Signed-digit representation

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. 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 −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).

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 = ⅔ = (1 . (0 −1) …)_NAF

and

(0 . 4 4 4 …)_(10bal) = 4⁄9 = (1 . -5 -5 -5 …)_(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 integral-base system.)

Negative numerals

In 1928 Florian Cajori noted the recurring theme of signed digits, starting with Colson (1726) and Cauchy (1840). In his book History of Mathematical Notations, Cajori titled the section "Negative numerals". Eduard Selling (1887) advocated inverting the digits 1, 2, 3, 4, and 5 to indicate the negative sign. He also suggested snie, jes, jerd, reff, and niff as names to use vocally. Most of the other early sources used a bar over a digit to indicate a negative sign for a it. For completeness, Colson uses examples and describes addition (pp 163,4), multiplication (pp 165,6) and division (pp 170,1) using a table of multiples of the divisor. He explains the convenience of approximation by truncation in multiplication. Colson also devised an instrument (Counting Table) that calculated using signed digits.

See also

• negative base
• redundant binary representation

Notes and references

• J. P. Balantine (1925) "A Digit for Negative One", American Mathematical Monthly 32:302.
• Cajori, Florian (1993) [1928-1929]. A History of Mathematical Notations. Dover Publications. p. 57. ISBN 0486677664. 
• Augustin-Louis Cauchy (16 Nov 1840) "Sur les moyens d'eviter les erreurs dans les calculs numerique", Comptes rendus 11:789. Also found in Oevres completes Ser. 1, vol. 5, pp. 434–42.
• John Colson (1726) "A Short Account of Negativo-Affirmativo Arithmetik", Philosophical Transactions of the Royal Society 34:161–73. Available as Early Journal Content from JSTOR.
• Lui Han, Dongdong Chen, Seok-Bum Ko, Khan A. Wahid "Non-speculative Decimal Signed Digit Adder" from Department of Electrical and Computer Engineering, University of Saskatchewan.
• Rudolf Mehmke (1902) "Numerisches Rechen", §4 Beschränkung in den verwendeten Ziffern, Klein's encyclopedia, I-2, p. 944.
• Eduard Selling (1887) Eine neue Rechenmachine, pp. 15–18, Berlin.