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.
[1] 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.
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).
[2] In his book History of Mathematical Notations, Cajori titled the section "Negative numerals".
[3] For completeness, Colson[4] uses examples and describes addition (pp.
Colson also devised an instrument (Counting Table) that calculated using signed digits.
Eduard Selling[5] 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.
Another German usage of signed-digits was described in 1902 in Klein's encyclopedia.
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" (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.
This convention is adopted in finite fields of odd prime order
, 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
The infinite series always converges to a finite real number.
numerals is represented by the formal power series ring
The set of all signed-digit representations of the Prüfer group is given by the Kleene star
The set of all signed-digit representations of the circle group is given by the Cantor space
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"):[8] Similarly, the Sesotho language utilizes negative numerals to form 8's and 9's.
In Classical Latin,[9] 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.
Hence, approaching thirty, numerals were expressed as:[10] 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.
Above list is no special case, it consequently appears in larger cardinals as well, e.g.: Emphasizing of these attributes stay present even in the shortest colloquial forms of numerals: ...
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.
For instance: The non-adjacent form (NAF) of Booth encoding does guarantee a unique representation for every integer value.