signed-digit representation

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 1928, Florian Cajori noted the recurring theme of signed digits, starting with Colson (1726) and Cauchy (1840).Augustin-Louis Cauchy (16 November 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. In his book History of Mathematical Notations, Cajori titled the section "Negative numerals".{{cite book |last= Cajori |first=Florian |author-link=Florian Cajori|title= A History of Mathematical Notations |page= [https://archive.org/details/historyofmathema00cajo_0/page/57 57] |publisher= Dover Publications |year= 1993 |orig-year= 1928-1929 |isbn= 978-0486677668 | url = https://archive.org/details/historyofmathema00cajo_0|url-access= registration }} For completeness, Colson{{Cite journal |last=Colson |first=John |date=1726 |title=A Short Account of Negativo-Affirmative Arithmetick, by Mr. John Colson, F. R. S. |url=https://www.jstor.org/stable/103469 |journal=Philosophical Transactions |volume=34 |pages=161–173 |jstor=103469 |bibcode=1726RSPT...34..161C |issn=0260-7085}} 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.

Eduard SellingEduard Selling (1887) Eine neue Rechenmachine, pp. 15–18, Berlin 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 it. Another German usage of signed-digits was described in 1902 in Klein's encyclopedia.Rudolf Mehmke (1902) "Numerisches Rechen", §4 Beschränkung in den verwendeten Ziffern, Klein's encyclopedia, I-2, p. 944.

Let $\backslash mathcal\{D\}$ be a finite set of numerical digits with cardinality $b\; >\; 1$ (If $b\; \backslash leq\; 1$, then the positional number system is trivial and only represents the trivial ring), with each digit denoted as $d\_i$ for $b$ is known as the radix or number base. $\backslash mathcal\{D\}$ can be used for a signed-digit representation if it's associated with a unique function $f\_\backslash mathcal\{D\}:\backslash mathcal\{D\}\backslash rightarrow\backslash mathbb\{Z\}$ such that $f\_\backslash mathcal\{D\}(d\_i)\; \backslash equiv\; i\; \backslash bmod\; b$ for all

This function, $f\_\{\backslash mathcal\{D\}\},$ is what rigorously and formally establishes how integer values are assigned to the symbols/glyphs in $\backslash mathcal\{D\}.$ One benefit of this formalism is that the definition of "the integers" (however they may be defined) is not conflated with any particular system for writing/representing them; in this way, these two distinct (albeit closely related) concepts are kept separate.

$\backslash mathcal\{D\}$ can be partitioned into three distinct sets $\backslash mathcal\{D\}\_\{+\}$, $\backslash mathcal\{D\}\_\{0\}$, and $\backslash mathcal\{D\}\_\{-\}$, representing the positive, zero, and negative digits respectively, such that all digits $d\_\{+\}\backslash in\backslash mathcal\{D\}\_\{+\}$ satisfy $f\_\backslash mathcal\{D\}(d\_\{+\})\; >\; 0$, all digits $d\_\{0\}\backslash in\backslash mathcal\{D\}\_\{0\}$ satisfy $f\_\backslash mathcal\{D\}(d\_\{0\})\; =\; 0$ and all digits $d\_\{-\}\backslash in\backslash mathcal\{D\}\_\{-\}$ satisfy . The cardinality of $\backslash mathcal\{D\}\_\{+\}$ is $b\_\{+\}$, the cardinality of $\backslash mathcal\{D\}\_\{0\}$ is $b\_\{0\}$, and the cardinality of $\backslash mathcal\{D\}\_\{-\}$ is $b\_\{-\}$, giving the number of positive and negative digits respectively, such that $b\; =\; b\_\{+\}\; +\; b\_\{0\}\; +\; b\_\{-\}$.

{{See also|Balanced ternary}}

Balanced form representations are representations where for every positive digit $d\_\{+\}$, there exist a corresponding negative digit $d\_\{-\}$ such that $f\_\backslash mathcal\{D\}(d\_\{+\})\; =\; -f\_\backslash mathcal\{D\}(d\_\{-\})$. It follows that $b\_\{+\}\; =\; b\_\{-\}$. Only odd bases can have balanced form representations, as otherwise $d\_\{b/2\}$ has to be the opposite of itself and hence 0, but $0\backslash ne\; \backslash frac\; b2$. In balanced form, the negative digits $d\_\{-\}\backslash in\backslash mathcal\{D\}\_\{-\}$ are usually denoted as positive digits with a bar over the digit, as $d\_\{-\}\; =\; \backslash bar\{d\}\_\{+\}$ for $d\_\{+\}\backslash in\backslash mathcal\{D\}\_\{+\}$. For example, the digit set of balanced ternary would be $\backslash mathcal\{D\}\_\{3\}\; =\; \backslash lbrace\backslash bar\{1\},0,1\backslash rbrace$ with $f\_\{\backslash mathcal\{D\}\_\{3\}\}(\backslash bar\{1\})\; =\; -1$, $f\_\{\backslash mathcal\{D\}\_\{3\}\}(0)\; =\; 0$, and $f\_\{\backslash mathcal\{D\}\_\{3\}\}(1)\; =\; 1$. This convention is adopted in finite fields of odd prime order $q$:{{Cite book|title=Projective Geometries Over Finite Fields|first1=J. W. P.|last1=Hirschfeld|author-link=J. W. P. Hirschfeld|publisher=Oxford University Press|year=1979|page=8|isbn=978-0-19-850295-1}}

:$\backslash mathbb\{F\}\_\{q\}\; =\; \backslash lbrace0,\; 1,\; \backslash bar\{1\}\; =\; -1,...\; d\; =\; \backslash frac\{q\; -\; 1\}\{2\},\backslash \; \backslash bar\{d\}\; =\; \backslash frac\{1-q\}\{2\}\backslash \; |\backslash \; q\; =\; 0\backslash rbrace.$

Every digit set $\backslash mathcal\{D\}$ has a dual digit set $\backslash mathcal\{D\}^\backslash operatorname\{op\}$ given by the inverse order of the digits with an isomorphism $g:\backslash mathcal\{D\}\backslash rightarrow\backslash mathcal\{D\}^\backslash operatorname\{op\}$ defined by $-f\_\backslash mathcal\{D\}\; =\; g\backslash circ\; f\_\{\backslash mathcal\{D\}^\backslash operatorname\{op\}\}$. As a result, for any signed-digit representations $\backslash mathcal\{N\}$ of a number system ring $N$ constructed from $\backslash mathcal\{D\}$ with valuation $v\_\backslash mathcal\{D\}:\backslash mathcal\{N\}\backslash rightarrow\; N$, there exists a dual signed-digit representations of $N$, $\backslash mathcal\{N\}^\backslash operatorname\{op\}$, constructed from $\backslash mathcal\{D\}^\backslash operatorname\{op\}$ with valuation $v\_\{\backslash mathcal\{D\}^\backslash operatorname\{op\}\}:\backslash mathcal\{N\}^\backslash operatorname\{op\}\backslash rightarrow\; N$, and an isomorphism $h:\backslash mathcal\{N\}\backslash rightarrow\backslash mathcal\{N\}^\backslash operatorname\{op\}$ defined by $-v\_\backslash mathcal\{D\}\; =\; h\backslash circ\; v\_\{\backslash mathcal\{D\}^\backslash operatorname\{op\}\}$, where $-$ is the additive inverse operator of $N$. The digit set for balanced form representations is self-dual.

Given the digit set $\backslash mathcal\{D\}$ and function $f:\backslash mathcal\{D\}\backslash rightarrow\backslash mathbb\{Z\}$ as defined above, let us define an integer endofunction $T:\backslash mathbb\{Z\}\backslash rightarrow\backslash mathbb\{Z\}$ as the following:

:$T(n)\; =$

\begin{cases}

\frac{n - f(d_i)}{b} &\text{if } n \equiv i \bmod b, 0 \leq i < b

\end{cases}

If the only periodic point of $T$ is the fixed point $0$, then the set of all signed-digit representations of the integers $\backslash mathbb\{Z\}$ using $\backslash mathcal\{D\}$ is given by the Kleene plus $\backslash mathcal\{D\}^+$, the set of all finite concatenated strings of digits $d\_n\; \backslash ldots\; d\_0$ with at least one digit, with $n\backslash in\backslash mathbb\{N\}$. Each signed-digit representation $m\; \backslash in\; \backslash mathcal\{D\}^+$ has a valuation $v\_\backslash mathcal\{D\}:\backslash mathcal\{D\}^+\backslash rightarrow\backslash mathbb\{Z\}$

:$v\_\backslash mathcal\{D\}(m)\; =\; \backslash sum\_\{i=0\}^\{n\}f\_\backslash mathcal\{D\}(d\_\{i\})b^\{i\}$.

Examples include balanced ternary with digits $\backslash mathcal\{D\}\; =\; \backslash lbrace\; \backslash bar\{1\},\; 0,\; 1\backslash rbrace$.

Otherwise, if there exist a non-zero periodic point of $T$, then there exist integers that are represented by an infinite number of non-zero digits in $\backslash mathcal\{D\}$. Examples include the standard decimal numeral system with the digit set $\backslash operatorname\{dec\}\; =\; \backslash lbrace\; 0,\; 1,\; 2,\; 3,\; 4,\; 5,\; 6,\; 7,\; 8,\; 9\; \backslash rbrace$, which requires an infinite number of the digit $9$ to represent the additive inverse $-1$, as $T\_\backslash operatorname\{dec\}(-1)\; =\; \backslash frac\{-1\; -\; 9\}\{10\}\; =\; -1$, and the positional numeral system with the digit set $\backslash mathcal\{D\}\; =\; \backslash lbrace\; \backslash text\{A\},\; 0,\; 1\backslash rbrace$ with $f(\backslash text\{A\})\; =\; -4$, which requires an infinite number of the digit $\backslash text\{A\}$ to represent the number $2$, as $T\_\backslash mathcal\{D\}(2)\; =\; \backslash frac\{2\; -\; (-4)\}\{3\}\; =\; 2$.

{{Main|Decimal representation}}

If the integers can be represented by the Kleene plus $\backslash mathcal\{D\}^+$, then the set of all signed-digit representations of the decimal fractions, or $b$-adic rationals $\backslash mathbb\{Z\}[1\backslash backslash\; b]$, is given by $\backslash mathcal\{Q\}\; =\; \backslash mathcal\{D\}^+\backslash times\backslash mathcal\{P\}\backslash times\backslash mathcal\{D\}^*$, the Cartesian product of the Kleene plus $\backslash mathcal\{D\}^+$, the set of all finite concatenated strings of digits $d\_n\; \backslash ldots\; d\_0$ with at least one digit, the singleton $\backslash mathcal\{P\}$ consisting of the radix point ($.$ or $,$), and the Kleene star $\backslash mathcal\{D\}^*$, the set of all finite concatenated strings of digits $d\_\{-1\}\; \backslash ldots\; d\_\{-m\}$, with $m,n\backslash in\backslash mathbb\{N\}$. Each signed-digit representation $q\; \backslash in\; \backslash mathcal\{Q\}$ has a valuation $v\_\backslash mathcal\{D\}:\backslash mathcal\{Q\}\backslash rightarrow\backslash mathbb\{Z\}[1\backslash backslash\; b]$

:$v\_\backslash mathcal\{D\}(q)\; =\; \backslash sum\_\{i=-m\}^\{n\}f\_\backslash mathcal\{D\}(d\_\{i\})b^\{i\}$

{{Main|Construction of the reals#Construction from Cauchy sequences}}

If the integers can be represented by the Kleene plus $\backslash mathcal\{D\}^+$, then the set of all signed-digit representations of the real numbers $\backslash mathbb\{R\}$ is given by $\backslash mathcal\{R\}\; =\; \backslash mathcal\{D\}^+\; \backslash times\; \backslash mathcal\{P\}\; \backslash times\; \backslash mathcal\{D\}^\backslash mathbb\{N\}$, the Cartesian product of the Kleene plus $\backslash mathcal\{D\}^+$, the set of all finite concatenated strings of digits $d\_n\; \backslash ldots\; d\_0$ with at least one digit, the singleton $\backslash mathcal\{P\}$ consisting of the radix point ($.$ or $,$), and the Cantor space $\backslash mathcal\{D\}^\backslash mathbb\{N\}$, the set of all infinite concatenated strings of digits $d\_\{-1\}\; d\_\{-2\}\; \backslash ldots$, with $n\backslash in\backslash mathbb\{N\}$. Each signed-digit representation $r\; \backslash in\; \backslash mathcal\{R\}$ has a valuation $v\_\backslash mathcal\{D\}:\backslash mathcal\{R\}\backslash rightarrow\backslash mathbb\{R\}$

:$v\_\backslash mathcal\{D\}(r)\; =\; \backslash sum\_\{i=-\backslash infty\}^\{n\}f\_\backslash mathcal\{D\}(d\_\{i\})b^\{i\}$.

The infinite series always converges to a finite real number.

All base-$b$ numerals can be represented as a subset of $\backslash mathcal\{D\}^\backslash mathbb\{Z\}$, the set of all doubly infinite sequences of digits in $\backslash mathcal\{D\}$, where $\backslash mathbb\{Z\}$ is the set of integers, and the ring of base-$b$ numerals is represented by the formal power series ring $\backslash mathbb\{Z\}b,b^\{-1\}$, the doubly infinite series

:$\backslash sum\_\{i\; =\; -\backslash infty\}^\{\backslash infty\}a\_i\; b^i$

where $a\_i\backslash in\backslash mathbb\{Z\}$ for $i\backslash in\backslash mathbb\{Z\}$.

The set of all signed-digit representations of the integers modulo $b^n$, $\backslash mathbb\{Z\}\backslash backslash\; b^n\backslash mathbb\{Z\}$ is given by the set $\backslash mathcal\{D\}^n$, the set of all finite concatenated strings of digits $d\_\{n\; -\; 1\}\; \backslash ldots\; d\_0$ of length $n$, with $n\backslash in\backslash mathbb\{N\}$. Each signed-digit representation $m\; \backslash in\; \backslash mathcal\{D\}^n$ has a valuation $v\_\backslash mathcal\{D\}:\backslash mathcal\{D\}^n\backslash rightarrow\backslash mathbb\{Z\}/b^n\backslash mathbb\{Z\}$

:$v\_\backslash mathcal\{D\}(m)\; \backslash equiv\; \backslash sum\_\{i=0\}^\{n\; -\; 1\}f\_\backslash mathcal\{D\}(d\_\{i\})b^\{i\}\; \backslash bmod\; b^n$

A Prüfer group is the quotient group $\backslash mathbb\{Z\}(b^\backslash infty)\; =\; \backslash mathbb\{Z\}[1\backslash backslash\; b]/\backslash mathbb\{Z\}$ of the integers and the $b$-adic rationals. The set of all signed-digit representations of the Prüfer group is given by the Kleene star $\backslash mathcal\{D\}^*$, the set of all finite concatenated strings of digits $d\_\{1\}\; \backslash ldots\; d\_\{n\}$, with $n\backslash in\backslash mathbb\{N\}$. Each signed-digit representation $p\; \backslash in\; \backslash mathcal\{D\}^*$ has a valuation $v\_\backslash mathcal\{D\}:\backslash mathcal\{D\}^*\backslash rightarrow\backslash mathbb\{Z\}(b^\backslash infty)$

:$v\_\backslash mathcal\{D\}(m)\; \backslash equiv\; \backslash sum\_\{i=1\}^\{n\}f\_\backslash mathcal\{D\}(d\_\{i\})b^\{-i\}\; \backslash bmod\; 1$

The circle group is the quotient group $\backslash mathbb\{T\}\; =\; \backslash mathbb\{R\}/\backslash mathbb\{Z\}$ of the integers and the real numbers. The set of all signed-digit representations of the circle group is given by the Cantor space $\backslash mathcal\{D\}^\backslash mathbb\{N\}$, the set of all right-infinite concatenated strings of digits $d\_\{1\}\; d\_\{2\}\; \backslash ldots$. Each signed-digit representation $m\; \backslash in\; \backslash mathcal\{D\}^n$ has a valuation $v\_\backslash mathcal\{D\}:\backslash mathcal\{D\}^\backslash mathbb\{N\}\backslash rightarrow\backslash mathbb\{T\}$

:$v\_\backslash mathcal\{D\}(m)\; \backslash equiv\; \backslash sum\_\{i=1\}^\{\backslash infty\}f\_\backslash mathcal\{D\}(d\_\{i\})b^\{-i\}\; \backslash bmod\; 1$

The infinite series always converges.

The set of all signed-digit representations of the $b$-adic integers, $\backslash mathbb\{Z\}\_b$ is given by the Cantor space $\backslash mathcal\{D\}^\backslash mathbb\{N\}$, the set of all left-infinite concatenated strings of digits $\backslash ldots\; d\_\{1\}\; d\_\{0\}$. Each signed-digit representation $m\; \backslash in\; \backslash mathcal\{D\}^n$ has a valuation $v\_\backslash mathcal\{D\}:\backslash mathcal\{D\}^\backslash mathbb\{N\}\backslash rightarrow\backslash mathbb\{Z\}\_\{b\}$

:$v\_\backslash mathcal\{D\}(m)\; =\; \backslash sum\_\{i=0\}^\{\backslash infty\}f\_\backslash mathcal\{D\}(d\_\{i\})b^\{i\}$

The set of all signed-digit representations of the $b$-adic solenoids, $\backslash mathbb\{T\}\_b$ is given by the Cantor space $\backslash mathcal\{D\}^\backslash mathbb\{Z\}$, the set of all doubly infinite concatenated strings of digits $\backslash ldots\; d\_\{1\}\; d\_\{0\}\; d\_\{-1\}\; \backslash ldots$. Each signed-digit representation $m\; \backslash in\; \backslash mathcal\{D\}^n$ has a valuation $v\_\backslash mathcal\{D\}:\backslash mathcal\{D\}^\backslash mathbb\{Z\}\backslash rightarrow\backslash mathbb\{T\}\_\{b\}$

:$v\_\backslash mathcal\{D\}(m)\; =\; \backslash sum\_\{i=-\backslash infty\}^\{\backslash infty\}f\_\backslash mathcal\{D\}(d\_\{i\})b^\{i\}$

The oral and written forms of numbers in the Indo-Aryan languages use a negative numeral (e.g., "un" in Hindi and Bengali, "un" or "unna" in Punjabi, "ekon" in Marathi) for the numbers between 11 and 90 that end with a nine. The numbers followed by their names are shown for Punjabi below (the prefix "ik" means "one"):[http://quizlet.com/16314536/punjabi-numbers-1-100-flash-cards/ Punjabi numbers] from Quizlet

• 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.

Similarly, the Sesotho language utilizes negative numerals to form 8's and 9's.

• 8 robeli (/Ro-bay-dee/) meaning "break two" i.e. two fingers down
• 9 robong (/Ro-bong/) meaning "break one" i.e. one finger down

In Classical Latin,J. Matthew Harrington (2016) [https://cpb-us-w2.wpmucdn.com/campuspress.yale.edu/dist/4/3253/files/2018/08/Harrington-Latin-Grammar-2016.pdf Synopsis of Ancient Latin Grammar] integers 18 and 19 did not even have a spoken, nor written form including corresponding parts for "eight" or "nine" in practice - despite them being in existence. Instead, in Classic Latin,

• 18 = duodēvīgintī ("two taken from twenty"), (IIXX or XIIX),
• 19 = ūndēvīgintī ("one taken from twenty"), (IXX or XIX)
• 20 = vīgintī ("twenty"), (XX).

For upcoming integer numerals [28, 29, 38, 39, ..., 88, 89] the additive form in the language had been much more common, however, for the listed numbers, the above form was still preferred. Hence, approaching thirty, numerals were expressed as:{{Citation |title=duodetriginta |date=2020-03-25 |work=Wiktionary, the free dictionary |url=https://en.wiktionary.org/w/index.php?title=duodetriginta&oldid=58999568 |access-date=2024-04-07 |language=en}}

• 28 = duodētrīgintā ("two taken from thirty"), less frequently also yet vīgintī octō / octō et vīgintī ("twenty eight / eight and twenty"), (IIXXX or XXIIX versus XXVIII, latter having been fully outcompeted.)
• 29 = ūndētrīgintā ("one taken from thirty") despite the less preferred form was also at their disposal.

This is one of the main foundations of contemporary historians' reasoning, explaining why the subtractive I- and II- was so common in this range of cardinals compared to other ranges. Numerals 98 and 99 could also be expressed in both forms, yet "two to hundred" might have sounded a bit odd - clear evidence is the scarce occurrence of these numbers written down in a subtractive fashion in authentic sources.

There is yet another language having this feature (by now, only in traces), however, still in active use today. This is the Finnish Language, where the (spelled out) numerals are used this way should a digit of 8 or 9 occur. The scheme is like this:{{Cite web |title=Kielitoimiston sanakirja |url=https://www.kielitoimistonsanakirja.fi/#/perusluku |access-date=2024-04-07 |website=www.kielitoimistonsanakirja.fi}}

• 1 = "yksi" (Note: yhd- or yht- mostly when about to be declined; e.g. "yhdessä" = "together, as one [entity]")
• 2 = "kaksi" (Also note: kahde-, kahte- when declined)
• 3 = "kolme"
• 4 = "neljä"

...

• 7 = "seitsemän"
• 8 = "kah(d)eksan" (two left [for it to reach it])
• 9 = "yh(d)eksän" (one left [for it to reach it])
• 10 = "kymmenen" (ten)

Above list is no special case, it consequently appears in larger cardinals as well, e.g.:

• 399 = "kolmesataayhdeksänkymmentäyhdeksän"

Emphasizing of these attributes stay present even in the shortest colloquial forms of numerals:

• 1 = "yy"
• 2 = "kaa"
• 3 = "koo"

...

• 7 = "seiska"
• 8 = "kasi"
• 9 = "ysi"
• 10 = "kymppi"

However, this phenomenon has no influence on written numerals, the Finnish use the standard Western-Arabic decimal notation.

In the English language it is common to refer to times as, for example, 'seven to three', 'to' performing the negation.

There exist other signed-digit bases such that the base $b\; \backslash neq\; b\_\{+\}\; +\; b\_\{-\}\; +\; 1$. A notable examples of this is Booth encoding, which has a digit set $\backslash mathcal\{D\}\; =\; \backslash lbrace\backslash bar\{1\},0,1\backslash rbrace$ with $b\_\{+\}\; =\; 1$ and $b\_\{-\}\; =\; 1$, but which uses a base . The standard binary numeral system would only use digits of value $\backslash lbrace0,1\backslash rbrace$.

Note that non-standard signed-digit representations are not unique. For instance:

: $0111\_\{\backslash mathcal\{D\}\}\; =\; 4\; +\; 2\; +\; 1\; =\; 7$

: $10\backslash bar\{1\}1\_\{\backslash mathcal\{D\}\}\; =\; 8\; -\; 2\; +\; 1\; =\; 7$

: $1\backslash bar\{1\}11\_\{\backslash mathcal\{D\}\}\; =\; 8\; -\; 4\; +\; 2\; +\; 1\; =\; 7$

: $100\backslash bar\{1\}\_\{\backslash mathcal\{D\}\}\; =\; 8\; -\; 1\; =\; 7$

The non-adjacent form (NAF) of Booth encoding does guarantee a unique representation for every integer value. However, this only applies for integer values. For example, consider the following repeating binary numbers in NAF,

: $\backslash frac\{2\}\{3\}\; =\; 0.\backslash overline\{10\}\_\{\backslash mathcal\{D\}\}\; =\; 1.\backslash overline\{0\backslash bar\{1\}\}\_\{\backslash mathcal\{D\}\}$

• Balanced ternary
• Negative base
• Redundant binary representation

{{reflist}}

• J. P. Balantine (1925) "A Digit for Negative One", American Mathematical Monthly 32:302.
• 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.

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{{DEFAULTSORT:Signed-Digit Representation}}

Category:Non-standard positional numeral systems

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