Signed-digit representation

Signed-digit representation of numbers indicates that digits can be prefixed with a − (minus) sign to indicate that they are negative.

Signed-digit representation can be used in low-level software and hardware to accomplish fast high speed 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, [http://citeseer.ist.psu.edu/phatak94hybrid.html] ] . 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 $k\; =\; leftlfloorfrac\{b\}\{2\}\; ight\; floor$. 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); another is "balanced decimal", where the digits range from −5 to +4.

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 −1 1)₂ = 8 − 2 + 1 = 7:(1 −1 1 1)₂ = 8 − 4 + 2 + 1 = 7:(1 0 0 −1)₂ = 8 − 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)…)_NAF = fraction|2|3 = (1 . (0 −1)…)_NAFand:(0 . 4 4 4 …)_(10bal) = fraction|4|9 = (1 . -5 -5 -5 …)_(10bal)

Such examples can be shown to exist by considering the largest and smallest possible representations with integral parts 0 and 1 respectively, and then noting that they are equal. (Indeed, this works with any integral-base system.)

ee also

* negative base (negabinary etc.)
* redundant binary representation

References