: We develop a formal account of digit serial number representations by describing them as strings from a language. A prefix of a string represents an interval approximating a number by enclosure. Standard on-line representations are shown to be a special case of the general digit serial representations. Matrices are introduced as representations of intervals and a finite-state transducer is used for mapping strings into intervals. Homographic and bi-homographic functions are used for representing basic arithmetic operations on digit serial numbers, and finally a digit serial representation of floating point numbers is introduced. Key Words: Computer Arithmetic, On-line Computation, Number Representations, Redundant Digit sets, Continued Fractions, Intervals. Category: B.2 1 Introduction A number is usually represented as a string of digits belonging to some digit set \Sigma . The number representation specifies a function that maps the string to its value. In the context of this pa...