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Notes - Chapter 2 DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERSITY OF NEW MEXICO ECE-238L: Computer Logic Design Fall 2013 Notes - Chapter 2 BINARY NUMBER CONVERSIONS DECIMAL NUMBER SYSTEM DIGIT . A decimal digit can take values from 0 to 9: 0 1 2 3 4 5 6 7 8 9 . Digit-by-digit representation of a positive integer number (powers of 10): Number: 9372 hundreds tens thousands units 9 3 7 2 9 thousands, 3 hundreds, 7 tens, and 2 units 103 102 101 100 9372 = 9×103 + 3×102 + 7×101 + 2×100 Positional Number Representation . Let’s consider the numbers from 0 to 999. We represent these numbers with 3 digits (each digit being a number between 0 and 9). We show a 3-digit number using the positional number representation: MATHEMATICAL REPRESENTATION EXAMPLE 3-digit d2 d1 d0 2 0 9 Third Digit Second Digit First Digit Third Digit Second Digit First Digit . The positional number representation allows us to express the decimal value using powers of ten: 2 1 0 퐷 = 푑2 × 10 + 푑1 × 10 + 푑0 × 10 . Example: 3-digit representation Decimal Powers of 10: 2 1 0 Number d2d1d0 d2 10 + d1 10 + d0 10 2 1 0 0 000 0 10 + 0 10 + 0 10 9 009 0 102 + 0 101 + 9 100 11 011 0 102 + 1 101 + 1 100 25 025 0 102 + 2 101 + 5 100 90 090 0 102 + 9 101 + 0 100 1 102 + 2 101 + 8 100 128 128 2 1 0 255 255 2 10 + 5 10 + 5 10 Exercise: Write down the 3-digit and the powers of ten representations for the following numbers: 2 1 0 Decimal Number 3-digit representation 푑2 × 10 + 푑1 × 10 + 푑0 × 10 5 254 100 99 Instructor: Daniel Llamocca DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERSITY OF NEW MEXICO ECE-238L: Computer Logic Design Fall 2013 General Case: . Positional number representation for an integer positive number with ‘n’ digits: dn-1dn-2 ... d1d0 n-1 n-2 1 0 Decimal Value: dn-1 10 + dn-2 10 + ... + d1 10 + d0 10 . Example: 1098324 (7 digits). 1098324 1106 0 105 9 104 8 103 3102 2 101 4 100 203476 (6 digits). 203476 2 105 0 104 3103 4 102 7 101 6 100 Maximum value: . The table presents the maximum attainable value for a given number of digits. What pattern do you find? Can you complete it for the highlighted cases (4 and 6)? Number of digits Maximum value Range 1 9 = 101-1 0 9 0 101-1 2 99 = 102-1 0 99 0 102-1 3 999 = 103-1 0 999 0 103-1 4 5 99999 = 105-1 0 99999 0 105-1 6 … n 999…999 = 10n-1 0 999…999 0 10n-1 . Maximum value for a number with ‘n’ digits: Based on the table, the maximum decimal value for a number with ‘n’ digits is given by: n-1 n-2 1 0 n D = 999...999 = 9 10 + 9 10 + ... + 9 10 + 9 10 = 10 -1 n digits With ‘n’ digits, we can represent positive integer numbers from 0 to 10n-1. With 7 digits, what is the range (starting from 0) of positive numbers that we can represent? How many different numbers can we represent? BINARY NUMBER SYSTEM . We are used to the decimal number system. However, DIGIT BIT there exist other number systems: octal, hexadecimal, vigesimal, binary, etc. In particular, binary numbers are 0 1 2 3 4 5 6 7 8 9 0 1 very practical as they are used by digital computers. For binary numbers, the counterpart of digit (that can take values from 0 to 9) is the bit (binary digit) that can take the value of 0 or 1. Bit: Unit of information that a computer uses to process and retrieve data. It can also be used as a logical variable (see Chapter 1). Binary representation: We use the positional number representation. Note that here, we are using a string of bits to represent a binary number. Converting a binary number into a decimal number: The following figure depicts two cases: 2-bit numbers and 3-bit numbers. Note that the positional representation with powers of two let us obtain the decimal value (integer positive) of the binary number. Instructor: Daniel Llamocca DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERSITY OF NEW MEXICO ECE-238L: Computer Logic Design Fall 2013 Binary number Powers of 2: Decimal 1 0 b1b0 b1 2 + b0 2 Number MATHEMATICAL REPRESENTATION 2-bit 1 0 b b 00 0 2 + 0 2 0 1 0 1 0 01 0 2 + 1 2 1 10 1 21 + 0 20 2 Second Bit First Bit 11 1 21 + 1 20 3 Binary number Powers of 2: Decimal 2 1 0 b2b1b0 b2 2 + b1 2 + b0 2 Number 2 1 0 MATHEMATICAL REPRESENTATION 000 0 2 + 0 2 + 0 2 0 2 1 0 3-bit 001 0 2 + 0 2 + 1 2 1 b b b 010 0 22 + 1 21 + 0 20 2 2 1 0 011 0 22 + 1 21 + 1 20 3 100 1 22 + 0 21 + 0 20 4 2 1 0 Third Bit Second Bit First Bit 101 1 2 + 0 2 + 1 2 5 110 1 22 + 1 21 + 0 20 6 111 1 22 + 1 21 + 1 20 7 General case: . Positional number representation for a binary number with ‘n’ bits: b b ... b b n-1 n-2 1 0 Most significant Least significant (leftmost) bit (rightmost) bit This formula converts a binary number into a decimal number: Decimal Value (integer positive): n-1 n-2 1 0 D = bn-1 2 + bn-2 2 + ... + b1 2 + b0 2 . To avoid confusion, we usually write a binary number and attach a suffix ‘2’: (푏푛−1푏푛−2 … 푏1푏0)2 5 4 3 2 1 0 . Example: 6 푏푖푡푠: (101011)2 ≡ 퐷 = 1 × 2 + 0 × 2 + 1 × 2 + 0 × 2 + 1 × 2 + 1 × 2 = 43 3 2 1 0 4 푏푖푡푠: (1011)2 ≡ 퐷 = 1 × 2 + 0 × 2 + 1 × 2 + 1 × 2 = 11 . Maximum value for a given number of bits. Complete the tables for the highlighted cases (4 and 6): Number of bits Maximum value Range 1 1 1 12 2 -1 0 12 0 2 -1 2 2 2 112 2 -1 0 112 0 2 -1 3 3 3 1112 2 -1 0 1112 0 2 -1 4 5 5 5 111112 2 -1 0 111112 0 2 -1 6 … n n n 111…1112 2 -1 0 111…1112 0 2 -1 . Maximum value for ‘n’ bits: The maximum decimal value with ‘n’ bits is given by: n-1 n-2 1 0 n D = 111...111 = 1 2 + 1 2 + ... + 1 2 + 1 2 = 2 -1 n bits With ‘n’ bits, we can represent positive integer numbers from 0 to 2n-1. (2n numbers) Instructor: Daniel Llamocca DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERSITY OF NEW MEXICO ECE-238L: Computer Logic Design Fall 2013 . The case n=8 bits is of particular interest, as a string of 8 bits is called a byte. Here, the range for 8-bit numbers is 0 to 28-1 0 to 255. b b b b b b b b 7 6 5 4 3 2 1 0 Most significant Least significant (leftmost) bit (rightmost) bit . The table shows some examples: Decimal 8-bit format b 27 + b 26 + b 25 + b 24 + b 23 + b 22 + b 21 + b 20 Number b7b6b5b4b3b2d1d0 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 0 00000000 0 2 + 0 2 + 0 2 + 0 2 + 0 2 + 0 2 + 0 2 + 0 2 9 00001001 0 27 + 0 26 + 0 25 + 0 24 + 1 23 + 0 22 + 0 21 + 1 20 11 00001011 0 27 + 0 26 + 0 25 + 0 24 + 1 23 + 0 22 + 1 21 + 1 20 25 00011001 0 27 + 0 26 + 0 25 + 1 24 + 1 23 + 0 22 + 0 21 + 1 20 90 01011010 0 27 + 1 26 + 0 25 + 1 24 + 1 23 + 0 22 + 1 21 + 0 20 128 10000000 1 27 + 0 26 + 0 25 + 0 24 + 0 23 + 0 22 + 0 21 + 0 20 255 11111111 1 27 + 1 26 + 1 25 + 1 24 + 1 23 + 1 22 + 1 21 + 1 20 Exercise: Convert the following binary numbers (positive integers) to their decimal values: 8-bit Decimal 7 6 5 4 3 2 1 0 representation 푏7 × 2 + 푏6 × 2 + 푏5 × 2 + 푏4 × 2 + 푏3 × 2 + 푏2 × 2 + 푏1 × 2 + 푏0 × 2 Number 00000001 00001001 10000101 10000111 11110011 . In general to convert a number of base 'r' (r = 2, 3,4 ,…) to decimal, we use the following formula: Number in base 'r': (푟푛−1푟푛−2 … 푟1푟0)푟 푛−1 푛−2 1 0 Conversion to decimal: 퐷 = 푟푛−1 × 푟 + 푟푛−2 × 푟 + ⋯ + 푟1 × 푟 + 푟0 × 푟 Also, the maximum decimal value for a number in base 'r' with 'n' digits is: 푛−1 푛−2 1 0 푛 퐷 = 푟푟푟 … 푟푟푟 = 푟 × 푟 + 푟푛−2 × 푟 + ⋯ + 푟 × 푟 + 푟 × 푟 = 푟 − 1 Example: Base-8: Number of digits Maximum value Range 1 1 1 78 8 -1 0 78 0 8 -1 2 2 2 778 8 -1 0 778 0 8 -1 3 3 3 7778 8 -1 0 7778 0 8 -1 … n n n 777…7778 8 -1 0 777…7778 0 8 -1 Examples: . (50632)8: Number in base 8 (octal system) Number of digits: n = 5 Conversion to decimal: 퐷 = 5 × 84 + 0 × 83 + 6 × 82 + 3 × 81 + 2 × 80 = 20890 .
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