Editing Signed-digit representation (section)

====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]].