Editing Signed-digit representation (section)

====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>