MINISTRY OF EDUCATION
DIPLOMA IN
INFORMATION COMMUNICATION
TECHNOLOGY
KENYA INSTITUTE OF CURRICULUM DEVELOPMENT
STUDY NOTES
Computational Mathematics
MODULE I: SUBJECT NO 5
Page 1 of 199
,Contents
CHAPTER 1: INTRODUCTION TO COMPUTATIONAL MATHEMATICS .................................. 6
Equations .................................................................................................................................................. 6
Algebraic equation .................................................................................................................................... 6
An introduction to MATRICES .............................................................................................................. 19
Definition of Matrix............................................................................................................................. 19
Types of Matrices ................................................................................................................................ 20
Operation of matrices ......................................................................................................................... 22
Determinant of a Matrix ..................................................................................................................... 28
Inverse of a Matrix .............................................................................................................................. 30
Algebraic Properties of Matrix Operations ......................................................................................... 38
Applications of Matrix Mathematics .................................................................................................. 42
CHAPTER 2: NUMBER SYSTEM AND BINARY ARITHMETIC................................................... 43
Introduction to Number System .............................................................................................................. 43
Bits & Bytes: ........................................................................................................................................... 43
Binary Number System: .......................................................................................................................... 43
Other Number System............................................................................................................................. 45
Conversion between Number System ..................................................................................................... 46
Introduction to Binary Arithmetic........................................................................................................... 49
Representation of negative numbers ....................................................................................................... 56
Computer data formats/coding ................................................................................................................ 58
CHAPTER 3: LOGIC GATES AND BOOLEAN ALGEBRA............................................................. 64
Introduction to Logic Mathematics ......................................................................................................... 64
Types of Sets ....................................................................................................................................... 64
Operation of a Set ............................................................................................................................... 70
Venn Diagrams ................................................................................................................................... 83
Boolean Algebra ..................................................................................................................................... 91
Terminologies in Boolean Algebra ..................................................................................................... 91
Logic Functions gates and circuitry .................................................................................................... 92
Theorems of Boolean Algebra ............................................................................................................ 95
Karnaugh maps (K-maps) ................................................................................................................... 98
CHAPTER 4: DISCRETE COUNTING .............................................................................................. 103
Page 2 of 199
, The Basic Rules/Principles of Counting ............................................................................................... 103
The Rules of Sum and Product .......................................................................................................... 103
Pigeonhole Principle ......................................................................................................................... 103
Examples ........................................................................................................................................... 104
The Inclusion-Exclusion principle ...................................................................................................... 104
Counting Techniques ............................................................................................................................ 105
Permutations..................................................................................................................................... 105
Combinations .................................................................................................................................... 107
Pascal's Identity................................................................................................................................. 108
Binomial Theorem ................................................................................................................................ 108
Exponents.......................................................................................................................................... 109
The Pattern ....................................................................................................................................... 110
Coefficients ....................................................................................................................................... 111
Formula ............................................................................................................................................. 113
Geometry .......................................................................................................................................... 116
Pascal's Triangle ................................................................................................................................ 118
CHAPTER 5: PROBABILITY .............................................................................................................. 122
Introduction to probability .................................................................................................................... 122
Probability Models ................................................................................................................................ 122
Equally likely outcomes ........................................................................................................................ 123
The Sum Rule ....................................................................................................................................... 125
Random Variables................................................................................................................................. 126
The Mean (Expected Value) of a Probability model ............................................................................ 126
The Law of Large Numbers .................................................................................................................. 127
Probability of Single or Combined Events ........................................................................................... 127
Independent and Dependent Events ...................................................................................................... 127
Conditional Probability ......................................................................................................................... 128
Probability Tree Diagrams .................................................................................................................... 129
Set Theory in Probability ...................................................................................................................... 132
Probability Distribution ........................................................................................................................ 135
CHAPTER 6: GRAPHS OF FUNCTIONS .......................................................................................... 137
Defining the Graph of a Function ..................................................................................................... 137
Page 3 of 199
, Vertical Line Test ................................................................................................................................. 138
Characteristics of Graphs ...................................................................................................................... 139
Continuous Functions ........................................................................................................................... 142
Interpolation and Extrapolation ............................................................................................................ 144
Area under a Curve ............................................................................................................................... 144
Inequality .............................................................................................................................................. 147
Common Graph Construction Errors .................................................................................................... 149
CHAPTER 7: DATA COLLECTION METHODS ............................................................................. 152
Introduction to Data Collection Methods.............................................................................................. 152
Data Edition: ......................................................................................................................................... 152
Types of Data Collection Methods ....................................................................................................... 152
Selecting Appropriate Data Collection Method .................................................................................... 153
Classification of Data ............................................................................................................................ 153
Data Tabulation and Presentation ......................................................................................................... 154
CHAPTER 8: MEASURES OF CENTRAL TENDENCY ................................................................. 162
Introduction to central tendency ........................................................................................................... 162
The significance property of measures of central tendency .................................................................. 162
Types and measures of central tendency............................................................................................... 162
Calculation in central tendency ............................................................................................................. 162
CHAPTER 9: MEASURES OF DISPERSION.................................................................................... 166
Introduction to measure of Dispersion .................................................................................................. 166
Properties of a good measure of Dispersion ......................................................................................... 166
Types of Dispersion .............................................................................................................................. 166
Methods of Dispersion and Calculations .............................................................................................. 167
Quartiles, deciles, percentiles and S.I.R. ............................................................................................... 167
Skewness and kurtosis in Dispersion interplatation .............................................................................. 170
CHAPTER 10: NUMERICAL ANALYSIS ......................................................................................... 172
Introduction to Numerical analysis ....................................................................................................... 172
General introduction ............................................................................................................................. 172
Approximation theory ........................................................................................................................... 172
Numerical methods ............................................................................................................................... 173
Direct and iterative methods ................................................................................................................. 173
Page 4 of 199
DIPLOMA IN
INFORMATION COMMUNICATION
TECHNOLOGY
KENYA INSTITUTE OF CURRICULUM DEVELOPMENT
STUDY NOTES
Computational Mathematics
MODULE I: SUBJECT NO 5
Page 1 of 199
,Contents
CHAPTER 1: INTRODUCTION TO COMPUTATIONAL MATHEMATICS .................................. 6
Equations .................................................................................................................................................. 6
Algebraic equation .................................................................................................................................... 6
An introduction to MATRICES .............................................................................................................. 19
Definition of Matrix............................................................................................................................. 19
Types of Matrices ................................................................................................................................ 20
Operation of matrices ......................................................................................................................... 22
Determinant of a Matrix ..................................................................................................................... 28
Inverse of a Matrix .............................................................................................................................. 30
Algebraic Properties of Matrix Operations ......................................................................................... 38
Applications of Matrix Mathematics .................................................................................................. 42
CHAPTER 2: NUMBER SYSTEM AND BINARY ARITHMETIC................................................... 43
Introduction to Number System .............................................................................................................. 43
Bits & Bytes: ........................................................................................................................................... 43
Binary Number System: .......................................................................................................................... 43
Other Number System............................................................................................................................. 45
Conversion between Number System ..................................................................................................... 46
Introduction to Binary Arithmetic........................................................................................................... 49
Representation of negative numbers ....................................................................................................... 56
Computer data formats/coding ................................................................................................................ 58
CHAPTER 3: LOGIC GATES AND BOOLEAN ALGEBRA............................................................. 64
Introduction to Logic Mathematics ......................................................................................................... 64
Types of Sets ....................................................................................................................................... 64
Operation of a Set ............................................................................................................................... 70
Venn Diagrams ................................................................................................................................... 83
Boolean Algebra ..................................................................................................................................... 91
Terminologies in Boolean Algebra ..................................................................................................... 91
Logic Functions gates and circuitry .................................................................................................... 92
Theorems of Boolean Algebra ............................................................................................................ 95
Karnaugh maps (K-maps) ................................................................................................................... 98
CHAPTER 4: DISCRETE COUNTING .............................................................................................. 103
Page 2 of 199
, The Basic Rules/Principles of Counting ............................................................................................... 103
The Rules of Sum and Product .......................................................................................................... 103
Pigeonhole Principle ......................................................................................................................... 103
Examples ........................................................................................................................................... 104
The Inclusion-Exclusion principle ...................................................................................................... 104
Counting Techniques ............................................................................................................................ 105
Permutations..................................................................................................................................... 105
Combinations .................................................................................................................................... 107
Pascal's Identity................................................................................................................................. 108
Binomial Theorem ................................................................................................................................ 108
Exponents.......................................................................................................................................... 109
The Pattern ....................................................................................................................................... 110
Coefficients ....................................................................................................................................... 111
Formula ............................................................................................................................................. 113
Geometry .......................................................................................................................................... 116
Pascal's Triangle ................................................................................................................................ 118
CHAPTER 5: PROBABILITY .............................................................................................................. 122
Introduction to probability .................................................................................................................... 122
Probability Models ................................................................................................................................ 122
Equally likely outcomes ........................................................................................................................ 123
The Sum Rule ....................................................................................................................................... 125
Random Variables................................................................................................................................. 126
The Mean (Expected Value) of a Probability model ............................................................................ 126
The Law of Large Numbers .................................................................................................................. 127
Probability of Single or Combined Events ........................................................................................... 127
Independent and Dependent Events ...................................................................................................... 127
Conditional Probability ......................................................................................................................... 128
Probability Tree Diagrams .................................................................................................................... 129
Set Theory in Probability ...................................................................................................................... 132
Probability Distribution ........................................................................................................................ 135
CHAPTER 6: GRAPHS OF FUNCTIONS .......................................................................................... 137
Defining the Graph of a Function ..................................................................................................... 137
Page 3 of 199
, Vertical Line Test ................................................................................................................................. 138
Characteristics of Graphs ...................................................................................................................... 139
Continuous Functions ........................................................................................................................... 142
Interpolation and Extrapolation ............................................................................................................ 144
Area under a Curve ............................................................................................................................... 144
Inequality .............................................................................................................................................. 147
Common Graph Construction Errors .................................................................................................... 149
CHAPTER 7: DATA COLLECTION METHODS ............................................................................. 152
Introduction to Data Collection Methods.............................................................................................. 152
Data Edition: ......................................................................................................................................... 152
Types of Data Collection Methods ....................................................................................................... 152
Selecting Appropriate Data Collection Method .................................................................................... 153
Classification of Data ............................................................................................................................ 153
Data Tabulation and Presentation ......................................................................................................... 154
CHAPTER 8: MEASURES OF CENTRAL TENDENCY ................................................................. 162
Introduction to central tendency ........................................................................................................... 162
The significance property of measures of central tendency .................................................................. 162
Types and measures of central tendency............................................................................................... 162
Calculation in central tendency ............................................................................................................. 162
CHAPTER 9: MEASURES OF DISPERSION.................................................................................... 166
Introduction to measure of Dispersion .................................................................................................. 166
Properties of a good measure of Dispersion ......................................................................................... 166
Types of Dispersion .............................................................................................................................. 166
Methods of Dispersion and Calculations .............................................................................................. 167
Quartiles, deciles, percentiles and S.I.R. ............................................................................................... 167
Skewness and kurtosis in Dispersion interplatation .............................................................................. 170
CHAPTER 10: NUMERICAL ANALYSIS ......................................................................................... 172
Introduction to Numerical analysis ....................................................................................................... 172
General introduction ............................................................................................................................. 172
Approximation theory ........................................................................................................................... 172
Numerical methods ............................................................................................................................... 173
Direct and iterative methods ................................................................................................................. 173
Page 4 of 199
