Numbers and Arithmetic

Abstract

Processing, display, and communication of digital information, that is, information encoded as numbers, is accomplished by arithmetic with various kinds of numbers. Such computations are performed with algorithms, which are sequences of operations with numbers such as addition, multiplication, and reading and writing digits. Only finite algorithms can be used: these are procedures in which •Every operation can be performed in a finite time; •The algorithm is guaranteed to stop after a finite number of operations.

For an algorithm to be finite, its arithmetic operations can only be carried out to a finite degree of precision. In reality, a computer can keep only a small number of digits for each number because memory, processing and data communication are costly resources. But this usually poses no problems since the digital information of multimedia signals is itself of low precision. For example, a “CD-quality” digital sound recording consists of a sequence of numbers measuring the electrical output of a microphone at sequential times, with a precision of 5 decimal digits or less per measurement. Images from typical scanners are even less precise, consisting of arrays of numbers measuring light intensity to 3 decimal digits. Physical measurement is always imprecise, so these low precisions cannot be improved much. But the result is that computation for multimedia signal processing can be done with low precision arithmetic.