Signed-digit representation

In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers.
Signed-digit representation can be used to accomplish fast addition of integers because it can eliminate chains of dependent carries. In the binary numeral system, a special case signed-digit representation is the non-adjacent form, which can offer speed benefits with minimal space overhead.

History

Challenges in calculation stimulated early authors Colson and Cauchy to use signed-digit representation. The further step of replacing negated digits with new ones was suggested by Selling and Cajori.
In 1928, Florian Cajori noted the recurring theme of signed digits, starting with Colson and Cauchy. In his book A History of Mathematical Notations, Cajori titled the section "Negative numerals". For completeness, Colson uses examples and describes addition, multiplication and division using a table of multiples of the divisor. He explains the convenience of approximation by truncation in multiplication. Colson also devised an instrument that calculated using signed digits.
Eduard Selling 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.

Definition and properties

Digit set

Let be a finite set of numerical digits with cardinality, with each digit denoted as for is known as the radix or number base. can be used for a signed-digit representation if it's associated with a unique function such that for all
This function, is what rigorously and formally establishes how integer values are assigned to the symbols/glyphs in One benefit of this formalism is that the definition of "the integers" is not conflated with any particular system for writing/representing them; in this way, these two distinct concepts are kept separate.
can be partitioned into three distinct sets,, and, representing the positive, zero, and negative digits respectively, such that all digits satisfy, all digits satisfy and all digits satisfy. The cardinality of is, the cardinality of is, and the cardinality of is, giving the number of positive and negative digits respectively, such that.

Balanced form representations

Balanced form representations are representations where for every positive digit, there exist a corresponding negative digit such that. It follows that. Only odd bases can have balanced form representations, as otherwise has to be the opposite of itself and hence 0, but. In balanced form, the negative digits are usually denoted as positive digits with a bar over the digit, as for. For example, the digit set of balanced ternary would be with,, and. This convention is adopted in finite fields of odd prime order :

Dual signed-digit representation

Every digit set has a dual digit set given by the inverse order of the digits with an isomorphism defined by. As a result, for any signed-digit representations of a number system ring constructed from with valuation, there exists a dual signed-digit representations of,, constructed from with valuation, and an isomorphism defined by, where is the additive inverse operator of. The digit set for balanced form representations is self-dual.

For integers

Given the digit set and function as defined above, let us define an integer endofunction as the following:
If the only periodic point of is the fixed point, then the set of all signed-digit representations of the integers using is given by the Kleene plus, the set of all finite concatenated strings of digits with at least one digit, with. Each signed-digit representation has a valuation
Examples include balanced ternary with digits.
Otherwise, if there exist a non-zero periodic point of, then there exist integers that are represented by an infinite number of non-zero digits in. Examples include the standard decimal numeral system with the digit set, which requires an infinite number of the digit to represent the additive inverse, as, and the positional numeral system with the digit set with, which requires an infinite number of the digit to represent the number, as.

For decimal fractions

If the integers can be represented by the Kleene plus, then the set of all signed-digit representations of the decimal fractions, or -adic rationals, is given by, the Cartesian product of the Kleene plus, the set of all finite concatenated strings of digits with at least one digit, the singleton consisting of the radix point, and the Kleene star, the set of all finite concatenated strings of digits, with. Each signed-digit representation has a valuation

For real numbers

If the integers can be represented by the Kleene plus, then the set of all signed-digit representations of the real numbers is given by, the Cartesian product of the Kleene plus, the set of all finite concatenated strings of digits with at least one digit, the singleton consisting of the radix point, and the Cantor space, the set of all infinite concatenated strings of digits, with. Each signed-digit representation has a valuation
The infinite series always converges to a finite real number.

For other number systems

All base- numerals can be represented as a subset of, the set of all infinite sequence">infinity">infinite sequences of digits in, where is the set of integers, and the ring of base- numerals is represented by the formal power series ring, the doubly infinite series
where for.

Integers modulo powers of

The set of all signed-digit representations of the integers modulo , is given by the set, the set of all finite concatenated strings of digits of length, with. Each signed-digit representation has a valuation

Prüfer groups

A Prüfer group is the quotient group of the integers and the -adic rationals. The set of all signed-digit representations of the Prüfer group is given by the Kleene star, the set of all finite concatenated strings of digits, with. Each signed-digit representation has a valuation

Circle group

The circle group is the quotient group of the integers and the real numbers. The set of all signed-digit representations of the circle group is given by the Cantor space, the set of all right-infinite concatenated strings of digits. Each signed-digit representation has a valuation
The infinite series always converges.

-adic integers

The set of all signed-digit representations of the -adic integers, is given by the Cantor space, the set of all left-infinite concatenated strings of digits. Each signed-digit representation has a valuation

-adic solenoids

The set of all signed-digit representations of the -adic solenoids, is given by the Cantor space, the set of all doubly infinite concatenated strings of digits. Each signed-digit representation has a valuation

In written and spoken language

Indo-Aryan languages

The oral and written forms of numbers in the Indo-Aryan languages use a negative numeral for the numbers between 11 and 90 that end with a nine. The numbers followed by their names are shown for Punjabi below :

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 meaning "break two" i.e. two fingers down
9 robong meaning "break one" i.e. one finger down

Classical Latin

In Classical Latin, integers 18 and 19 had in practice neither spoken nor written form, even corresponding parts for "eight" or "nine despite them being in existence. Instead, in Classic Latin,

18 = duodēvīgintī,,
19 = ūndēvīgintī,
20 = vīgintī,.

For upcoming integer numerals 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:

28 = duodētrīgintā, less frequently also yet vīgintī octō / octō et vīgintī,
29 = ūndētrīgintā 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 of this is the scarce occurrence of these numbers written in such a fashion in authentic sources.

Finnish Language

There is yet another language having this feature, however, still in active use today. This is the Finnish Language, where the numerals are used this way should a digit of 8 or 9 occur. The scheme is like this:

1 = "yksi"
2 = "kaksi"
3 = "kolme"
4 = "neljä"

...

7 = "seitsemän"
8 = "kaheksan"
9 = "yheksän"
10 = "kymmenen"

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"

This phenomenon has no influence on written numerals, however; the Finnish use the standard Western-Arabic decimal notation.

Time keeping

In the English language it is common to refer to a time as, for example, "seven to three" in which 'to' performs the subtraction.

Other systems

There exist other signed-digit bases such that the base. A notable examples of this is Booth encoding, which has a digit set with and, but which uses a base. The standard binary numeral system would only use digits of value.
Note that non-standard signed-digit representations are not unique. For instance:
The non-adjacent form 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,