Digital Algebra and Circuits

Abstract. Digital numbers D are the world’s most popular data representation: nearly all texts, sounds and images are coded somewhere in time and space by binary sequences. The mathematical construction of the fixed-point D Z2 and floating-point D Q2 digital numbers is a dual to the classical constructions of the real numbers R. The domain D contains the binary integers N and Z, as well as Q. The arithmetic operations in D are the usual ones when restricted to integers or rational numbers. Similarly, the polynomial operations in D are the usual ones when applied to finite binary polynomials F2[z] or their quotients F2(z). Finally, the set operations in D are the usual ones over finite or infinite subsets of N. The resulting algebraic structure is rich, and we identify over a dozen rings, fields and Boolean algebras in D . Each structure is well-known in its own right. The unique nature of D is to combine all into a single algebraic structure, where operations of different natu...

• Type: Conference
• Year: 2003
• Where: BIRTHDAY
• Authors: Jean Vuillemin
• Keywords: Algebraic Digital Number, Applied Computing, BIRTHDAY 2003, Digital Number, Usual Ones