Negative base

A negative base (or negative radix) may be used to construct a non-standard positional numeral system. Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that is to say, the base $b$ is equal to $-r$ for some natural number $r$ ( $r \geq 2$ ).

Negative-base systems can accommodate all the same numbers as standard place-value systems, but both positive and negative numbers are represented without the use of a minus sign (or, in computer representation, a sign bit); this advantage is countered by an increased complexity of arithmetic operations. The need to store the information normally contained by a negative sign often results in a negative-base number being one digit longer than its positive-base equivalent.

The common names for negative-base positional numeral systems are formed by prefixing nega- to the name of the corresponding positive-base system; for example, negadecimal (base −10) corresponds to decimal (base 10), negabinary (base −2) to binary (base 2), negaternary (base −3) to ternary (base 3), and negaquaternary (base −4) to quaternary (base 4).^[1]^[2]

^ Knuth, Donald (1998), The Art of Computer Programming, Volume 2 (3rd ed.), pp. 204–205. Knuth mentions both negabinary and negadecimal.
^ The negaternary system is discussed briefly in Petkovšek, Marko (1990), "Ambiguous numbers are dense", The American Mathematical Monthly, 97 (5): 408–411, doi:10.2307/2324393, ISSN 0002-9890, JSTOR 2324393, MR 1048915.

[1] Knuth, Donald (1998), The Art of Computer Programming, Volume 2 (3rd ed.), pp. 204–205. Knuth mentions both negabinary and negadecimal.

[2] The negaternary system is discussed briefly in Petkovšek, Marko (1990), "Ambiguous numbers are dense", The American Mathematical Monthly, 97 (5): 408–411, doi:10.2307/2324393, ISSN 0002-9890, JSTOR 2324393, MR 1048915.

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