is , and the cardinality of is , giving the number of positive and negative digits respectively, such that . Balanced form representations Balanced form representations are representations where for every positive digit , there exist a corresponding negative digit such that . It follows that . Only odd bases can have balanced form representations, as otherwise has to be the opposite of itself and hence 0, but . In balanced form, the negative digits are usually denoted as positive digits with a bar over the digit, as for . For example, the digit set of balanced ternary would be with , , and . This convention is adopted in finite fields of odd prime order : Dual signed-digit representation Every digit set has a dual digit set given by the inverse order of the digits with an isomorphism defined by . As a result, for any signed-digit representations of a number system ring constructed from with valuation , there exists a dual signed-digit representations of , , constructed from with valuation , and an isomorphism defined by , where is the additive inverse operator of . The digit set for balanced form representations is self-dual. For integers Given the digit set and function as defined above, let us define an integer endofunction as the following: If the only periodic point of is the fixed point , then the set of all signed-digit representations of the integers using is given by the Kleene plus , the set of all finite concatenated strings of digits with at least one digit, with . Each signed-digit representation has a valuation . Examples include balanced ternary with digits . Otherwise, if there exist a non-zero periodic point of , then there exist integers that are represented by an infinite number of non-zero digits in . Examples include the standard decimal numeral system with the digit set , which requires an infinite number of the digit to represent the additive inverse , as , and the positional numeral system with the digit set with , which requires an infinite number of the digit to represent the number , as . For decimal fractions If the integers can be represented by the Kleene plus , then the set of all signed-digit representations of the decimal fractions, or -adic rationals , is given by , the Cartesian product of the Kleene plus , the set of all