Editing Signed-digit representation (section)

===Digit set===
Let <math>\mathcal{D}</math> be a [[finite set]] of [[numerical digits]] with [[cardinality]] <math>b > 1</math> (If <math>b \leq 1</math>, then the positional number system is [[Triviality (mathematics)|trivial]] and only represents the [[trivial ring]]), with each digit denoted as <math>d_i</math> for <math>0 \leq i < b.</math> <math>b</math> is known as the <em>[[radix]]</em> or <em>[[number base]]</em>. <math>\mathcal{D}</math> can be used for a signed-digit representation if it's associated with a unique [[Function (mathematics)|function]] <math>f_\mathcal{D}:\mathcal{D}\rightarrow\mathbb{Z}</math> such that <math>f_\mathcal{D}(d_i) \equiv i \bmod b</math> for all <math>0 \leq i < b.</math> 
This function, <math>f_{\mathcal{D}},</math> is what rigorously and formally establishes how integer values are assigned to the symbols/glyphs in <math>\mathcal{D}.</math> 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. 

<math>\mathcal{D}</math> can be [[Partition of a set|partitioned]] into three distinct sets <math>\mathcal{D}_{+}</math>, <math>\mathcal{D}_{0}</math>, and <math>\mathcal{D}_{-}</math>, representing the positive, zero, and negative digits respectively, such that all digits <math>d_{+}\in\mathcal{D}_{+}</math> satisfy <math>f_\mathcal{D}(d_{+}) > 0</math>, all digits <math>d_{0}\in\mathcal{D}_{0}</math> satisfy <math>f_\mathcal{D}(d_{0}) = 0</math> and all digits <math>d_{-}\in\mathcal{D}_{-}</math> satisfy <math>f_\mathcal{D}(d_{-}) < 0</math>. The cardinality of <math>\mathcal{D}_{+}</math> is <math>b_{+}</math>, the cardinality of <math>\mathcal{D}_{0}</math> is <math>b_{0}</math>, and the cardinality of <math>\mathcal{D}_{-}</math> is <math>b_{-}</math>, giving the number of positive and negative digits respectively, such that <math>b = b_{+} + b_{0} + b_{-}</math>. 

====Balanced form representations====
{{See also|Balanced ternary}}
Balanced form representations are representations where for every positive digit <math>d_{+}</math>, there exist a corresponding negative digit <math>d_{-}</math> such that <math>f_\mathcal{D}(d_{+}) = -f_\mathcal{D}(d_{-})</math>. It follows that <math>b_{+} = b_{-}</math>. Only [[odd number|odd]] bases can have balanced form representations, as otherwise <math>d_{b/2}</math> has to be the opposite of itself and hence 0, but <math>0\ne \frac b2</math>. In balanced form, the negative digits <math>d_{-}\in\mathcal{D}_{-}</math> are usually denoted as positive digits with a bar over the digit, as <math>d_{-} = \bar{d}_{+}</math> for <math>d_{+}\in\mathcal{D}_{+}</math>. For example, the digit set of [[balanced ternary]] would be <math>\mathcal{D}_{3} = \lbrace\bar{1},0,1\rbrace</math> with <math>f_{\mathcal{D}_{3}}(\bar{1}) = -1</math>, <math>f_{\mathcal{D}_{3}}(0) = 0</math>, and <math>f_{\mathcal{D}_{3}}(1) = 1</math>. This convention is adopted in [[finite field]]s of odd [[Prime number|prime]] order <math>q</math>:<ref>{{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}}</ref>
:<math>\mathbb{F}_{q} = \lbrace0, 1, \bar{1} = -1,... d = \frac{q - 1}{2},\ \bar{d} = \frac{1-q}{2}\ |\ q = 0\rbrace.</math>

====Dual signed-digit representation====
Every digit set <math>\mathcal{D}</math> has a [[Duality (order theory)|dual]] digit set <math>\mathcal{D}^\operatorname{op}</math> given by the [[inverse order]] of the digits with an [[isomorphism]] <math>g:\mathcal{D}\rightarrow\mathcal{D}^\operatorname{op}</math> defined by <math>-f_\mathcal{D} = g\circ f_{\mathcal{D}^\operatorname{op}}</math>. As a result, for any signed-digit representations <math>\mathcal{N}</math> of a number system [[Ring (mathematics)|ring]] <math>N</math> constructed from <math>\mathcal{D}</math> with [[Valuation (algebra)|valuation]] <math>v_\mathcal{D}:\mathcal{N}\rightarrow N</math>, there exists a dual signed-digit representations of <math>N</math>, <math>\mathcal{N}^\operatorname{op}</math>, constructed from <math>\mathcal{D}^\operatorname{op}</math> with [[Valuation (algebra)|valuation]] <math>v_{\mathcal{D}^\operatorname{op}}:\mathcal{N}^\operatorname{op}\rightarrow N</math>, and an isomorphism <math>h:\mathcal{N}\rightarrow\mathcal{N}^\operatorname{op}</math> defined by <math>-v_\mathcal{D} = h\circ v_{\mathcal{D}^\operatorname{op}}</math>, where <math>-</math> is the additive inverse operator of <math>N</math>. The digit set for balanced form representations is [[self-dual]].