Chapter 4. Binary Data Representation and Binary Arithmetic
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1 Christian Jacob Chapter 4 Binary Data Representation and Binary Arithmetic 4.1 Binary Data Representation 4.2 Important Number Systems for Computers Number System Basics Useful Number Systems for Computers Decimal Number System Binary Number System Octal Number System Hexadecimal Number System Comparison of Number Systems 4.3 Powers of 2 Chapter Overview
2 Christian Jacob 4.4 Conversions Between Number Systems Conversion of Natural Numbers Conversion of Rational Numbers 4.5 Binary Logic 4.6 Binary Arithmetic 4.7 Negative Numbers and Complements Problems of Signed Number Representation Complement ((B-1)-Complement) Complement (B-Complement) 4.8 Floating Point Numbers Mantissa and Exponent Floating Point Arithmetics Limited Precision 4.9 Representation of Strings Representation of Characters Representation of Strings Chapter Overview
3 Page 3 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.1 Binary Data Representation Bit: smallest unit of information yes / no, on / off, L / 0, 1 / 0, 5V / 0V Byte: group of 8 bits --> 2 8 = 256 different states Word: the number of bits (word length) which can be processed by a computer in a single step (e.g., 32 or 64) --> machine dependent Representation: n n First Back TOC Prev Next Last
4 Page 4 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob The word size in any given computer is fixed. Example: 16-bit word every word (memory location) can hold a 16-bit pattern, with each bit either 0 or 1. How many distinct patterns are there in a 16-bit word? Each bit has 2 possible values: 0 or 1 1 bit has 2 distinct patterns: 0, 1 With 2 bits, each position has 2 possible values: 00, 01, 10, = 4 distinct bit patterns First Back TOC Prev Next Last
5 Page 5 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob With 3 bits, again each position can be either 0 or 1: = 8 distinct bit patterns In general, for n bits (a word of length n) we have 2 n distinct bit patterns. NOTE: What these bit patterns mean depends entirely on the context in which the patterns are used. First Back TOC Prev Next Last
6 Page 6 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.2 Important Number Systems for Computers Number System Basics Number systems, as we use them, consist of - a basic set Z of digits (or letters); example: Z = { 012,,,, 9} - a base B = Z (how many digits); example: B = Z = 10 A number is a linear sequence of digits. The value of a digit at a specific position depends on its assigned meaning and on its position. The value of a number N is the sum of these values. Number: N = B d n 1 d n 2 d 1 d 0 with word length n Value: N B = n 1 d i B i = d n 1 B n 1 + d n 2 B n d 1 B 1 + d 0 B 0 i = 0 First Back TOC Important Number Systems for Computers Prev Next Last
7 Page 7 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Useful Number Systems for Computers Name Base Digits dual binary 2 0, 1 octal 8 0, 1, 2, 3, 4, 5, 6, 7 decimal 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 hexadecimal sedecimal 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F First Back TOC Important Number Systems for Computers Prev Next Last
8 Page 8 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Decimal Number System Z = { ,,,,,,,,, }; B = 10 Each position to the left of a digit increases by a power of 10. Each position to the right of a digit decreases by a power of 10. Example: = = First Back TOC Important Number Systems for Computers Prev Next Last
9 Page 9 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Binary Number System Z = { 01, }; B = 2 Each position to the left of a digit increases by a power of 2. Each position to the right of a digit decreases by a power of 2. Example: = = = First Back TOC Important Number Systems for Computers Prev Next Last
10 Page 10 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Counting in Binary Decimal Dual Decimal Dual First Back TOC Important Number Systems for Computers Prev Next Last
11 Page 11 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Octal Number System Z = { ,,,,,,, }; B = 8 Each position to the left of a digit increases by a power of 8. Each position to the right of a digit decreases by a power of 8. Example: = = = First Back TOC Important Number Systems for Computers Prev Next Last
12 Page 12 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Counting in Octal Decimal Octal Decimal Octal First Back TOC Important Number Systems for Computers Prev Next Last
13 Page 13 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Hexadecimal Number System Z = { A,,,,,,,,,,, B, C, D, E, F} ; B = 16 Each position to the left of a digit increases by a power of 16. Each position to the right of a digit decreases by a power of 16. Example: FB40A 16 = = = 1,029, First Back TOC Important Number Systems for Computers Prev Next Last
14 Page 14 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Counting in Hexadecimal Decimal Hexadecimal Decimal Hexadecimal A 26 1 A 11 0 B 27 1 B 12 0 C 28 1 C 13 0 D 29 1 D 14 0 E 30 1 E 15 0 F 31 1 F First Back TOC Important Number Systems for Computers Prev Next Last
15 Page 15 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Comparison of Number Systems Decimal Dual Octal Hexadecimal Decimal Dual Octal Hexadecimal A A B B C C D D E E F F First Back TOC Important Number Systems for Computers Prev Next Last
16 Page 16 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.3 Powers of 2 N 2 N N 2 N , , , ,048, ,097, ,194, ,388, ,777, ,554, ,108, , ,217, , ,435, , ,870, , ,073,741, , ,147,483, , ,294,967, , ,589,934,592 First Back TOC Powers of 2 Prev Next Last
17 Page 17 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.4 Conversions Between Number Systems Conversion of Natural Numbers Base 10 Base B Conversion We use the metod of iterated division to convert a number of base 10 into a number N B of base B. N Divide the number N 10 by B: whole number N & remainder R. - Take N as the next number to be divided by B. - Keep R as the next left-most digit of the new number N B. 2. If N is zero then STOP, else set N 10 = N and go to step 1. First Back TOC Conversions Between Number Systems Prev Next Last
18 Page 18 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Example: =? 8 Number to be converted: N 10 = 1020; target base B = 8 Step 1: 1020 : 8 = 127 Remainder: = 1016 Step 2: 127 : 8 = 15 Remainder: = 120 Step 3: 15 : 8 = 1 Remainder: = 8 Step 4: 1 : 8 = 0 Remainder: = First Back TOC Conversions Between Number Systems Prev Next Last
19 Page 19 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Base B Base 10 Conversion We use the metod of iterated multiplication to convert a number of base B into a number N 10 of base 10. N B We simply add the contribution of each digit. Example: 1F2 16 =? 10 Number to be converted: N 16 = 1F2; target base B = 10 1F2 16 = = F2 16 = First Back TOC Conversions Between Number Systems Prev Next Last
20 Page 20 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Conversions between Numbers 2 n and 2 m Conversion 2 n 2 1 n = 3: octal dual: Replace 1 octal digit by 3 binary digits. Example: = n = 4: hexadecimal dual Replace 1 hexadecimal digit by 4 binary digits. Example: 1 F 2 16 = First Back TOC Conversions Between Number Systems Prev Next Last
21 Page 21 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Conversion n n = 3: dual octal: Replace 3 binary digits by 1 octal digit. Example: = = n = 4: hexadecimal dual Replace 4 binary digits by 1 hexadecimal digit. Example: = = 3 1 F 2 16 First Back TOC Conversions Between Number Systems Prev Next Last
22 Page 22 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Conversion 2 n 2 m 2 n m Example: octal hexadecimal = octal to binary: = rearrange binary: = = 2 C E 9 16 First Back TOC Conversions Between Number Systems Prev Next Last
23 Page 23 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Conversion of Rational Numbers Note: An exact conversion of rational numbers R 10 R B or R B R 10 is not always possible. We want: R 10 = R B + ε, where ε is a sufficiently small number. Base B Base 10 Conversion Example: =? = = = = First Back TOC Conversions Between Number Systems Prev Next Last
24 Page 24 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Base 10 Base B Conversion Example: =? 2 with k = 9 bit precision Step i N Operation R z = = = = = = = = = ( i) = ε Multiplication! First Back TOC Conversions Between Number Systems Prev Next Last
25 Page 25 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.5 Binary Logic Logical AND ( x y) AND Logical OR ( x y) OR First Back TOC Binary Logic Prev Next Last
26 Page 26 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Logical NOT ( x, negation) x x Logical XOR (exclusive OR) XOR Logical NAND (negated AND) NAND First Back TOC Binary Logic Prev Next Last
27 Page 27 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Examples: AND NAND OR XOR First Back TOC Binary Logic Prev Next Last
28 Page 28 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.6 Binary Arithmetic Elementary Rules for addition, subtraction, and multiplication Addition Subtraction Multiplication Operation Result Carry First Back TOC Binary Arithmetic Prev Next Last
29 Page 29 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Examples: Addition: Subtraction: carry: carry: First Back TOC Binary Arithmetic Prev Next Last
30 Page 30 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Multiplication: carry: carry: carry: First Back TOC Binary Arithmetic Prev Next Last
31 Page 31 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.7 Negative Numbers and Complements Integer Numbers: A possible representation sign value Number = sign & value sign = 0 +, positive number sign = 1 -, negative number Range of integer numbers for n bits: - 2 n n-1-1 First Back TOC Negative Numbers and Complements Prev Next Last
32 Page 32 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Example (1): n = 3 Sign Value Number ? Negative zero? Range: = = = First Back TOC Negative Numbers and Complements Prev Next Last
33 Page 33 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Problems of Signed Number Representation Given 2 binary numbers: one positive, one negative (base 2) 12 (base 10) (= ) : 1-1 = 1 + (-1) = 0? (= ) Clearly this won t work! Solution: 1-Complement and 2-Complement First Back TOC Negative Numbers and Complements Prev Next Last
34 Page 34 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Complement ((B-1)-Complement) We represent negative numbers by negating the positive representation of the number (and vice versa): any 1 is changed to a 0 any 0 is changed to a 1 Example: Binary Decimal One s Complement Decimal ? Problem! First Back TOC Negative Numbers and Complements Prev Next Last
35 Page 35 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Complement (B-Complement) Positive numbers are represented as usual. Negative numbers are created by - negating the positive value and - adding one (this avoids the negative zero). Examples: Binary Decimal Two s Complement Decimal = = = = ( ) = 1 (1 1110) = First Back TOC Negative Numbers and Complements Prev Next Last
36 Page 36 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Addition Number ring for 5-digit 2-complement negative numbers positive numbers Subtraction First Back TOC Negative Numbers and Complements Prev Next Last
37 Page 37 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob With the 2-complement, subtractions can be performed as simple additions: Example: 5 digit precision, base 10 Example 1 Example 2 B = Complement 7: (-7) B-1 : (-7) B : Addition 14: (-7) B : : : (-13) B-1 : (-13) B : : (-13) B : : First Back TOC Negative Numbers and Complements Prev Next Last
38 Page 38 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.8 Floating Point Numbers Mantissa and Exponent Recall the general form of numbers expressed in scientific notation: x Printed out by a computer these numbers usually look like this: e0 6.02e23 This representation saves space and reduces redundancy. Hence, the usual format for a 32-bit real (floating point) number is: mantissa = 21 bits exponent = 11 bits First Back TOC Floating Point Numbers Prev Next Last
39 Page 39 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Floating Point Arithmetics Example: Operands Compare Exponents Adjust Exponents and Mantissae e 1 - e 2 = Result: Addition: 1. Identify mantissae and exponents 2. Compare exponents 3. Adjust exponents if necessary 4. Adjust the shorter mantissa 5. Add mantissae 6. Normalize mantissa 7. Adjust exponent First Back TOC Floating Point Numbers Prev Next Last
40 Page 40 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Example: Multiplication 1. Multiply of mantissae 2. Normalize mantissae 3. Adjust exponents 4. Add exponents Example: ( ) ( ) Step 1: multiplication of the mantissae: = First Back TOC Floating Point Numbers Prev Next Last
41 Page 41 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Step 1 (cont.): step-by-step multiplication of the mantissae Step 2: Add exponents 5 + (-3) + (-1) = 1 Result: (with 3 digits of precision) in the mantissa) First Back TOC Floating Point Numbers Prev Next Last
42 Page 42 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Limited Precision Note: Not all values can be represented! Example: - mantissa: 2 decimal digits - exponent: 1 decimal digit Sample number: = 7400 What is the next higher value? = 7500 What about values 7400 < x < 7500? They cannot be represented!!! Remedy, but not a perfect solution: Increase the number of mantissa digits. First Back TOC Floating Point Numbers Prev Next Last
43 Page 43 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.9 Representation of Strings Representation of Characters Usually: single characters are represented by 1 byte different standards: ASCII 1, EBCDIC 2 Currently: internationalization with Unicode a single character is represented by 2 bytes 1. ASCII: American Standard Code for Information Interchange 2. EBCDIC: Extended Binary Coded Decimal Interchange Code (derived from punch card standards) First Back TOC Representation of Strings Prev Next Last
44 Page 44 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob ASCII Character Encoding D C D C D C D C D C D C D C D C 0 NUL 16 DLE 32 SP P p 1 SCH 17 DC1 33! A 81 Q 97 a 113 q 2 STX 18 DC B 82 R 98 b 114 r 3 ETX 19 DC3 35 # C 83 S 99 c 115 s 4 EOT 20 DC4 36 $ D 84 T 100 d 116 t 5 ENQ 21 NAK 37 % E 85 U 101 e 117 u 6 ACK 22 SYN 38 & F 86 V 102 f 118 v 7 BEL 23 ETB G 87 W 103 g 119 w 8 BS 24 CAN 40 ( H 88 X 104 h 120 x 9 HT 25 EM 41 ) I 89 Y 105 i 121 y 10 LF 26 SUB 42 * 58 : 74 J 90 Z 106 j 122 z 11 VT 27 ESC ; 75 K 91 [ 107 k 123 { 12 FF 28 FS 44, 60 < 76 L 92 \ 108 l CR 29 GS = 77 M 93 ] 109 m 125 } 14 S0 30 RS > 78 N 94 ^ 110 n 126 ~ 15 S1 31 US 47 / 63? 79 O 95 _ 111 o 127 DEL Compare the Unicode character encodings First Back TOC Representation of Strings Prev Next Last
45 Page 45 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.10 Representation of Strings Strings are represented by sequences of characters ( usually: byte sequence) 1. Fixed length strings (example: length = 7) P e t e r SP SP C o m p u t e r 2. Variable length strings n-1 n P e t e r NUL 3. Encoding with string length at the beginning n-1 n 5 P e t e r First Back TOC Representation of Strings Prev Next Last
46 Page 46 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob With ASCII charcter encoding string comparisons are possible: AA < AD because: < AA < Aa < A1 < AA < Hence, sorting of strings becomes easy: Compare their numerical representations! First Back TOC Representation of Strings Prev Next Last
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