Editing Signed-digit representation (section)

===For other number systems===
All base-<math>b</math> numerals can be represented as a subset of <math>\mathcal{D}^\mathbb{Z}</math>, the set of all [[doubly infinite sequence]]s of digits in <math>\mathcal{D}</math>, where <math>\mathbb{Z}</math> is the set of [[integers]], and the [[Ring (mathematics)|ring]] of base-<math>b</math> numerals is represented by the [[formal power series ring]] <math>\mathbb{Z}[[b,b^{-1}]]</math>, the doubly infinite series
:<math>\sum_{i = -\infty}^{\infty}a_i b^i</math>
where <math>a_i\in\mathbb{Z}</math> for <math>i\in\mathbb{Z}</math>. 

====Integers modulo powers of {{math|''b''}}====
The set of all signed-digit representations of the [[Integers modulo n|integers modulo <math>b^n</math>]], <math>\mathbb{Z}\backslash b^n\mathbb{Z}</math> is given by the set <math>\mathcal{D}^n</math>, the set of all finite [[concatenation|concatenated]] strings of digits <math>d_{n - 1} \ldots d_0</math> of length <math>n</math>, with <math>n\in\mathbb{N}</math>. Each signed-digit representation <math>m \in \mathcal{D}^n</math> has a [[Valuation (algebra)|valuation]] <math>v_\mathcal{D}:\mathcal{D}^n\rightarrow\mathbb{Z}/b^n\mathbb{Z}</math>
:<math>v_\mathcal{D}(m) \equiv \sum_{i=0}^{n - 1}f_\mathcal{D}(d_{i})b^{i} \bmod b^n</math>

====Prüfer groups====
A [[Prüfer group]] is the [[quotient group]] <math>\mathbb{Z}(b^\infty) = \mathbb{Z}[1\backslash b]/\mathbb{Z}</math> of the integers and the <math>b</math>-adic rationals. The set of all signed-digit representations of the [[Prüfer group]] is given by the [[Kleene star]] <math>\mathcal{D}^*</math>, the set of all finite [[concatenation|concatenated]] strings of digits <math>d_{1} \ldots d_{n}</math>, with <math>n\in\mathbb{N}</math>. Each signed-digit representation <math>p \in \mathcal{D}^*</math> has a [[Valuation (algebra)|valuation]] <math>v_\mathcal{D}:\mathcal{D}^*\rightarrow\mathbb{Z}(b^\infty)</math>
:<math>v_\mathcal{D}(m) \equiv \sum_{i=1}^{n}f_\mathcal{D}(d_{i})b^{-i} \bmod 1</math>

====Circle group====
The [[circle group]] is the quotient group <math>\mathbb{T} = \mathbb{R}/\mathbb{Z}</math> of the integers and the real numbers. The set of all signed-digit representations of the [[circle group]] is given by the [[Cantor space]] <math>\mathcal{D}^\mathbb{N}</math>, the set of all right-infinite concatenated strings of digits <math>d_{1} d_{2} \ldots</math>. Each signed-digit representation <math>m \in \mathcal{D}^n</math> has a [[Valuation (algebra)|valuation]] <math>v_\mathcal{D}:\mathcal{D}^\mathbb{N}\rightarrow\mathbb{T}</math>
:<math>v_\mathcal{D}(m) \equiv \sum_{i=1}^{\infty}f_\mathcal{D}(d_{i})b^{-i} \bmod 1</math>
The [[infinite series]] always [[Convergent series|converges]].

===={{math|''b''}}-adic integers====
The set of all signed-digit representations of the [[p-adic integers|<math>b</math>-adic integers]], <math>\mathbb{Z}_b</math> is given by the [[Cantor space]] <math>\mathcal{D}^\mathbb{N}</math>, the set of all left-infinite concatenated strings of digits <math>\ldots d_{1} d_{0}</math>. Each signed-digit representation <math>m \in \mathcal{D}^n</math> has a [[Valuation (algebra)|valuation]] <math>v_\mathcal{D}:\mathcal{D}^\mathbb{N}\rightarrow\mathbb{Z}_{b}</math>
:<math>v_\mathcal{D}(m) = \sum_{i=0}^{\infty}f_\mathcal{D}(d_{i})b^{i}</math>

===={{math|''b''}}-adic solenoids====
The set of all signed-digit representations of the [[Solenoid (mathematics)#p-adic solenoids|<math>b</math>-adic solenoids]], <math>\mathbb{T}_b</math> is given by the [[Cantor space]] <math>\mathcal{D}^\mathbb{Z}</math>, the set of all [[doubly infinite]] concatenated strings of digits <math>\ldots d_{1} d_{0} d_{-1} \ldots</math>. Each signed-digit representation <math>m \in \mathcal{D}^n</math> has a [[Valuation (algebra)|valuation]] <math>v_\mathcal{D}:\mathcal{D}^\mathbb{Z}\rightarrow\mathbb{T}_{b}</math>
:<math>v_\mathcal{D}(m) = \sum_{i=-\infty}^{\infty}f_\mathcal{D}(d_{i})b^{i}</math>