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/
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