finite concatenated strings of digits with at least one digit, the singleton consisting of the radix point ( or ), and the Kleene star , the set of all finite concatenated strings of digits , with . Each signed-digit representation has a valuation For real numbers If the integers can be represented by the Kleene plus , then the set of all signed-digit representations of the real numbers is given by , the Cartesian product of the Kleene plus , the set of all finite concatenated strings of digits with at least one digit, the singleton consisting of the radix point ( or ), and the Cantor space , the set of all infinite concatenated strings of digits , with . Each signed-digit representation has a valuation . The infinite series always converges to a finite real number. For other number systems All base- numerals can be represented as a subset of , the set of all doubly infinite sequences of digits in , where is the set of integers, and the ring of base- numerals is represented by the formal power series ring , the doubly infinite series where for . Integers modulo powers of The set of all signed-digit representations of the integers modulo , is given by the set , the set of all finite concatenated strings of digits of length , with . Each signed-digit representation has a valuation Prüfer groups A Prüfer group is the quotient group of the integers and the -adic rationals. The set of all signed-digit representations of the Prüfer group is given by the Kleene star , the set of all finite concatenated strings of digits , with . Each signed-digit representation has a valuation Circle group The circle group is the quotient group of the integers and the real numbers. The set of all signed-digit representations of the circle group is given by the Cantor space , the set of all right-infinite concatenated strings of digits . Each signed-digit representation has a valuation The infinite series always converges. -adic integers The set of all signed-digit representations of the -adic integers, is given by the Cantor space , the set of all left-infinite concatenated strings of digits . Each signed-digit representation has a valuation -adic solenoids The set of all signed-digit representations of the -adic solenoids, is given by the Cantor space , the set of all doubly infinite concatenated strings