IT 1204 Section 2.0. Data Representation and Arithmetic. 2009, University of Colombo School of Computing 1


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1 IT 1204 Section 2.0 Data Representation and Arithmetic 2009, University of Colombo School of Computing 1
2 What is Analog and Digital The interpretation of an analog signal would correspond to a signal whose key characteristic would be a continuous signal A digital signal is one whose key characteristic (e.g. voltage or current) fall into discrete ranges of values Most digital systems utilize two voltage levels 2009, University of Colombo School of Computing 2
3 Advantage of Digital over Analog 2009, University of Colombo School of Computing 3
4 What is a bit A bit is a binary digit, the smallest increment of data on a machine. A bit can hold only one of two values: 0 or 1 Because bits are so small, you rarely work with information one bit at a time. 2009, University of Colombo School of Computing 4
5 Real World Problem Abstract to What is a bit Apply to Algorithm Code for Knowledge Leading to Use to make Operate on Machine Interpret as Model Encode as Data Information 2009, University of Colombo School of Computing 5
6 Bit Storage  Capacitor plates discharged plates charging plates charged 4 5 plates discharging plates discharged Can store 1 of 2 possible states 2009, University of Colombo School of Computing 6
7 What is a bit Byte is an abbreviation for "binary term". A single byte is composed of 8 consecutive bits capable of storing a single character 2009, University of Colombo School of Computing 7
8 Storage Hierarchy 8 Bits = 1 Byte 1024 Bytes = 1 Kilobyte (KB) 1024 KB = 1 Megabyte (MB) 1024 MB = 1 Gigabyte (GB) A word is the default data size for a processor 2009, University of Colombo School of Computing 8
9 Numbering System Decimal System Alphabet = { 0,1,2,3,4,5,6,7,8,9 } Octal System Alphabet = { 0,1,2,3,4,5,6,7 } Hexadecimal System Alphabet = { 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F } Binary System Alphabet = { 0,1 } 2009, University of Colombo School of Computing 9
10 Converting decimal to binary remainder remainder remainder remainder remainder remainder remainder , University of Colombo School of Computing 10
11 Converting decimal to binary remainder remainder remainder remainder remainder remainder 1 2 Most 0 remainder 1 significant bit 2009, University of Colombo School of Computing 11
12 Converting decimal to binary remainder remainder remainder remainder remainder remainder remainder 1 Least significant bit 2009, University of Colombo School of Computing 12
13 Your turn Convert the number to binary 2009, University of Colombo School of Computing 13
14 Bit position Converting binary to decimal Decimal value 2009, University of Colombo School of Computing 14
15 Bit position Converting binary to decimal Decimal value 2009, University of Colombo School of Computing 15
16 Converting binary to decimal Example: Convert the unsigned binary number to decimal , University of Colombo School of Computing 16
17 Converting binary to decimal , University of Colombo School of Computing 17
18 Converting binary to decimal , University of Colombo School of Computing 18
19 Converting binary to decimal = 154 So, in unsigned binary is 154 in decimal 2009, University of Colombo School of Computing 19
20 Converting binary to decimal Example: Convert the decimal number 105 to unsigned binary 2009, University of Colombo School of Computing 20
21 Converting binary to decimal Q. Does 128 fit into 105? A. No Next, consider the difference: 1050*128 = , University of Colombo School of Computing 21
22 Converting binary to decimal Q. Does 64 fit into 105? A. Yes Next, consider the difference: 1051*64 = , University of Colombo School of Computing 22
23 Q. Does 32 fit into 41? A. Yes Converting binary to decimal Next, consider the difference: = , University of Colombo School of Computing 23
24 Your turn Convert the number to decimal 2009, University of Colombo School of Computing 24
25 Converting binary numbers Decimal System 0,1,2,3,4,5,6,7,8,9,10,11,12,13 Binary System 0,1,10,11,100,101,110,111,1000,1001,1010,1011,1100, , University of Colombo School of Computing 25
26 Your turn Using 5 binary digits how many numbers you can represent? 2009, University of Colombo School of Computing 26
27 Hexadecimal Notation HEX Bit Pattern HEX Bit Pattern A B C D E F , University of Colombo School of Computing 27
28 Your turn How many binary digits need to represent a hexadecimal digit? 2009, University of Colombo School of Computing 28
29 Converting hexadecimal numbers Decimal System 0,1,2,3,4,5,6,7,8,9,10,11,12,13 Hexadecimal System 0,1,,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10,11,12,13,14,15, 16,17,18,19,1A,1B,1C,1D,1E,1F , University of Colombo School of Computing 29
30 Binary to Hexadecimal Conversion = 96 Hexadecimal 2009, University of Colombo School of Computing 30
31 Binary to Hexadecimal Conversion D B = DB Hexadecimal 2009, University of Colombo School of Computing 31
32 Your turn Convert the following binary string to Hexadecimal , University of Colombo School of Computing 32
33 Binary to Hexadecimal Conversion F = 29F5 Hex 2009, University of Colombo School of Computing 33
34 Computer Number System 2009, University of Colombo School of Computing 34
35 ASCII Codes American Standard Code for Information Interchange (ASCII) Use bit patterns of length seven to represent Letters of English alphabet: a  z and A  Z Digits: 0 9 Punctuation symbols: (, ), [, ], {, },,,!, /, \ Arithmetic Operation symbols: +, , *, <, >, = Special symbols: (space), %, $, #, ^ 2 7 = 128 characters can be represented by ASCII 2009, University of Colombo School of Computing 35
36 Character Representation: ASCII Table Symbol ASCII Symbol ASCII Symbol ASCII Symbol ASCII (space) A a ! B b C c # D d $ E e % F f & G g , University of Colombo School of Computing 36
37 Character Representation: ASCII Table 2009, University of Colombo School of Computing 37
38 Character Representation: ASCII Table As computers became more reliable the need for parity bit faded. Computer manufacturers extended ASCII to provide more characters, e.g., international characters Used ranges (2 7 ) (2 81) 2009, University of Colombo School of Computing 38
39 The BINARY string can have two meanings! the CHARACTER 5 in ASCII AND... Your turn the DECIMAL NUMBER 53 in BINARY Notation 2009, University of Colombo School of Computing 39
40 Character Representation: Unicode EBCDIC and ASCII are built around the Latin alphabet Are restricted in their ability for representing non Latin alphabet Countries developed their own codes for native languages Unicode: 16bit system that can encode the characters of most languages 16 bits = 2 16 = 65,636 characters 2009, University of Colombo School of Computing 40
41 Character Representation: Unicode The Java programming language and some operating systems now use Unicode as their default character code Unicode codespace is divided into six parts The first part is for Western alphabet codes, including English, Greek, and Russian Downward compatible with ASCII and Latin1 character sets 2009, University of Colombo School of Computing 41
42 Character Representation: Unicode 2009, University of Colombo School of Computing 42
43 Character Representation: Example English section of Unicode Table ACSII equivalent of A is Unicode is equivalent of A: Full chart list: , University of Colombo School of Computing 43
44 Performing Arithmetic 2009, University of Colombo School of Computing 44
45 Binary Addition = = = = 10 (carry: 1) E.g (carry) = , University of Colombo School of Computing 45
46 Binary Subtraction 00 = = 1 (with borrow) 10 = = 0 E.g. * * * (borrow) = , University of Colombo School of Computing 46
47 Binary Multiplication E.g x = , University of Colombo School of Computing 47
48 Binary Division E.g , University of Colombo School of Computing 48
49 Representing Numbers Problems of number representation Positive and negative Radix point Range of representation Different ways to represent numbers Unsigned representation: nonnegative integers Signed representation: integers Floatingpoint representation: fractions 2009, University of Colombo School of Computing 49
50 Unsigned and Signed Numbers Unsigned binary numbers Have 0 and 1 to represent numbers Only positive numbers stored in binary The Smallest binary number would be which equals to 0 The largest binary number would be which equals = 255 = Therefore the range is (256 numbers) 2009, University of Colombo School of Computing 50
51 Unsigned and Signed Numbers Signed binary numbers Have 0 and 1 to represent numbers The leftmost bit is a sign bit 0 for positive 1 for negative Sign bit 2009, University of Colombo School of Computing 51
52 Unsigned and Signed Numbers Signed binary numbers The Smallest positive binary number is which equals to 0 The largest positive binary number is which equals = 127 = Therefore the range for positive numbers is (128 numbers) 2009, University of Colombo School of Computing 52
53 Negative Numbers in Binary Problems with simple signed representation Two representation of zero: + 0 and and Need to consider both sign and magnitude in arithmetic E.g. 5 3 = 5 + (3) = = = , University of Colombo School of Computing 53
54 Negative Numbers in Binary Problems with simple signed representation Need to consider both sign and magnitude in arithmetic E.g. = 18 + (18) = = = , University of Colombo School of Computing 54
55 Negative Numbers in Binary The representation of a negative integer (Two s Complement) is established by: Start from the signed binary representation of its positive value Copy the bit pattern from right to left until a 1 has been copied Complement the remaining bits: all the 1 s with 0 s, and all the 0 s with 1 s An exception: = , University of Colombo School of Computing 55
56 Your turn What is the SMALLEST and LARGEST signed binary numbers that can be stored in 1 BYTE 2009, University of Colombo School of Computing 56
57 Two s Compliment (8 bit pattern) = = = = = = = = = = , University of Colombo School of Computing 57
58 Two s Compliment benefits One representation of zero Arithmetic works easily Negating is fairly easy 2009, University of Colombo School of Computing 58
59 Ranges of Integer Representation 8bit unsigned binary representation Largest number: = Smallest number: = bit two s complement representation Largest number: = Smallest number: = The problem of overflow = in two s complement 2009, University of Colombo School of Computing 59
60 Geometric Depiction of Two s Complement Integers 2009, University of Colombo School of Computing 60
61 Integer Data Types in C++ Type Size in Bits Range unsigned int int unsigned long int 32 0 to 4,294,967,295 long int 322,147,483,648 to 2,147,483, , University of Colombo School of Computing 61
62 = the SUM of... Fractions in Decimal 7 * 103 = 7 / * 102 = 5 / * 101 = 3 / 10 6 * 10 0 = 6 1 * 10 1 = 10 7 / / / = / , University of Colombo School of Computing 62
63 = the SUM of... Fractions in Binary 1 * 23 = 1 / 8 1 * 22 = 1 / 4 0 * 21 = 0 0 * 2 0 = 0 1 * 2 1 = 2 1 / / = 2 3 / 8 i.e = 2 3 / 8 in Decimal (Base 10) 2009, University of Colombo School of Computing 63
64 Your turn What is in Base 10? 2009, University of Colombo School of Computing 64
65 Fractions in Binary = the SUM of... 1 * 24 = 1 / 16 0 * 23 = 0 1 * 22 = 1 / 4 0 * 21 = 0 1 * 2 0 = 1 1 * 2 1 = 2 0 * 2 2 = 0 1 / / = / , University of Colombo School of Computing 65
66 Decimal Scientific Notation Consider the following representation in decimal number = x = x =.2452 x x 10 3 has the following components: a Mantissa = an Exponent = 3 a Base = , University of Colombo School of Computing 66
67 Floating Point Representation of Fractions Scientific notation for binary. Examples = x = x = x , University of Colombo School of Computing 67
68 Floating Point Format in 1 Byte RADIX POINT SIGN = 0 (+ve) 1 (ve) EXPONENT in EXCESS FOUR Notation MANTISSA EXPONENT SIGN 2009, University of Colombo School of Computing 68
69 Floating Point Format in 1 Byte To STORE the number / 8 = in FLOATING POINT NOTATION STORE the SIGN BIT 2009, University of Colombo School of Computing 69
70 Floating Point Format in 1 Byte To STORE the number / 8 = in FLOATING POINT NOTATION STORE the SIGN BIT , University of Colombo School of Computing 70
71 Floating Point Format in 1 Byte To STORE the number / 8 = in FLOATING POINT NOTATION STORE the MANTISSA BITS , University of Colombo School of Computing 71
72 Floating Point Format in 1 Byte To STORE the number / 8 = in FLOATING POINT NOTATION STORE the MANTISSA BITS , University of Colombo School of Computing 72
73 Floating Point Format in 1 Byte To STORE the number / 8 = in FLOATING POINT NOTATION STORE the EXPONENT BITS , University of Colombo School of Computing 73
74 Bit Pattern Excessk Representation Value Representation EXCESS THREE NOTATION An excess notation system using bit pattern of length three 2009, University of Colombo School of Computing 74
75 Excessk Representation For N bit numbers, k is 2 N11 E.g., for 4bit integers, k is 7 The actual value of each bit string is its unsigned value minus k To represent a number in excessk, add k 2009, University of Colombo School of Computing 75
76 Excessk Representation sliding ruler Unsigned k = 7 Excessk , University of Colombo School of Computing 76
77 Floating Point Format in 1 Byte To STORE the number / 8 = in FLOATING POINT NOTATION STORE the EXPONENT BITS , University of Colombo School of Computing 77
78 Floating Point Format in 1 Byte To STORE the number / 4 = in FLOATING POINT NOTATION STORE the SIGN BIT 2009, University of Colombo School of Computing 78
79 Floating Point Format in 1 Byte To STORE the number / 4 = in FLOATING POINT NOTATION STORE the SIGN BIT , University of Colombo School of Computing 79
80 Floating Point Format in 1 Byte To STORE the number / 4 = in FLOATING POINT NOTATION STORE the MANTISSA BITS , University of Colombo School of Computing 80
81 Floating Point Format in 1 Byte To STORE the number / 4 = in FLOATING POINT NOTATION STORE the MANTISSA BITS , University of Colombo School of Computing 81
82 Floating Point Format in 1 Byte To STORE the number / 4 = in FLOATING POINT NOTATION STORE the EXPONENT BITS , University of Colombo School of Computing 82
83 Floating Point Format in 1 Byte To STORE the number / 4 = in FLOATING POINT NOTATION STORE the EXPONENT BITS , University of Colombo School of Computing 83
84 Your turn Write down the FLOATING POINT form for the number + 11 / 64? 2009, University of Colombo School of Computing 84
85 SOLUTION: + 11 / 64 ( ) 1. STORE the SIGN BIT 2009, University of Colombo School of Computing 85
86 SOLUTION: + 11 / 64 ( ) 1. STORE the SIGN BIT , University of Colombo School of Computing 86
87 SOLUTION: + 11 / 64 ( ) 2. STORE the MANTISSA BITS , University of Colombo School of Computing 87
88 SOLUTION: + 11 / 64 ( ) 2. STORE the MANTISSA BITS , University of Colombo School of Computing 88
89 SOLUTION: + 11 / 64 ( ) 3. STORE the EXPONENT BITS , University of Colombo School of Computing 89
90 Converting FP Binary to Decimal Example... CONVERT to decimal steps Convert EXPONENT (EXCESS 4) 2. Apply EXPONENT to MANTISSA 3. Convert BINARY Fraction 4. Apply SIGN 2009, University of Colombo School of Computing 90
91 SOLUTION: CONVERT THE EXPONENT = , University of Colombo School of Computing 91
92 SOLUTION: APPLY the EXPONENT to the MANTISSA , University of Colombo School of Computing 92
93 SOLUTION: CONVERT from BINARY FRACTION / / 8 = 1 5 / , University of Colombo School of Computing 93
94 SOLUTION: APPLY the SIGN ( / / 8 ) = / , University of Colombo School of Computing 94
95 ROUNDOFF ERRORS CONSIDER the FLOATING POINT Form of the number / , University of Colombo School of Computing 95
96 ROUNDOFF OFF ERRORS +2 5 / 8 1. CONVERT to BINARY FRACTION / 8 = i.e / / , University of Colombo School of Computing 96
97 ROUNDOFF OFF ERRORS +2 5 / 8 = STORE THE SIGN BIT , University of Colombo School of Computing 97
98 ROUNDOFF OFF ERRORS +2 5 / 8 = STORE THE SIGN BIT , University of Colombo School of Computing 98
99 ROUNDOFF OFF ERRORS +2 5 / 8 = STORE THE MANTISSA , University of Colombo School of Computing 99
100 ROUNDOFF OFF ERRORS +2 5 / 8 = STORE THE MANTISSA , University of Colombo School of Computing 100
101 ROUNDOFF OFF ERRORS +2 5 / 8 = STORE THE MANTISSA , University of Colombo School of Computing 101
102 ROUNDOFF OFF ERRORS +2 5 / 8 = STORE THE MANTISSA , University of Colombo School of Computing 102
103 ROUNDOFF OFF ERRORS +2 5 / 8 = STORE THE EXPONENT , University of Colombo School of Computing 103
104 ROUNDOFF OFF ERRORS +2 5 / 8 = STORE THE EXPONENT Converting this back to DECIMAL we get / 4 i.e. a ROUND OFF ERROR of 1 / , University of Colombo School of Computing 104
105 Range of FP Representation What is the BIGGEST and SMALLEST can be represented by onebyte floating point notation 2009, University of Colombo School of Computing 105
106 Range of FP Representation The biggest number can be represented by onebyte floating point notation is: = x 2 4 = = , University of Colombo School of Computing 106
107 Range of FP Representation The Smallest positive number can be represented by onebyte floating point notation is: = x 23 = = + 1 / , University of Colombo School of Computing 107
108 Range of FP Representation The largest negative number can be represented by onebyte floating point notation is: = x 23 = =  1 / , University of Colombo School of Computing 108
109 Range of FP Representation The smallest number can be represented by onebyte floating point notation is: = x 2 4 = = , University of Colombo School of Computing 109
110 Range of FP Representation What is the SOLUTION for this??? 2009, University of Colombo School of Computing 110
111 FloatingPoint Data types in C++ Type Size in Bits Range float E38 to 3.4E+38 Six digits of precision double E308 to 1.7E+308 Ten digits of precision long double E4932 to 3.4E+4932 Ten digits of precision 2009, University of Colombo School of Computing 111
112 FloatingPoint Representation Sign bit Biased Exponent Significand or Mantissa +/. Mantissa x 2 exponent Point is actually fixed between sign bit and body of Mantissa Exponent indicates place value (point position) 2009, University of Colombo School of Computing 112
113 FloatingPoint Representation Mantissa is stored in 2 s compliment Exponent is in excess notation 8 bit exponent field Pure range is Subtract 127 to get correct value Range 127 to , University of Colombo School of Computing 113
114 FloatingPoint Representation Floating Point numbers are usually normalized i.e. exponent is adjusted so that leading bit (MSB) of mantissa is 1 Since it is always 1 there is no need to store it Where numbers are normalized to give a single digit before the decimal point E.g x , University of Colombo School of Computing 114
115 FloatingPoint Representation 2009, University of Colombo School of Computing 115
116 Floating Point Representation: Expressible Numbers 2009, University of Colombo School of Computing 116
117 Representing the Mantissa The mantissa has to be in the range 1 mantissa < base Therefore If we use base 2, the digit before the point must be a 1 So we don't have to worry about storing it We get 24 bits of precision using 23 bits 24 bits of precision are equivalent to a little over 7 decimal digits: 24 log , University of Colombo School of Computing 117
118 Representing the Mantissa Suppose we want to represent π: That means that we can only represent it as: (if we truncate) (if we round) 2009, University of Colombo School of Computing 118
119 Representing the Mantissa The IEEE standard restricts exponents to the range: 126 exponent +127 The exponents 127 and +128 have special meanings: If exponent = 127, the stored value is 0 If exponent = 128, the stored value is 2009, University of Colombo School of Computing 119
120 Floating Point Overflow Floating point representations can overflow, e.g., = , University of Colombo School of Computing 120
121 Floating Point Underflow Floating point numbers can also get too small, e.g., = , University of Colombo School of Computing 121
122 Floating Point Representation: Double Precision IEEE754 Double Precision Standard 64 bits: 1 bit sign 52 bit mantissa 11 bit exponent Exponent range is to k = = , University of Colombo School of Computing 122
123 Limitations Floatingpoint representations only approximate real numbers Using a greater number of bits in a representation can reduce errors but can never eliminate them Floating point errors Overflow/underflow can cause programs to crash Can lead to erroneous results / hard to detect 2009, University of Colombo School of Computing 123
124 Floating Point Addition Five steps to add two floating point numbers: 1. Express the numbers with the same exponent (denormalize) 2. Add the mantissas 3. Adjust the mantissa to one digit/bit before the point (renormalize) 4. Round or truncate to required precision 5. Check for overflow/underflow 2009, University of Colombo School of Computing 124
125 Thank You 2009, University of Colombo School of Computing 125
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