lets you play with bases:

  http://www.uacch.cc.ar.us/science/numnotation.htm    

Here's an article by Keith Devlin on bases in number systems that 
plays with a "negadecimal" system.

  http://www.maa.org/devlin/devlinfeb.html   

- Doctor Steve, The Math Forum
  http://mathforum.org/dr.math/