Signed-digit representation

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

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

Balanced form representations are representations where for every positive digit ${\displaystyle d_{+}}$, there exist a corresponding negative digit ${\displaystyle d_{-}}$ such that ${\displaystyle f_{\mathcal {D}}(d_{+})=-f_{\mathcal {D}}(d_{-})}$. It follows that ${\displaystyle b_{+}=b_{-}}$. Only odd bases can have balanced form representations, as when ${\displaystyle b_{+}=b_{-}}$ then ${\displaystyle b=b_{+}+b_{-}+1=2b_{+}+1}$ will be an odd number. In balanced form, the negative digits ${\displaystyle d_{-}\in {\mathcal {D}}_{-}}$ are usually denoted as positive digits with a bar over the digit, as ${\displaystyle d_{-}={\bar {d}}_{+}}$ for ${\displaystyle d_{+}\in {\mathcal {D}}_{+}}$. For example, the digit set of balanced ternary would be ${\displaystyle {\mathcal {D}}_{3}=\lbrace {\bar {1}},0,1\rbrace }$ with ${\displaystyle f_{{\mathcal {D}}_{3}}({\bar {1}})=-1}$, ${\displaystyle f_{{\mathcal {D}}_{3}}(0)=0}$, and ${\displaystyle f_{{\mathcal {D}}_{3}}(1)=1}$. This convention is adopted in finite fields of odd prime order ${\displaystyle q}$:[7]

${\displaystyle \mathbb {F} _{q}=\lbrace 0,1,{\bar {1}}=-1,...d={\frac {q-1}{2}},\ {\bar {d}}={\frac {1-q}{2}}\ |\ q=0\rbrace .}$

Every digit set ${\displaystyle {\mathcal {D}}}$ has a dual digit set ${\displaystyle {\mathcal {D}}^{\operatorname {op} }}$ given by the inverse order of the digits with an isomorphism ${\displaystyle g:{\mathcal {D}}\rightarrow {\mathcal {D}}^{\operatorname {op} }}$ defined by ${\displaystyle -f_{\mathcal {D}}=g\circ f_{{\mathcal {D}}^{\operatorname {op} }}}$. As a result, for any signed-digit representations ${\displaystyle {\mathcal {N}}}$ of a number system ring ${\displaystyle N}$ constructed from ${\displaystyle {\mathcal {D}}}$ with valuation ${\displaystyle v_{\mathcal {D}}:{\mathcal {N}}\rightarrow N}$, there exists a dual signed-digit representations of ${\displaystyle N}$, ${\displaystyle {\mathcal {N}}^{\operatorname {op} }}$, constructed from ${\displaystyle {\mathcal {D}}^{\operatorname {op} }}$ with valuation ${\displaystyle v_{{\mathcal {D}}^{\operatorname {op} }}:{\mathcal {N}}^{\operatorname {op} }\rightarrow N}$, and an isomorphism ${\displaystyle h:{\mathcal {N}}\rightarrow {\mathcal {N}}^{\operatorname {op} }}$ defined by ${\displaystyle -v_{\mathcal {D}}=h\circ v_{{\mathcal {D}}^{\operatorname {op} }}}$, where ${\displaystyle -}$ is the additive inverse operator of ${\displaystyle N}$. The digit set for balanced form representations is self-dual.

Given the digit set ${\displaystyle {\mathcal {D}}}$ and function ${\displaystyle f:{\mathcal {D}}\rightarrow \mathbb {Z} }$ as defined above, let us define an integer endofunction ${\displaystyle T:\mathbb {Z} \rightarrow \mathbb {Z} }$ as the following:

${\displaystyle T(n)={\begin{cases}{\frac {n-f(d_{i})}{b}}&{\text{if }}n\equiv i{\bmod {b}},0\leq i<b\end{cases}}}$

If the only periodic point of ${\displaystyle T}$ is the fixed point ${\displaystyle 0}$, then the set of all signed-digit representations of the integers ${\displaystyle \mathbb {Z} }$ using ${\displaystyle {\mathcal {D}}}$ is given by the Kleene plus ${\displaystyle {\mathcal {D}}^{+}}$, the set of all finite concatenated strings of digits ${\displaystyle d_{n}\ldots d_{0}}$ with at least one digit, with ${\displaystyle n\in \mathbb {N} }$. Each signed-digit representation ${\displaystyle m\in {\mathcal {D}}^{+}}$ has a valuation ${\displaystyle v_{\mathcal {D}}:{\mathcal {D}}^{+}\rightarrow \mathbb {Z} }$

${\displaystyle v_{\mathcal {D}}(m)=\sum _{i=0}^{n}f_{\mathcal {D}}(d_{i})b^{i}}$.

Examples include balanced ternary with digits ${\displaystyle {\mathcal {D}}=\lbrace {\bar {1}},0,1\rbrace }$.

Otherwise, if there exist a non-zero periodic point of ${\displaystyle T}$, then there exist integers that are represented by an infinite number of non-zero digits in ${\displaystyle {\mathcal {D}}}$. Examples include the standard decimal numeral system with the digit set ${\displaystyle \operatorname {dec} =\lbrace 0,1,2,3,4,5,6,7,8,9\rbrace }$, which requires an infinite number of the digit ${\displaystyle 9}$ to represent the additive inverse ${\displaystyle -1}$, as ${\displaystyle T_{\operatorname {dec} }(-1)={\frac {-1-9}{10}}=-1}$, and the positional numeral system with the digit set ${\displaystyle {\mathcal {D}}=\lbrace {\text{A}},0,1\rbrace }$ with ${\displaystyle f({\text{A}})=-4}$, which requires an infinite number of the digit ${\displaystyle {\text{A}}}$ to represent the number ${\displaystyle 2}$, as ${\displaystyle T_{\mathcal {D}}(2)={\frac {2-(-4)}{3}}=2}$.

If the integers can be represented by the Kleene plus ${\displaystyle {\mathcal {D}}^{+}}$, then the set of all signed-digit representations of the decimal fractions, or ${\displaystyle b}$-adic rationals ${\displaystyle \mathbb {Z} [1\backslash b]}$, is given by ${\displaystyle {\mathcal {Q}}={\mathcal {D}}^{+}\times {\mathcal {P}}\times {\mathcal {D}}^{*}}$, the Cartesian product of the Kleene plus ${\displaystyle {\mathcal {D}}^{+}}$, the set of all finite concatenated strings of digits ${\displaystyle d_{n}\ldots d_{0}}$ with at least one digit, the singleton ${\displaystyle {\mathcal {P}}}$ consisting of the radix point (${\displaystyle .}$ or ${\displaystyle ,}$), and the Kleene star ${\displaystyle {\mathcal {D}}^{*}}$, the set of all finite concatenated strings of digits ${\displaystyle d_{-1}\ldots d_{-m}}$, with ${\displaystyle m,n\in \mathbb {N} }$. Each signed-digit representation ${\displaystyle q\in {\mathcal {Q}}}$ has a valuation ${\displaystyle v_{\mathcal {D}}:{\mathcal {Q}}\rightarrow \mathbb {Z} [1\backslash b]}$

${\displaystyle v_{\mathcal {D}}(q)=\sum _{i=-m}^{n}f_{\mathcal {D}}(d_{i})b^{i}}$

If the integers can be represented by the Kleene plus ${\displaystyle {\mathcal {D}}^{+}}$, then the set of all signed-digit representations of the real numbers ${\displaystyle \mathbb {R} }$ is given by ${\displaystyle {\mathcal {R}}={\mathcal {D}}^{+}\times {\mathcal {P}}\times {\mathcal {D}}^{\mathbb {N} }}$, the Cartesian product of the Kleene plus ${\displaystyle {\mathcal {D}}^{+}}$, the set of all finite concatenated strings of digits ${\displaystyle d_{n}\ldots d_{0}}$ with at least one digit, the singleton ${\displaystyle {\mathcal {P}}}$ consisting of the radix point (${\displaystyle .}$ or ${\displaystyle ,}$), and the Cantor space ${\displaystyle {\mathcal {D}}^{\mathbb {N} }}$, the set of all infinite concatenated strings of digits ${\displaystyle d_{-1}d_{-2}\ldots }$, with ${\displaystyle n\in \mathbb {N} }$. Each signed-digit representation ${\displaystyle r\in {\mathcal {R}}}$ has a valuation ${\displaystyle v_{\mathcal {D}}:{\mathcal {R}}\rightarrow \mathbb {R} }$

${\displaystyle v_{\mathcal {D}}(r)=\sum _{i=-\infty }^{n}f_{\mathcal {D}}(d_{i})b^{i}}$.

The infinite series always converges to a finite real number.

All base-${\displaystyle b}$ numerals can be represented as a subset of ${\displaystyle {\mathcal {D}}^{\mathbb {Z} }}$, the set of all doubly infinite sequences of digits in ${\displaystyle {\mathcal {D}}}$, where ${\displaystyle \mathbb {Z} }$ is the set of integers, and the ring of base-${\displaystyle b}$ numerals is represented by the formal power series ring ${\displaystyle \mathbb {Z} [[b,b^{-1}]]}$, the doubly infinite series

${\displaystyle \sum _{i=-\infty }^{\infty }a_{i}b^{i}}$

where ${\displaystyle a_{i}\in \mathbb {Z} }$ for ${\displaystyle i\in \mathbb {Z} }$.

The set of all signed-digit representations of the integers modulo ${\displaystyle b^{n}}$, ${\displaystyle \mathbb {Z} \backslash b^{n}\mathbb {Z} }$ is given by the set ${\displaystyle {\mathcal {D}}^{n}}$, the set of all finite concatenated strings of digits ${\displaystyle d_{n-1}\ldots d_{0}}$ of length ${\displaystyle n}$, with ${\displaystyle n\in \mathbb {N} }$. Each signed-digit representation ${\displaystyle m\in {\mathcal {D}}^{n}}$ has a valuation ${\displaystyle v_{\mathcal {D}}:{\mathcal {D}}^{n}\rightarrow \mathbb {Z} /b^{n}\mathbb {Z} }$

${\displaystyle v_{\mathcal {D}}(m)\equiv \sum _{i=0}^{n-1}f_{\mathcal {D}}(d_{i})b^{i}{\bmod {b}}^{n}}$

A Prüfer group is the quotient group ${\displaystyle \mathbb {Z} (b^{\infty })=\mathbb {Z} [1\backslash b]/\mathbb {Z} }$ of the integers and the ${\displaystyle b}$-adic rationals. The set of all signed-digit representations of the Prüfer group is given by the Kleene star ${\displaystyle {\mathcal {D}}^{*}}$, the set of all finite concatenated strings of digits ${\displaystyle d_{1}\ldots d_{n}}$, with ${\displaystyle n\in \mathbb {N} }$. Each signed-digit representation ${\displaystyle p\in {\mathcal {D}}^{*}}$ has a valuation ${\displaystyle v_{\mathcal {D}}:{\mathcal {D}}^{*}\rightarrow \mathbb {Z} (b^{\infty })}$

${\displaystyle v_{\mathcal {D}}(m)\equiv \sum _{i=1}^{n}f_{\mathcal {D}}(d_{i})b^{-i}{\bmod {1}}}$

The circle group is the quotient group ${\displaystyle \mathbb {T} =\mathbb {R} /\mathbb {Z} }$ of the integers and the real numbers. The set of all signed-digit representations of the circle group is given by the Cantor space ${\displaystyle {\mathcal {D}}^{\mathbb {N} }}$, the set of all right-infinite concatenated strings of digits ${\displaystyle d_{1}d_{2}\ldots }$. Each signed-digit representation ${\displaystyle m\in {\mathcal {D}}^{n}}$ has a valuation ${\displaystyle v_{\mathcal {D}}:{\mathcal {D}}^{\mathbb {N} }\rightarrow \mathbb {T} }$

${\displaystyle v_{\mathcal {D}}(m)\equiv \sum _{i=1}^{\infty }f_{\mathcal {D}}(d_{i})b^{-i}{\bmod {1}}}$

The infinite series always converges.

The set of all signed-digit representations of the ${\displaystyle b}$-adic integers, ${\displaystyle \mathbb {Z} _{b}}$ is given by the Cantor space ${\displaystyle {\mathcal {D}}^{\mathbb {N} }}$, the set of all left-infinite concatenated strings of digits ${\displaystyle \ldots d_{1}d_{0}}$. Each signed-digit representation ${\displaystyle m\in {\mathcal {D}}^{n}}$ has a valuation ${\displaystyle v_{\mathcal {D}}:{\mathcal {D}}^{\mathbb {N} }\rightarrow \mathbb {Z} _{b}}$

${\displaystyle v_{\mathcal {D}}(m)=\sum _{i=0}^{\infty }f_{\mathcal {D}}(d_{i})b^{i}}$

The set of all signed-digit representations of the ${\displaystyle b}$-adic solenoids, ${\displaystyle \mathbb {T} _{b}}$ is given by the Cantor space ${\displaystyle {\mathcal {D}}^{\mathbb {Z} }}$, the set of all doubly infinite concatenated strings of digits ${\displaystyle \ldots d_{1}d_{0}d_{-1}\ldots }$. Each signed-digit representation ${\displaystyle m\in {\mathcal {D}}^{n}}$ has a valuation ${\displaystyle v_{\mathcal {D}}:{\mathcal {D}}^{\mathbb {Z} }\rightarrow \mathbb {T} _{b}}$

${\displaystyle v_{\mathcal {D}}(m)=\sum _{i=-\infty }^{\infty }f_{\mathcal {D}}(d_{i})b^{i}}$