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### Negative Bases

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

Are there such things as negative bases?
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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/
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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

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

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

- Doctor Steve, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Negative Numbers
High School Number Theory
Middle School Negative Numbers