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{{numeral systems}} | |||
In [[mathematical notation]] for [[number]]s, '''signed-digit representation''' indicates that each [[numerical digit|digit]] is associated with a [[sign (mathematics)|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.<ref>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]</ref> 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 <math>-k</math> to <math>(b-1) - k</math>, where typically <math>k = \left\lfloor\frac{b}{2}\right\rfloor</math>. 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 <math>b=3</math>, 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 <math>b=2</math>, 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)<sub>2</sub> = 4 + 2 + 1 = 7 | |||
: (1 0 −1 1)<sub>2</sub> = 8 − 2 + 1 = 7 | |||
: (1 −1 1 1)<sub>2</sub> = 8 − 4 + 2 + 1 = 7 | |||
: (1 0 0 −1)<sub>2</sub> = 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) …)<sub>NAF</sub> = {{fraction|2|3}} = (1 . (0 −1) …)<sub>NAF</sub> {{Clarify|reason=What base is this? Signed unbalanced base3?|date=May 2013}} | |||
and | |||
: (0 . 4 4 4 …)<sub>(10bal)</sub> = {{fraction|4|9}} = (1 . -5 -5 -5 …)<sub>(10bal)</sub> | |||
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== | |||
The oral and written forms of numbers in the [[Punjabi language]] use a form of a negative numeral one written as ''una'' or ''un''.<ref>[http://quizlet.com/16314536/punjabi-numbers-1-100-flash-cards/ Punjabi numbers] from [[Quizlet]]</ref> This negative one is used to form 19, 29,…89 from the root for 20, 30,…90. Explicitly, here are the numbers: | |||
* 19 unni, 20 vih, 21 ikki | |||
* 29 unatti, 30 tih, 31 ikatti | |||
* 39 untali, 40 chali, 41 iktali | |||
* 49 unanja, 50 panjah, 51 ikvanja | |||
* 59 unahat, 60 sath, 61 ikahat | |||
* 69 unattar, 70 sattar, 71 ikhattar | |||
* 79 unasi, 80 assi, 81 ikiasi | |||
* 89 unanve, 90 nabbe, 91 ikinnaven. | |||
In 1928 [[Florian Cajori]] noted the recurring theme of signed digits, starting with Colson (1726) and [[Augustin-Louis Cauchy|Cauchy]] (1840). In his book ''History of Mathematical Notations'', Cajori titled the section "Negative numerals".<ref>{{cite book |last= Cajori |first=Florian |authorlink=Florian Cajori|title= A History of Mathematical Notations |page= 57 |publisher= [[Dover Publications]] |year= 1993 |origyear= 1928-1929 |isbn= 0486677664 | url = http://books.google.com/?id=7juWmvQSTvwC&printsec=frontcover}}</ref> Eduard Selling<ref>Eduard Selling (1887) ''Eine neue Rechenmachine'', pp. 15–18, Berlin</ref> 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<ref>[[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 [http://www.jstor.org/stable/103469 JSTOR]</ref> uses examples and describes [[addition]] (pp 163,4), [[multiplication]] (pp 165,6) and [[division (mathematics)|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== | |||
{{reflist}} | |||
* J. P. Balantine (1925) "A Digit for Negative One", [[American Mathematical Monthly]] 32:302. | |||
* [[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. | |||
* Lui Han, Dongdong Chen, Seok-Bum Ko, Khan A. Wahid [http://homepage.usask.ca/~doc220/index_files/doc/C12.pdf "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. | |||
{{DEFAULTSORT:Signed-Digit Representation}} | |||
[[Category:Non-standard positional numeral systems]] | |||
Revision as of 23:23, 2 February 2014
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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.[1] 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 to , 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 , 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 , 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 = Template:Fraction = (1 . (0 −1) …)NAF Template:Clarify
and
- (0 . 4 4 4 …)(10bal) = Template:Fraction = (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
The oral and written forms of numbers in the Punjabi language use a form of a negative numeral one written as una or un.[2] This negative one is used to form 19, 29,…89 from the root for 20, 30,…90. Explicitly, here are the numbers:
- 19 unni, 20 vih, 21 ikki
- 29 unatti, 30 tih, 31 ikatti
- 39 untali, 40 chali, 41 iktali
- 49 unanja, 50 panjah, 51 ikvanja
- 59 unahat, 60 sath, 61 ikahat
- 69 unattar, 70 sattar, 71 ikhattar
- 79 unasi, 80 assi, 81 ikiasi
- 89 unanve, 90 nabbe, 91 ikinnaven.
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".[3] Eduard Selling[4] 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[5] 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
Notes and references
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- J. P. Balantine (1925) "A Digit for Negative One", American Mathematical Monthly 32:302.
- 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.
- 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.
- ↑ Dhananjay Phatak, I. Koren, Hybrid Signed-Digit Number Systems: A Unified Framework for Redundant Number Representations with Bounded Carry Propagation Chains, 1994, [1]
- ↑ Punjabi numbers from Quizlet
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Eduard Selling (1887) Eine neue Rechenmachine, pp. 15–18, Berlin
- ↑ 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