Question

History: I read from one of Knuth’s algorithm book that first computers used the base of 10. Then, it switched to two’s complement here.

Question: Why does the base could not be -2 in at least a monoid?

Examples:

(-2)^1 = -2 
(-2)^3 = -8

Answer 1

The problem is that with a negabinary (base -2) system, it’s more difficult to understand, and the number of possible positive and negative values are different. To see this latter point, consider a simple 3 bit case.

Here
the first (rightmost) bit represents the decimal 1;
the middle bit represents the decimal -2; and
the third (leftmost) bit represents the decimal 4

So

000 -> 0

001 -> 1

010 -> -2

011 -> -1

100 -> 4

101 -> 5

110 -> 2

111 -> 3

Thus the range of expressable values is -2 to 5, i.e. non-symmetric.