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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

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: 105-0*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: 105-1*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 8-1) 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: 16-bit 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 Latin-1 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 0-0 = = 1 (with borrow) 1-0 = = 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: non-negative integers Signed representation: integers Floating-point 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 8-bit 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 32-2,147,483,648 to 2,147,483, , University of Colombo School of Computing 61

62 = the SUM of... Fractions in Decimal 7 * 10-3 = 7 / * 10-2 = 5 / * 10-1 = 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 * 2-3 = 1 / 8 1 * 2-2 = 1 / 4 0 * 2-1 = 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 * 2-4 = 1 / 16 0 * 2-3 = 0 1 * 2-2 = 1 / 4 0 * 2-1 = 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 Excess-k 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 Excess-k Representation For N bit numbers, k is 2 N-1-1 E.g., for 4-bit integers, k is 7 The actual value of each bit string is its unsigned value minus k To represent a number in excess-k, add k 2009, University of Colombo School of Computing 75

76 Excess-k Representation sliding ruler Unsigned k = 7 Excess-k , 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 ROUND-OFF ERRORS CONSIDER the FLOATING POINT Form of the number / , University of Colombo School of Computing 95

96 ROUND-OFF OFF ERRORS +2 5 / 8 1. CONVERT to BINARY FRACTION / 8 = i.e / / , University of Colombo School of Computing 96

97 ROUND-OFF OFF ERRORS +2 5 / 8 = STORE THE SIGN BIT , University of Colombo School of Computing 97

98 ROUND-OFF OFF ERRORS +2 5 / 8 = STORE THE SIGN BIT , University of Colombo School of Computing 98

99 ROUND-OFF OFF ERRORS +2 5 / 8 = STORE THE MANTISSA , University of Colombo School of Computing 99

100 ROUND-OFF OFF ERRORS +2 5 / 8 = STORE THE MANTISSA , University of Colombo School of Computing 100

101 ROUND-OFF OFF ERRORS +2 5 / 8 = STORE THE MANTISSA , University of Colombo School of Computing 101

102 ROUND-OFF OFF ERRORS +2 5 / 8 = STORE THE MANTISSA , University of Colombo School of Computing 102

103 ROUND-OFF OFF ERRORS +2 5 / 8 = STORE THE EXPONENT , University of Colombo School of Computing 103

104 ROUND-OFF 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 one-byte floating point notation 2009, University of Colombo School of Computing 105

106 Range of FP Representation The biggest number can be represented by one-byte 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 one-byte floating point notation is: = x 2-3 = = + 1 / , University of Colombo School of Computing 107

108 Range of FP Representation The largest negative number can be represented by one-byte floating point notation is: = x 2-3 = = - 1 / , University of Colombo School of Computing 108

109 Range of FP Representation The smallest number can be represented by one-byte 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 Floating-Point Data types in C++ Type Size in Bits Range float E-38 to 3.4E+38 Six digits of precision double E-308 to 1.7E+308 Ten digits of precision long double E-4932 to 3.4E+4932 Ten digits of precision 2009, University of Colombo School of Computing 111

112 Floating-Point 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 Floating-Point 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 Floating-Point 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 Floating-Point 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 IEEE-754 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 Floating-point 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

### Module 2: Computer Arithmetic

Module 2: Computer Arithmetic 1 B O O K : C O M P U T E R O R G A N I Z A T I O N A N D D E S I G N, 3 E D, D A V I D L. P A T T E R S O N A N D J O H N L. H A N N E S S Y, M O R G A N K A U F M A N N

### 10.1. Unit 10. Signed Representation Systems Binary Arithmetic

0. Unit 0 Signed Representation Systems Binary Arithmetic 0.2 BINARY REPRESENTATION SYSTEMS REVIEW 0.3 Interpreting Binary Strings Given a string of s and 0 s, you need to know the representation system

### Floating Point Arithmetic

Floating Point Arithmetic CS 365 Floating-Point What can be represented in N bits? Unsigned 0 to 2 N 2s Complement -2 N-1 to 2 N-1-1 But, what about? very large numbers? 9,349,398,989,787,762,244,859,087,678

### CHAPTER 2 Data Representation in Computer Systems

CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 37 2.2 Positional Numbering Systems 38 2.3 Decimal to Binary Conversions 38 2.3.1 Converting Unsigned Whole Numbers 39 2.3.2 Converting

### Inf2C - Computer Systems Lecture 2 Data Representation

Inf2C - Computer Systems Lecture 2 Data Representation Boris Grot School of Informatics University of Edinburgh Last lecture Moore s law Types of computer systems Computer components Computer system stack

### CHAPTER 2 Data Representation in Computer Systems

CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 37 2.2 Positional Numbering Systems 38 2.3 Decimal to Binary Conversions 38 2.3.1 Converting Unsigned Whole Numbers 39 2.3.2 Converting

### Floating-point Arithmetic. where you sum up the integer to the left of the decimal point and the fraction to the right.

Floating-point Arithmetic Reading: pp. 312-328 Floating-Point Representation Non-scientific floating point numbers: A non-integer can be represented as: 2 4 2 3 2 2 2 1 2 0.2-1 2-2 2-3 2-4 where you sum

### 15213 Recitation 2: Floating Point

15213 Recitation 2: Floating Point 1 Introduction This handout will introduce and test your knowledge of the floating point representation of real numbers, as defined by the IEEE standard. This information

### Digital Computers and Machine Representation of Data

Digital Computers and Machine Representation of Data K. Cooper 1 1 Department of Mathematics Washington State University 2013 Computers Machine computation requires a few ingredients: 1 A means of representing

### DRAM uses a single capacitor to store and a transistor to select. SRAM typically uses 6 transistors.

Data Representation Data Representation Goal: Store numbers, characters, sets, database records in the computer. What we got: Circuit that stores 2 voltages, one for logic 0 (0 volts) and one for logic

### Data Representation. DRAM uses a single capacitor to store and a transistor to select. SRAM typically uses 6 transistors.

Data Representation Data Representation Goal: Store numbers, characters, sets, database records in the computer. What we got: Circuit that stores 2 voltages, one for logic ( volts) and one for logic (3.3

### Number Systems. Binary Numbers. Appendix. Decimal notation represents numbers as powers of 10, for example

Appendix F Number Systems Binary Numbers Decimal notation represents numbers as powers of 10, for example 1729 1 103 7 102 2 101 9 100 decimal = + + + There is no particular reason for the choice of 10,

### A complement number system is used to represent positive and negative integers. A complement number system is based on a fixed length representation

Complement Number Systems A complement number system is used to represent positive and negative integers A complement number system is based on a fixed length representation of numbers Pretend that integers

### Floating point. Today! IEEE Floating Point Standard! Rounding! Floating Point Operations! Mathematical properties. Next time. !

Floating point Today! IEEE Floating Point Standard! Rounding! Floating Point Operations! Mathematical properties Next time! The machine model Chris Riesbeck, Fall 2011 Checkpoint IEEE Floating point Floating

### 1. NUMBER SYSTEMS USED IN COMPUTING: THE BINARY NUMBER SYSTEM

1. NUMBER SYSTEMS USED IN COMPUTING: THE BINARY NUMBER SYSTEM 1.1 Introduction Given that digital logic and memory devices are based on two electrical states (on and off), it is natural to use a number

### Data Representation Floating Point

Data Representation Floating Point CSCI 2400 / ECE 3217: Computer Architecture Instructor: David Ferry Slides adapted from Bryant & O Hallaron s slides via Jason Fritts Today: Floating Point Background:

### Chapter 2 Float Point Arithmetic. Real Numbers in Decimal Notation. Real Numbers in Decimal Notation

Chapter 2 Float Point Arithmetic Topics IEEE Floating Point Standard Fractional Binary Numbers Rounding Floating Point Operations Mathematical properties Real Numbers in Decimal Notation Representation

### Chapter 2 Bits, Data Types, and Operations

Chapter 2 Bits, Data Types, and Operations How do we represent data in a computer? At the lowest level, a computer is an electronic machine. works by controlling the flow of electrons Easy to recognize

### Signed umbers. Sign/Magnitude otation

Signed umbers So far we have discussed unsigned number representations. In particular, we have looked at the binary number system and shorthand methods in representing binary codes. With m binary digits,

### ECE 2020B Fundamentals of Digital Design Spring problems, 6 pages Exam Two 26 February 2014

Instructions: This is a closed book, closed note exam. Calculators are not permitted. If you have a question, raise your hand and I will come to you. Please work the exam in pencil and do not separate

### Floating Point Numbers. Lecture 9 CAP

Floating Point Numbers Lecture 9 CAP 3103 06-16-2014 Review of Numbers Computers are made to deal with numbers What can we represent in N bits? 2 N things, and no more! They could be Unsigned integers:

### Logic, Words, and Integers

Computer Science 52 Logic, Words, and Integers 1 Words and Data The basic unit of information in a computer is the bit; it is simply a quantity that takes one of two values, 0 or 1. A sequence of k bits

### Number Systems and Computer Arithmetic

Number Systems and Computer Arithmetic Counting to four billion two fingers at a time What do all those bits mean now? bits (011011011100010...01) instruction R-format I-format... integer data number text

### Computer Arithmetic Ch 8

Computer Arithmetic Ch 8 ALU Integer Representation Integer Arithmetic Floating-Point Representation Floating-Point Arithmetic 1 Arithmetic Logical Unit (ALU) (2) Does all work in CPU (aritmeettis-looginen

Chapter 1 Data Storage 2007 Pearson Addison-Wesley. All rights reserved Chapter 1: Data Storage 1.1 Bits and Their Storage 1.2 Main Memory 1.3 Mass Storage 1.4 Representing Information as Bit Patterns

### Digital Fundamentals

Digital Fundamentals Tenth Edition Floyd Chapter 1 Modified by Yuttapong Jiraraksopakun Floyd, Digital Fundamentals, 10 th 2008 Pearson Education ENE, KMUTT ed 2009 Analog Quantities Most natural quantities

### UNIT - I: COMPUTER ARITHMETIC, REGISTER TRANSFER LANGUAGE & MICROOPERATIONS

UNIT - I: COMPUTER ARITHMETIC, REGISTER TRANSFER LANGUAGE & MICROOPERATIONS (09 periods) Computer Arithmetic: Data Representation, Fixed Point Representation, Floating Point Representation, Addition and

### Scientific Computing. Error Analysis

ECE257 Numerical Methods and Scientific Computing Error Analysis Today s s class: Introduction to error analysis Approximations Round-Off Errors Introduction Error is the difference between the exact solution

### 1010 2?= ?= CS 64 Lecture 2 Data Representation. Decimal Numbers: Base 10. Reading: FLD Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

CS 64 Lecture 2 Data Representation Reading: FLD 1.2-1.4 Decimal Numbers: Base 10 Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 3271 = (3x10 3 ) + (2x10 2 ) + (7x10 1 ) + (1x10 0 ) 1010 10?= 1010 2?= 1

### CS101 Lecture 04: Binary Arithmetic

CS101 Lecture 04: Binary Arithmetic Binary Number Addition Two s complement encoding Briefly: real number representation Aaron Stevens (azs@bu.edu) 25 January 2013 What You ll Learn Today Counting in binary

### Characters, Strings, and Floats

Characters, Strings, and Floats CS 350: Computer Organization & Assembler Language Programming 9/6: pp.8,9; 9/28: Activity Q.6 A. Why? We need to represent textual characters in addition to numbers. Floating-point

### C NUMERIC FORMATS. Overview. IEEE Single-Precision Floating-point Data Format. Figure C-0. Table C-0. Listing C-0.

C NUMERIC FORMATS Figure C-. Table C-. Listing C-. Overview The DSP supports the 32-bit single-precision floating-point data format defined in the IEEE Standard 754/854. In addition, the DSP supports an

### 1. Which of the following Boolean operations produces the output 1 for the fewest number of input patterns?

This is full of Test bank for Computer Science An Overview 12th Edition by Brookshear SM https://getbooksolutions.com/download/computer-science-an-overview-12th-editionby-brookshear-sm Test Bank Chapter

### Signed Binary Numbers

Signed Binary Numbers Unsigned Binary Numbers We write numbers with as many digits as we need: 0, 99, 65536, 15000, 1979, However, memory locations and CPU registers always hold a constant, fixed number

### CS429: Computer Organization and Architecture

CS429: Computer Organization and Architecture Dr. Bill Young Department of Computer Sciences University of Texas at Austin Last updated: September 18, 2017 at 12:48 CS429 Slideset 4: 1 Topics of this Slideset

### 3.5 Floating Point: Overview

3.5 Floating Point: Overview Floating point (FP) numbers Scientific notation Decimal scientific notation Binary scientific notation IEEE 754 FP Standard Floating point representation inside a computer

### Floating point. Today. IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Next time.

Floating point Today IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Next time The machine model Fabián E. Bustamante, Spring 2010 IEEE Floating point Floating point

### IBM 370 Basic Data Types

IBM 370 Basic Data Types This lecture discusses the basic data types used on the IBM 370, 1. Two s complement binary numbers 2. EBCDIC (Extended Binary Coded Decimal Interchange Code) 3. Zoned Decimal

### Number System (Different Ways To Say How Many) Fall 2016

Number System (Different Ways To Say How Many) Fall 2016 Introduction to Information and Communication Technologies CSD 102 Email: mehwish.fatima@ciitlahore.edu.pk Website: https://sites.google.com/a/ciitlahore.edu.pk/ict/

### Floating Point Puzzles The course that gives CMU its Zip! Floating Point Jan 22, IEEE Floating Point. Fractional Binary Numbers.

class04.ppt 15-213 The course that gives CMU its Zip! Topics Floating Point Jan 22, 2004 IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Floating Point Puzzles For

### Systems I. Floating Point. Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties

Systems I Floating Point Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties IEEE Floating Point IEEE Standard 754 Established in 1985 as uniform standard for

### CSC201, SECTION 002, Fall 2000: Homework Assignment #2

1 of 7 11/8/2003 7:34 PM CSC201, SECTION 002, Fall 2000: Homework Assignment #2 DUE DATE Monday, October 2, at the start of class. INSTRUCTIONS FOR PREPARATION Neat, in order, answers easy to find. Staple

### CS321. Introduction to Numerical Methods

CS31 Introduction to Numerical Methods Lecture 1 Number Representations and Errors Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 40506 0633 August 5, 017 Number

### umber Systems bit nibble byte word binary decimal

umber Systems Inside today s computers, data is represented as 1 s and 0 s. These 1 s and 0 s might be stored magnetically on a disk, or as a state in a transistor. To perform useful operations on these

### Fundamentals of Programming Session 2

Fundamentals of Programming Session 2 Instructor: Reza Entezari-Maleki Email: entezari@ce.sharif.edu 1 Fall 2013 Sharif University of Technology Outlines Programming Language Binary numbers Addition Subtraction

### IEEE-754 floating-point

IEEE-754 floating-point Real and floating-point numbers Real numbers R form a continuum - Rational numbers are a subset of the reals - Some numbers are irrational, e.g. π Floating-point numbers are an

### System Programming CISC 360. Floating Point September 16, 2008

System Programming CISC 360 Floating Point September 16, 2008 Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Powerpoint Lecture Notes for Computer Systems:

### Five classic components

CS/COE0447: Computer Organization and Assembly Language Chapter 3 modified by Bruce Childers original slides by Sangyeun Cho Dept. of Computer Science Five classic components I am like a control tower

### Chapter 2: Number Systems

Chapter 2: Number Systems Logic circuits are used to generate and transmit 1s and 0s to compute and convey information. This two-valued number system is called binary. As presented earlier, there are many

### Data Storage. Slides derived from those available on the web site of the book: Computer Science: An Overview, 11 th Edition, by J.

Data Storage Slides derived from those available on the web site of the book: Computer Science: An Overview, 11 th Edition, by J. Glenn Brookshear Copyright 2012 Pearson Education, Inc. Data Storage Bits

### 4 Operations On Data 4.1. Foundations of Computer Science Cengage Learning

4 Operations On Data 4.1 Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: List the three categories of operations performed on data.

### CS311 Lecture: Representing Information in Binary Last revised 8/2707

CS311 Lecture: Representing Information in Binary Last revised 8/2707 Objectives: 1. To review binary representation for unsigned integers 2. To introduce octal and hexadecimal shorthands 3. To review

### Unit 3. Analog vs. Digital. Analog vs. Digital ANALOG VS. DIGITAL. Binary Representation

3.1 3.2 Unit 3 Binary Representation ANALOG VS. DIGITAL 3.3 3.4 Analog vs. Digital The analog world is based on continuous events. Observations can take on (real) any value. The digital world is based

### 9/23/15. Agenda. Goals of this Lecture. For Your Amusement. Number Systems and Number Representation. The Binary Number System

For Your Amusement Number Systems and Number Representation Jennifer Rexford Question: Why do computer programmers confuse Christmas and Halloween? Answer: Because 25 Dec = 31 Oct -- http://www.electronicsweekly.com

### Declaring Floating Point Data

Declaring Floating Point Data There are three ways to declare floating point storage. These are E D L Single precision floating point, Double precision floating point, and Extended precision floating point.

### ECE 372 Microcontroller Design Assembly Programming Arrays. ECE 372 Microcontroller Design Assembly Programming Arrays

Assembly Programming Arrays Assembly Programming Arrays Array For Loop Example: unsigned short a[]; for(j=; j

### Numbers and Representations

Çetin Kaya Koç http://koclab.cs.ucsb.edu/teaching/cs192 koc@cs.ucsb.edu Çetin Kaya Koç http://koclab.cs.ucsb.edu Fall 2016 1 / 38 Outline Computational Thinking Representations of integers Binary and decimal

### Floating Point Numbers

Floating Point Numbers Summer 8 Fractional numbers Fractional numbers fixed point Floating point numbers the IEEE 7 floating point standard Floating point operations Rounding modes CMPE Summer 8 Slides

### Binary Addition & Subtraction. Unsigned and Sign & Magnitude numbers

Binary Addition & Subtraction Unsigned and Sign & Magnitude numbers Addition and subtraction of unsigned or sign & magnitude binary numbers by hand proceeds exactly as with decimal numbers. (In fact this

### CMSC 313 Lecture 03 Multiple-byte data big-endian vs little-endian sign extension Multiplication and division Floating point formats Character Codes

Multiple-byte data CMSC 313 Lecture 03 big-endian vs little-endian sign extension Multiplication and division Floating point formats Character Codes UMBC, CMSC313, Richard Chang 4-5 Chapter

### JAVA Programming Fundamentals

Chapter 4 JAVA Programming Fundamentals By: Deepak Bhinde PGT Comp.Sc. JAVA character set Character set is a set of valid characters that a language can recognize. It may be any letter, digit or any symbol

### EEM 232 Digital System I

EEM 232 Digital System I Instructor : Assist. Prof. Dr. Emin Germen egermen@anadolu.edu.tr Course Book : Logic and Computer Design Fundamentals by Mano & Kime Third Ed/Fourth Ed.. Pearson Grading 1 st

### Appendix. Numbering Systems. In this Appendix

Numbering Systems ppendix n this ppendix ntroduction... inary Numbering System... exadecimal Numbering System... Octal Numbering System... inary oded ecimal () Numbering System... 5 Real (Floating Point)

### Digital Arithmetic. Digital Arithmetic: Operations and Circuits Dr. Farahmand

Digital Arithmetic Digital Arithmetic: Operations and Circuits Dr. Farahmand Binary Arithmetic Digital circuits are frequently used for arithmetic operations Fundamental arithmetic operations on binary

### Name: CMSC 313 Fall 2001 Computer Organization & Assembly Language Programming Exam 1. Question Points I. /34 II. /30 III.

CMSC 313 Fall 2001 Computer Organization & Assembly Language Programming Exam 1 Name: Question Points I. /34 II. /30 III. /36 TOTAL: /100 Instructions: 1. This is a closed-book, closed-notes exam. 2. You

### Lecture (03) Binary Codes Registers and Logic Gates

Lecture (03) Binary Codes Registers and Logic Gates By: Dr. Ahmed ElShafee Binary Codes Digital systems use signals that have two distinct values and circuit elements that have two stable states. binary

### The Sign consists of a single bit. If this bit is '1', then the number is negative. If this bit is '0', then the number is positive.

IEEE 754 Standard - Overview Frozen Content Modified by on 13-Sep-2017 Before discussing the actual WB_FPU - Wishbone Floating Point Unit peripheral in detail, it is worth spending some time to look at

### TUTORIAL -1 COMPUTER ORGANISATION TIT-402

TUTORIAL -1 COMPUTER ORGANISATION TIT-402 Q1.(43) 10 = (? ) 2 s (GATE 2000) a)01010101 b)11010101 c)00101011 d)10101011 Q2.The 2 s complement representation of (-539) 10 in hexadecimal is (GATE 2001) a)abe

### Principles of Computer Architecture. Chapter 3: Arithmetic

3-1 Chapter 3 - Arithmetic Principles of Computer Architecture Miles Murdocca and Vincent Heuring Chapter 3: Arithmetic 3-2 Chapter 3 - Arithmetic 3.1 Overview Chapter Contents 3.2 Fixed Point Addition

### More Programming Constructs -- Introduction

More Programming Constructs -- Introduction We can now examine some additional programming concepts and constructs Chapter 5 focuses on: internal data representation conversions between one data type and

Octal and Hexadecimal Integers CS 350: Computer Organization & Assembler Language Programming A. Why? Octal and hexadecimal numbers are useful for abbreviating long bitstrings. Some operations on octal

### FLOATING POINT NUMBERS

FLOATING POINT NUMBERS Robert P. Webber, Longwood University We have seen how decimal fractions can be converted to binary. For instance, we can write 6.25 10 as 4 + 2 + ¼ = 2 2 + 2 1 + 2-2 = 1*2 2 + 1*2

### On a 64-bit CPU. Size/Range vary by CPU model and Word size.

On a 64-bit CPU. Size/Range vary by CPU model and Word size. unsigned short x; //range 0 to 65553 signed short x; //range ± 32767 short x; //assumed signed There are (usually) no unsigned floats or doubles.

### CS101 Introduction to computing Floating Point Numbers

CS101 Introduction to computing Floating Point Numbers A. Sahu and S. V.Rao Dept of Comp. Sc. & Engg. Indian Institute of Technology Guwahati 1 Outline Need to floating point number Number representation

### CPS311 Lecture: Representing Information in Binary Last revised Sept. 25, 2017

CPS311 Lecture: Representing Information in Binary Last revised Sept. 25, 2017 Objectives: 1. To review binary representation for unsigned integers 2. To introduce octal and hexadecimal shorthands 3. To

### Chapter 2 Binary Values and Number Systems

Chapter 2 Binary Values and Number Systems Chapter Goals 10 進位 2 / 8 / 16 進位 進位系統間轉換 各進位系統小數表示 各進位系統加減法 各進位系統乘除法 2 24 6 Numbers Natural Numbers Zero and any number obtained by repeatedly adding one to

### P( Hit 2nd ) = P( Hit 2nd Miss 1st )P( Miss 1st ) = (1/15)(15/16) = 1/16. P( Hit 3rd ) = (1/14) * P( Miss 2nd and 1st ) = (1/14)(14/15)(15/16) = 1/16

CODING and INFORMATION We need encodings for data. How many questions must be asked to be certain where the ball is. (cases: avg, worst, best) P( Hit 1st ) = 1/16 P( Hit 2nd ) = P( Hit 2nd Miss 1st )P(

### Computer Systems C S Cynthia Lee

Computer Systems C S 1 0 7 Cynthia Lee 2 Today s Topics LECTURE: Floating point! Real Numbers and Approximation MATH TIME! Some preliminary observations on approximation We know that some non-integer numbers

### IEEE Standard 754 Floating Point Numbers

IEEE Standard 754 Floating Point Numbers Steve Hollasch / Last update 2005-Feb-24 IEEE Standard 754 floating point is the most common representation today for real numbers on computers, including Intel-based

### 4 Operations On Data 4.1. Foundations of Computer Science Cengage Learning

4 Operations On Data 4.1 Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: List the three categories of operations performed on data.

### CS/EE1012 INTRODUCTION TO COMPUTER ENGINEERING SPRING 2013 HOMEWORK I. Solve all homework and exam problems as shown in class and sample solutions

CS/EE2 INTRODUCTION TO COMPUTER ENGINEERING SPRING 23 DUE : February 22, 23 HOMEWORK I READ : Related portions of the following chapters : È Chapter È Chapter 2 È Appendix E ASSIGNMENT : There are eight

### Data Representation and Networking

Data Representation and Networking Instructor: Dmitri A. Gusev Spring 2007 CSC 120.02: Introduction to Computer Science Lecture 3, January 30, 2007 Data Representation Topics Covered in Lecture 2 (recap+)

### Administrivia. CMSC 216 Introduction to Computer Systems Lecture 24 Data Representation and Libraries. Representing characters DATA REPRESENTATION

Administrivia CMSC 216 Introduction to Computer Systems Lecture 24 Data Representation and Libraries Jan Plane & Alan Sussman {jplane, als}@cs.umd.edu Project 6 due next Friday, 12/10 public tests posted

### CSCI 2212: Intermediate Programming / C Chapter 15

... /34 CSCI 222: Intermediate Programming / C Chapter 5 Alice E. Fischer October 9 and 2, 25 ... 2/34 Outline Integer Representations Binary Integers Integer Types Bit Operations Applying Bit Operations

### NUMBER SYSTEMS AND CODES

C H A P T E R 69 Learning Objectives Number Systems The Decimal Number System Binary Number System Binary to Decimal Conversion Binary Fractions Double-Dadd Method Decimal to Binary Conversion Shifting

### Description Hex M E V smallest value > largest denormalized negative infinity number with hex representation 3BB0 ---

CSE2421 HOMEWORK #2 DUE DATE: MONDAY 11/5 11:59pm PROBLEM 2.84 Given a floating-point format with a k-bit exponent and an n-bit fraction, write formulas for the exponent E, significand M, the fraction

### Classes of Real Numbers 1/2. The Real Line

Classes of Real Numbers All real numbers can be represented by a line: 1/2 π 1 0 1 2 3 4 real numbers The Real Line { integers rational numbers non-integral fractions irrational numbers Rational numbers

### Learning the Binary System

Learning the Binary System www.brainlubeonline.com/counting_on_binary/ Formated to L A TEX: /25/22 Abstract This is a document on the base-2 abstract numerical system, or Binary system. This is a VERY

### Chapter 3. Fundamental Data Types

Chapter 3. Fundamental Data Types Byoung-Tak Zhang TA: Hanock Kwak Biointelligence Laboratory School of Computer Science and Engineering Seoul National Univertisy http://bi.snu.ac.kr Variable Declaration

### Building a Test Suite

Program #3 Is on the web Exam #1 Announcements Today, 6:00 7:30 in Armory 0126 Makeup Exam Friday March 9, 2:00 PM room TBA Reading Notes (Today) Chapter 16 (Tuesday) 1 API: Building a Test Suite Int createemployee(char

### Independent Representation

Independent Representation 1 Integer magnitude + 1 2 3 4 5 6 7 8 9 0 Float sign sign of exponent mantissa On standard 32 bit machines: INT_MAX = 2147483647 which gives 10 digits of precision, (i.e. the

### Numeric Precision 101

www.sas.com > Service and Support > Technical Support TS Home Intro to Services News and Info Contact TS Site Map FAQ Feedback TS-654 Numeric Precision 101 This paper is intended as a basic introduction

### EE 109 Unit 6 Binary Arithmetic

EE 109 Unit 6 Binary Arithmetic 1 2 Semester Transition Point At this point we are going to start to transition in our class to look more at the hardware organization and the low-level software that is

### Computer Systems Programming. Practice Midterm. Name:

Computer Systems Programming Practice Midterm Name: 1. (4 pts) (K&R Ch 1-4) What is the output of the following C code? main() { int i = 6; int j = -35; printf( %d %d\n,i++, ++j); i = i >

### 1.2 Round-off Errors and Computer Arithmetic

1.2 Round-off Errors and Computer Arithmetic 1 In a computer model, a memory storage unit word is used to store a number. A word has only a finite number of bits. These facts imply: 1. Only a small set

### Lecture Notes: Floating-Point Numbers

Lecture Notes: Floating-Point Numbers CS227-Scientific Computing September 8, 2010 What this Lecture is About How computers represent numbers How this affects the accuracy of computation Positional Number

### Lecture 5. Computer Arithmetic. Ch 9 [Stal10] Integer arithmetic Floating-point arithmetic

Lecture 5 Computer Arithmetic Ch 9 [Stal10] Integer arithmetic Floating-point arithmetic ALU ALU = Arithmetic Logic Unit (aritmeettis-looginen yksikkö) Actually performs operations on data Integer and