ABSTRACT
In conventional digital computers, integers are represented as binary numbers of fixed length. A binary number of length n is an ordered sequence
(xn-1, xn-2 , N N N , x1, xo ) of binary digits where each digit xi (also known as a bit) can assume one of the values 0 or 1. The length n of the sequence is of significance, since binary numbers in digital computers are stored in registers of a fixed length, n. The above sequence of n digits (or n-tuple) represents the integer value
X ) xn-12n-1 + xn-2 2n-2 + N N N + x12 + xo ) n-1E
xi2i . (1.1) i-o
Upper case letters are used in this book to represent numerical values or sequences of digits while lower case letters, usually indexed, represent individual digits. The weight of the digit xi in (1.1) is the ith power of 2, which is called the radix of the number system. The interpretation rule in Equation (1.1) is similar to the rule used for the ordinary decimal numbers. There are, however, two differences between these interpretation rules. First, the radix 10 is used instead of 2 in Equation (1.1) and consequently, the allowed digits in the decimal case are xi ; \0, 1, 2 , N N N , 9i instead of xi ; \0, 1i. We call the decimal numbers radix-10
numbers and the binary numbers radix-2 numbers. We indicate the radix to be used when interpreting a given sequence of digits by writing it as a subscript. Thus, the sequence (101)10 represents the decimal value 101, while the sequence (101)2 represents the decimal value 5.
