Chapter 2 Number Systems and Codes 2.1 Positional Number Systems The traditional number system that we learned in school and use every day in positional number business is called a positional number system.In such a system,a number is rep- resented by a string of digits where each digit position has an associated weight. weight The value of a number is a weighted sum of the digits,for example: 1734=1.1000+7-100+3.10+41 Each weight is a power of 10 corresponding to the digit's position.A decimal point allows negative as well as positive powers of 10 to be used: 5185.68=5·1000+1-100+8-10+5-1+6-0.1+8-0.01 In general,a number D of the form dido.dd2 has the value D=d110+d010+d1101+d210 Here,10 is called the base or radix of the number system.In a general positional radix number system,the radix may be any integerr2,and a digit in position i has weight r.The general form of a number in such a system is dp-dp-2.d0.dd2.dn where there are p digits to the left of the point and n digits to the right of the radix point point,called the radix point.If the radix point is missing,it is assumed to be to the right of the rightmost digit.The value of the number is the sum of each digit multiplied by the corresponding power of the radix: D=∑df i=-n high-order digit Except for possible leading and trailing zeroes,the representation of a most significant digit number in a positional number system is unique.(Obviously,0185.6300 equals low-order digit 185.63,and so on.)The leftmost digit in such a number is called the high-order least significant digit or most significant digit,the rightmost is the low-order or least significant digit. As we'll learn in Chapter 3,digital circuits have signals that are normally binary digit in one of only two conditions-low or high,charged or discharged,off or on The signals in these circuits are interpreted to represent binary digits(or bits) binary radix that have one of two values,0 and 1.Thus,the binary radix is normally used to represent numbers in a digital system.The general form of a binary number is bp-1bp-2.bbo.b1b-2·bn and its value is B= b·2 Copyright 1999 by John F.Wakerly Copying Prohibited 22 Chapter 2 Number Systems and Codes DO NOT COPY DO NOT COPY DO NOT COPY DO NOT COPY DO NOT COPY DO NOT COPY DO NOT COPY DO NOT COPY DO NOT COPY Copyright © 1999 by John F. Wakerly Copying Prohibited 2.1 Positional Number Systems The traditional number system that we learned in school and use every day in business is called a positional number system. In such a system, a number is rep￾resented by a string of digits where each digit position has an associated weight. The value of a number is a weighted sum of the digits, for example: 1734 = 1·1000 + 7·100 + 3·10 + 4·1 Each weight is a power of 10 corresponding to the digit’s position. A decimal point allows negative as well as positive powers of 10 to be used: 5185.68 = 5·1000 + 1·100 + 8·10 + 5·1 + 6·0.1 + 8·0.01 In general, a number D of the form d1d0.d−1d−2 has the value D = d1·101 + d0·100 + d–1·10–1 + d–2·10–2 Here, 10 is called the base or radix of the number system. In a general positional number system, the radix may be any integer r ≥ 2, and a digit in position i has weight ri . The general form of a number in such a system is dp–1dp–2···d1d0 . d–1d–2···d–n where there are p digits to the left of the point and n digits to the right of the point, called the radix point. If the radix point is missing, it is assumed to be to the right of the rightmost digit. The value of the number is the sum of each digit multiplied by the corresponding power of the radix: Except for possible leading and trailing zeroes, the representation of a number in a positional number system is unique. (Obviously, 0185.6300 equals 185.63, and so on.) The leftmost digit in such a number is called the high-order or most significant digit; the rightmost is the low-order or least significant digit. As we’ll learn in Chapter 3, digital circuits have signals that are normally in one of only two conditions—low or high, charged or discharged, off or on. The signals in these circuits are interpreted to represent binary digits (or bits) that have one of two values, 0 and 1. Thus, the binary radix is normally used to represent numbers in a digital system. The general form of a binary number is bp–1bp–2···b1b0 . b–1b–2···b–n and its value is positional number system weight base radix radix point D di r i ⋅ i n = – p – 1 = ∑ high-order digit most significant digit low-order digit least significant digit binary digit bit binary radix B bi 2i ⋅ i n = – p – 1 = ∑