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Negative Bases

Date: 05/13/2002 at 14:53:48
From: S. James
Subject: Base -2

Dear Sir,

I recently ran across a problem where it was desirable to attempt to 
express a number in base -2 (minus two) format! Just what values might 
such a base allow? 0 and -1?  What would 16 (base 10) then yield as 
the result of base -2?


S. James

Date: 05/13/2002 at 16:12:03
From: Doctor Peterson
Subject: Re: Nth Root, Base -2

Hi, S.

See this page in our archives, which I found by searching for the 
phrase "negative bases":

You use digits 0 and 1 for base -2, though I suppose 0 and -1 could 
be used just as well. To convert, you use the same method as for 
positive bases, but you have to be careful how you think about 
remainders. Dividing repeatedly by the base and taking the positive 
remainders as digits (starting at the right), we convert 10 to base 
-2 this way:

    10 = -2*-5 + 0

    -5 = -2*3  + 1  (the quotient is 3 rather than 2 
                     so the remainder is positive)
     3 = -2*-1 + 1

    -1 = -2*1  + 1  (again, the quotient is NOT 0!)

     1 = -2*0  + 1

So 10 (base 10) = 11110 (base -2).

I'll let you work out 16 the same way.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Negative Numbers
High School Number Theory

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