Signed-digit representation

In mathematical notation for numbers, signed-digit representation is a positional system with signed digits; the representation may not be unique.

Signed-digit representation can be used to accomplish fast addition of integers because it can eliminate chains of dependent carries. In the binary numeral system, a special case signed-digit representation is the non-adjacent form, which can offer speed benefits with minimal space overhead.

Challenges in calculation stimulated early authors Colson (1726) and Cauchy (1840) to use signed-digit representation. The further step of replacing negated digits with new ones was suggested by Selling (1887) and Cajori (1928).

In balanced form, the digits are drawn from a range −k to (b − 1) − k, where typically

For balanced forms, odd base numbers are advantageous. With an odd base number, truncation and rounding become the same operation, and all the digits except 0 are used in both positive and negative form.

A notable example is balanced ternary, where the base is b = 3, and the numerals have the values −1, 0 and +1 (rather than 0, 1, and 2 as in the standard ternary numeral system). Balanced ternary uses the minimum number of digits in a balanced form. Balanced decimal uses digits from −5 to +4. Balanced base nine, with digits from −4 to +4 provides the advantages of an odd-base balanced form with a similar number of digits, and is easy to convert to and from balanced ternary.

Other notable examples include Booth encoding and non-adjacent form, both of which use a base of b = 2, and both of which use numerals with the values −1, 0, and +1 (rather than 0 and 1 as in the standard binary numeral system).

Note that signed-digit representation is not necessarily unique. For instance:

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