negativebase

Negative base numeral systems use a base -b where b is an integer greater than 1. In such a system, every integer can be represented as a finite sum of digits d_i times (-b)^i, with digits 0 ≤ d_i < b. Unlike ordinary decimal or binary, the base is negative, so positive and negative values do not require a separate sign. Representations are unique for each integer once the digit set is fixed.

To convert a decimal integer to base -b, repeatedly divide by -b, recording remainders that fall in 0..b-1. If a division yields a negative remainder, adjust by adding b to the remainder and increasing the quotient accordingly. The sequence of remainders, read in reverse, gives the base -b representation. Example: 5 in base -2 is 101(-2) because 1(-2)^2 + 0(-2)^1 + 1(-2)^0 = 5. For -3, the representation is 1101(-2).

Any base -b with b > 1 yields a signless representation of all integers. The base -2 system, called negabinary, is the best known example and is used in demonstrations and theoretical discussions. Negative-base representations are mainly of mathematical and educational interest; they are not widely used in mainstream computing due to arithmetic complexity, but they illustrate that numeral bases are not restricted to positive numbers.