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The Foundation of MATHEMATICS

Author Michael Colin Roberts B.Sc. MIED

Edition 2 published 2017

ISBN 978-0-9566315-5-8

Book size 6 x 9 inches, 224 interior pages, softback (paperback), Perfect binding.

CONTENTS

1 Introduction

1.1 Purpose of the book

1.2 Biographical sketches

2 Arithmetic

2.1 Introduction

2.2 Definitions and axioms

2.3 Fundamental concepts

2.4 Addition

2.5 Method of addition

2.6 Subtraction

2.7 Multiplication

2.8 Method of multiplication

2.9 Rule of signs

2.10 Division

2.11 Fraction operations

2.12 Decimal fractions

2.13 Decimal operations

2.14 Money and metrology

3 Algebra

3.1 Introduction

3.2 Simultaneous equations

3.3 Powers

3.4 Sequences and series

3.5 Roots and surds

3.6 Precision in calculations

3.7 Extraction of roots by iteration

3.8 Extraction of roots by sequence

3.9 Extraction of roots by series

3.10 Algebraic equations

3.11 Transcendental numbers

3.12 Infinity

3.13 Complex numbers

3.14 Inequalities

3.15 Proportion and ratio

3.16 Logarithms

3.17 Greek alphabet

4 Coordinate Geometry

4.1 Introduction

4.2 Axioms

4.3 Parametric line

4.4 Line defined by equations

4.5 Line through two points

4.6 Line in 1D

4.7 Intersecting lines

4.8 Scale factor

4.9 Collinear lines

4.10 Parametric plane

4.11 Plane defined by equation

4.12 Line in a plane

4.13 Plane through three points

4.14 Plane in 2D

4.15 Parallel lines

4.16 Coplanar planes

4.17 Parametric block

4.18 Parametric curve

4.19 Curve length

4.20 Parametric surface

4.21 Parametric solid

4.22 Gradient

4.23 Angle

5 Greek Geometry

5.1 Vertically opposite angles

5.2 Trigonometry

5.3 Trigonometric lemma

5.4 Parallel and orthogonal lines

5.5 Corresponding angles

5.6 The triangle

5.7 The rectangle

5.8 Congruence

5.9 The square

5.10 Congruent triangles

5.11 Isosceles triangle

5.12 Pythagoras’ theorem

5.13 3D Pythagoras

5.14 Similar triangles

5.15 Similar plane figures

5.16 Similar solids

5.17 The circle

5.18 The value of Pi

5.19 Triangles with circles

6 Geometric Equations

6.1 Trigonometric theorems

6.2 Obtuse and reflex angles

6.3 Trigonometric relations

6.4 Equations to curves

6.5 Vectors

7 Number Theory

7.1 Mathematical induction

7.2 The binomial theorem

7.3 The division algorithm

7.4 Prime numbers

7.5 Fundamental theorem of arithmetic

7.6 Odd and even numbers

7.7 Surds

7.8 The highest common factor

7.9 Euclid’s algorithm

8 Probability Theory

8.1 Introduction

8.2 Permutations

8.3 Combinations

8.4 Probability

8.5 Statistics

9 Units & Conversion Factors

10 Worked Examples

11 Figures

Bibliography

Index

21st Century ALGEBRAIC EQUATIONS

Author: Michael Colin Roberts B.Sc. MIED

Edition 3 published 2017

ISBN 978-0-9566315-6-5

Book size 6 x 9 inches, 191 interior pages, softback (paperback), Perfect binding.

CONTENTS

1 Introduction

1.1 Purpose of the book

1.2 Historical perspective

2 Fundamental Principles

2.1 Revision

2.2 Algebraic equations

2.3 Algebraic functions

2.4 Differential calculus

2.5 Integral calculus

3 The Process of Iteration

3.1 The concept of iteration

3.2 The bisection method

3.3 Linear extrapolation

3.4 Newton’s approximation

4 The General Numerical Solution

4.1 The loder method

4.2 End roots

4.3 The cubic equation

4.4 Complex roots

4.5 The quartic equation

4.6 Repeated roots

4.7 Adjacent inflection points

5 Exact Solutions

5.1 Tartaglia’s solution to the cubic equation

5.2 The cubic algorithm

5.3 Ferraro’s solution to the quartic equation

5.4 Descartes’ solution to the quartic equation

5.5 Reduction method

5.6 Taylor’s series

6 Complex Arithmetic

6.1 Arithmetic operations

6.2 The Argand diagram

6.3 De Moivre’s theorem

6.4 Extraction of roots

6.5 Exponential functions

7 Theorems of Algebraic Equations

7.1 The concept of theorems

7.2 The fundamental theorem of algebra

7.3 Taylor’s theorem

7.4 The RST theorem

8 Computer Programs

8.1 Discussion

8.2 Spreadsheet # 1, general AE

8.3 Spreadsheet # 2, the quintic equation

8.4 Spreadsheet # 3, to reduce the AE degree

8.5 Spreadsheet # 4, to find the y-ordinates

8.6 Program AER1.bas

9 Worked Examples

10 Figures

Glossary

Appendix

Bibliography

Index

Author: Michael Colin Roberts B.Sc. MIED

Edition 3 published 2017

ISBN 978-0-9566315-6-5

Book size 6 x 9 inches, 191 interior pages, softback (paperback), Perfect binding.

CONTENTS

1 Introduction

1.1 Purpose of the book

1.2 Historical perspective

2 Fundamental Principles

2.1 Revision

2.2 Algebraic equations

2.3 Algebraic functions

2.4 Differential calculus

2.5 Integral calculus

3 The Process of Iteration

3.1 The concept of iteration

3.2 The bisection method

3.3 Linear extrapolation

3.4 Newton’s approximation

4 The General Numerical Solution

4.1 The loder method

4.2 End roots

4.3 The cubic equation

4.4 Complex roots

4.5 The quartic equation

4.6 Repeated roots

4.7 Adjacent inflection points

5 Exact Solutions

5.1 Tartaglia’s solution to the cubic equation

5.2 The cubic algorithm

5.3 Ferraro’s solution to the quartic equation

5.4 Descartes’ solution to the quartic equation

5.5 Reduction method

5.6 Taylor’s series

6 Complex Arithmetic

6.1 Arithmetic operations

6.2 The Argand diagram

6.3 De Moivre’s theorem

6.4 Extraction of roots

6.5 Exponential functions

7 Theorems of Algebraic Equations

7.1 The concept of theorems

7.2 The fundamental theorem of algebra

7.3 Taylor’s theorem

7.4 The RST theorem

8 Computer Programs

8.1 Discussion

8.2 Spreadsheet # 1, general AE

8.3 Spreadsheet # 2, the quintic equation

8.4 Spreadsheet # 3, to reduce the AE degree

8.5 Spreadsheet # 4, to find the y-ordinates

8.6 Program AER1.bas

9 Worked Examples

10 Figures

Glossary

Appendix

Bibliography

Index

21st Century SIMULTANEOUS EQUATIONS

Author: Michael Colin Roberts B.Sc. MIED

Edition 2 published 2017

ISBN 978-0-9566315-7-2

Book size 6 x 9 inches, 190 interior pages, softback (paperback), Perfect binding.

CONTENTS

1 Introduction

2 Linear Algebraic Relations

2.1 Definitions

2.2 Simple linear relations

2.3 Determinant solution

2.4 Matrix solution

2.5 Graphical solution

3 Non-Linear Algebraic Relations

3.1 SE in two unknowns

3.2 SE in three unknowns

3.3 Complex algebraic equations

3.4 Complex simultaneous equations

4 Circular Relations

4.1 Introduction

4.2 3D trigonometry

4.3 The linear trig equation

4.4 The non-linear trig equation

4.5 Simultaneous trig equations

4.6 Miscellaneous equations

5 SE Analysis

5.1 Linear algebraic relations

5.2 Non-linear algebraic relations

5.3 Degree of the eliminant

5.4 Precision in calculations

6 Determinants

6.1 Definitions and laws

6.2 Cofactors

6.3 Computations

6.4 Multiplication

7 Matrices

7.1 Definitions

7.2 Addition and multiplication

7.3 Laws

7.4 Application to equations

7.5 Transpose matrix

7.6 Inverse matrix

7.7 Adjoint matrix

7.8 Transformations

7.9 Eigenvectors and eigenvalues

8 Vectors

8.1 Introduction

8.2 Addition of vectors

8.3 Scaling vectors

8.4 Change of frame

8.5 Base vectors

8.6 Points, lines and curves

8.7 Planes, surfaces and solids

8.8 Scalar product

8.9 Vector product

8.10 Triple products

8.11 3D distance

8.12 Vector equation

9 Computer Programs

9.1 Discussion

9.2 General SE spreadsheet

9.3 Spreadsheet for solving two unknowns

9.4 Spreadsheet for solving three unknowns

9.5 Spreadsheet to multiply polynomials

9.6 Program SER1.bas

10 Worked Examples

11 Figures

Bibliography

Index

Author: Michael Colin Roberts B.Sc. MIED

Edition 2 published 2017

ISBN 978-0-9566315-7-2

Book size 6 x 9 inches, 190 interior pages, softback (paperback), Perfect binding.

CONTENTS

1 Introduction

2 Linear Algebraic Relations

2.1 Definitions

2.2 Simple linear relations

2.3 Determinant solution

2.4 Matrix solution

2.5 Graphical solution

3 Non-Linear Algebraic Relations

3.1 SE in two unknowns

3.2 SE in three unknowns

3.3 Complex algebraic equations

3.4 Complex simultaneous equations

4 Circular Relations

4.1 Introduction

4.2 3D trigonometry

4.3 The linear trig equation

4.4 The non-linear trig equation

4.5 Simultaneous trig equations

4.6 Miscellaneous equations

5 SE Analysis

5.1 Linear algebraic relations

5.2 Non-linear algebraic relations

5.3 Degree of the eliminant

5.4 Precision in calculations

6 Determinants

6.1 Definitions and laws

6.2 Cofactors

6.3 Computations

6.4 Multiplication

7 Matrices

7.1 Definitions

7.2 Addition and multiplication

7.3 Laws

7.4 Application to equations

7.5 Transpose matrix

7.6 Inverse matrix

7.7 Adjoint matrix

7.8 Transformations

7.9 Eigenvectors and eigenvalues

8 Vectors

8.1 Introduction

8.2 Addition of vectors

8.3 Scaling vectors

8.4 Change of frame

8.5 Base vectors

8.6 Points, lines and curves

8.7 Planes, surfaces and solids

8.8 Scalar product

8.9 Vector product

8.10 Triple products

8.11 3D distance

8.12 Vector equation

9 Computer Programs

9.1 Discussion

9.2 General SE spreadsheet

9.3 Spreadsheet for solving two unknowns

9.4 Spreadsheet for solving three unknowns

9.5 Spreadsheet to multiply polynomials

9.6 Program SER1.bas

10 Worked Examples

11 Figures

Bibliography

Index

Synopsis of the CALCULUS

Author: Michael Colin Roberts B.Sc. MIED

Edition 2 published 2017

ISBN 978-0-9566315-8-9

Book size 6 x 9 inches, 213 interior pages, softback (paperback), perfect binding.

CONTENTS

1 Introduction

2 Differential Calculus

2.1 The principle of differentiation

2.2 Techniques of differentiation

2.3 Maxima & minima

3 Mathematical Analysis

3.1 Limits and continuity

3.2 Formal differentiation

3.3 Theorems

3.4 Power series

3.5 The exponential function

3.6 The logarithmic function

3.7 Hyperbolic functions

4 Integral Calculus

4.1 Indefinite integration

4.2 The definite integral

4.3 Improper integrals

4.4 Substitution (change of variable)

4.5 Trigonometric integration

4.6 Integration by parts

4.7 Reduction formulas

4.8 The remainder theorem

4.9 Partial fractions

4.10 Methods of approximation

5 Applications of Calculus

5.1 mensuration

5.2 Mechanics

5.3 Dynamics

5.4 Coordinate systems

5.5 Fourier series

5.6 Multiple integrals

6 Partial Differentiation

6.1 Introduction

6.2 Polar coordinates

6.3 Higher derivatives

6.4 Function of two functions

6.5 The calculus of variations

7 Vector Calculus

7.1 Differentiation

7.2 Curves

7.3 Surfaces

7.4 Kinematics

7.5 Integration

7.6 Grad, Div & Curl

8 Differential Equations

8.1 Introduction

8.2 Primal techniques

8.3 Linear equations

8.4 Simultaneous linear differential equations

8.5 Partial differential equations

9 Construction of Figures

9.1 The straight line

9.2 Linear programming

9.3 Graphs of logarithms

9.4 Linear regression

9.5 Conic sections

9.6 Figures in 3D

10 Worked Examples

11 Figures

12 Parex Expansions

13 Standard Relations & Expressions

Bibliography

Index

Author: Michael Colin Roberts B.Sc. MIED

Edition 2 published 2017

ISBN 978-0-9566315-8-9

Book size 6 x 9 inches, 213 interior pages, softback (paperback), perfect binding.

CONTENTS

1 Introduction

2 Differential Calculus

2.1 The principle of differentiation

2.2 Techniques of differentiation

2.3 Maxima & minima

3 Mathematical Analysis

3.1 Limits and continuity

3.2 Formal differentiation

3.3 Theorems

3.4 Power series

3.5 The exponential function

3.6 The logarithmic function

3.7 Hyperbolic functions

4 Integral Calculus

4.1 Indefinite integration

4.2 The definite integral

4.3 Improper integrals

4.4 Substitution (change of variable)

4.5 Trigonometric integration

4.6 Integration by parts

4.7 Reduction formulas

4.8 The remainder theorem

4.9 Partial fractions

4.10 Methods of approximation

5 Applications of Calculus

5.1 mensuration

5.2 Mechanics

5.3 Dynamics

5.4 Coordinate systems

5.5 Fourier series

5.6 Multiple integrals

6 Partial Differentiation

6.1 Introduction

6.2 Polar coordinates

6.3 Higher derivatives

6.4 Function of two functions

6.5 The calculus of variations

7 Vector Calculus

7.1 Differentiation

7.2 Curves

7.3 Surfaces

7.4 Kinematics

7.5 Integration

7.6 Grad, Div & Curl

8 Differential Equations

8.1 Introduction

8.2 Primal techniques

8.3 Linear equations

8.4 Simultaneous linear differential equations

8.5 Partial differential equations

9 Construction of Figures

9.1 The straight line

9.2 Linear programming

9.3 Graphs of logarithms

9.4 Linear regression

9.5 Conic sections

9.6 Figures in 3D

10 Worked Examples

11 Figures

12 Parex Expansions

13 Standard Relations & Expressions

Bibliography

Index

21st Century ALGEBRAIC RELATIONS

Author: Michael Colin Roberts B.Sc. MIED

Edition 2 published 2017

ISBN 978-0-9566315-9-6

Book size 6 x 9 inches, 222 interior pages, softback (paperback), Perfect binding.

CONTENTS

1 Introduction

2 Sequences & Series

2.1 Review

2.2 Sequences

2.3 Series

2.4 Convergence of infinite series

3 The general Algebraic relation

3.1 Derivation of the algebraic relation

3.2 Characteristics of the AR

3.3 The multiple roots method

3.4 The super-function method

3.5 Multiple variables

4 The General Solution in Calculus

4.1 The general equations

4.2 Patternation

4.3 Primal methods

4.4 The B0-Replica method

4.5 The A0-Replica method

4.6 The eliminant method

4.7 Serial methods

4.8 Factors method

4.9 Negative powers

4.10 Partial differential equations

4.11 Simultaneous differential equations

4.12 Computer programs

5 Worked Examples

6 Figures & Tables

7 Register of Classes

Bibliography

Index

Author: Michael Colin Roberts B.Sc. MIED

Edition 2 published 2017

ISBN 978-0-9566315-9-6

Book size 6 x 9 inches, 222 interior pages, softback (paperback), Perfect binding.

CONTENTS

1 Introduction

2 Sequences & Series

2.1 Review

2.2 Sequences

2.3 Series

2.4 Convergence of infinite series

3 The general Algebraic relation

3.1 Derivation of the algebraic relation

3.2 Characteristics of the AR

3.3 The multiple roots method

3.4 The super-function method

3.5 Multiple variables

4 The General Solution in Calculus

4.1 The general equations

4.2 Patternation

4.3 Primal methods

4.4 The B0-Replica method

4.5 The A0-Replica method

4.6 The eliminant method

4.7 Serial methods

4.8 Factors method

4.9 Negative powers

4.10 Partial differential equations

4.11 Simultaneous differential equations

4.12 Computer programs

5 Worked Examples

6 Figures & Tables

7 Register of Classes

Bibliography

Index