Find an algorithm for writing any positive integer in negabinary, i.e., as a sum of powers of -2.

For example,

7 = 11011 = (-2)^4 + (-2)^3 + (-2)^1 + (-2)^0
19 = 10111 = (-2)^4 + (-2)^2 + (-2)^1 + (-2)^0

the algorithm works the same as it does for binary except with a little twist.
Digits are limited to {0,1} just like with binary.
To convert a base 10 number like 10 to binary you keep dividing by 2 and keeping track of the remainders.
10/2 = 5 rem 0
5/2 = 2 rem 1
2/2= 1 rem 0
1/2=0 rem 1
then you read off the remainders in reverse order to get the digits, which in this case would be 1010.  We can verify that 2^3+2^1=10

Now for NegaBinary we divide by -2 instead of 2.  Calculating the dividend and remainder is little bit trickier because we are dividing by a negative number.  For binary what we are actually doing is trying to find a,b such that a*2+b=n with b restricted to {0,1}.  For NegaBinary it is the same except 2 is replaced by -2.  b is easy to find as it is simply n mod 2, then a is simply (n-b)/(-2).  Thus for finding the NegaBinary digits of 10 we have
10/(-2) = -5 rem 0
(-5)/(-2) = 3 rem 1
3/(-2) = -1 rem 1
(-1)/(-2) = 1 rem 1
1/(-2) = 0 rem 1
thus the digits are 11110 and we can verify that
(-2)^4+(-2)^3+(-2)^2+(-2)^1=16-8+4-2=10.

The proof that this representation is unique is left as an exercise for the reader ;-)

Posted by Daniel on 2013-08-14 21:20:31