 in mathematical notation for numbers, signed-digit representation indicates that each digit is associated with a sign, positive or negative. 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). signed-digit representation can be used in low-level software and hardware to accomplish fast addition of integers because it can eliminate carries.dhananjay phatak, i. koren, hybrid signed-digit number systems: a unified framework for redundant number representations with bounded carry propagation chains, 1994,  in the binary numeral system one special case of signed-digit representation is the non-adjacent form which can offer speed benefits with minimal space overhead. ==balanced form== in balanced form, the digits are drawn from a range -k to (b-1) - k, where typically k = \left\lfloor\frac{b}{2}\right\rfloor. 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 â1, 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 â5 to +4. balanced base nine, with digits from â4 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 â1, 0, and +1 (rather than 0 and 1 as in the standard binary numeral system). ==non-unique representations== note that signed-digit representation is not necessarily unique. for instance: : (0 1 1 1)2 = 4 + 2 + 1 = 7 : (1 0 â1 1)2 = 8 â 2 + 1 = 7 : (1 â1 1 1)2 = 8 â 4 + 2 + 1 = 7 : (1 0 0 â1)2 = 8 â 1 = 7 the non-adjacent form does guarantee a unique representation for every integer value, as do balanced forms. when representations are extended to fractional numbers, uniqueness is lost for non-adjacent and balanced forms; for example, : (0 . (1 0) â¦)naf =  = (1 . (0 &minus;1) â¦)naf  and : (0 . 4 4 4 â¦)(10bal) =  = (1 . -5 -5 -5 â¦)(10bal) such examples can be shown to exist by considering the greatest and smallest possible representations with integral parts 0 and 1 respectively, and then noting that they are equal. (indeed, this works with any integr
Non-standard_positional_numeral_systems