## Description

This **digital book contains 9 lectures** on practically everything STEM students need to know about **Elementary Functions and Complex Numbers** to succeed in their courses. The Lecture Notes contain useful Summaries (Cheat Sheets) and description of necessary Study Skills, including tips for preparing for tests and exam. The Notes are also supplied with answers to Socratic questions dispersed throughout and Solutions to suggested Home Exercises (Worked Examples).

The emphasis is on disentangling and explaining the necessary concepts. Teachers using these notes would enhance the quality of their teaching and by implication, students success. The Notes can be used on their own or to complement standard classroom materials.

**Product Summary:**

- Comprehensive Lecture Notes on College Algebra and Basic Calculus
- Socratic Questions and Suggested Answers
- Useful Summaries (also known as Cheat Sheets)
- A mathematics-specific Study Skills Guide, including tips for exam preparation.
- Solutions to Exercises suggested for Self-study (Worked Examples)

**Table of Contents:**

I. INTRODUCTION

II. CONCEPT MAPS

III. LECTURES

Lecture 1. ALGEBRA: Addition, Subtraction, Multiplication and Division of Rational Numbers

1.1 Variables

1.2 Variables and operations on variables

1.3 General remarks

1.4 Glossary of terms introduced in this Lecture

1.5 Historical notes

1.6 Instructions for self-study

Lecture 2. Applications of Elementary ALGEBRA: Solving Simple Equations

2.1 Revision: Factorisation

2.2 Revision: Adding fractions

2.3 Decision Tree for Solving Simple Equations

2.4 Applications of equations

2.5 A historical note

2.6 Instructions for self-study

Lecture 3. ALGEBRA: Exponentiation, Roots and Logarithms of Real Numbers

3.1 Types of variables and operations on variables (ctd.)

3.2 Applications

3.3 Instructions for self-study

Lecture 4. FUNCTIONS

4.1 Variables

4.2 Functions

4.3 Elementary operations on functions

4.4 Order of Operations (extended)

4.5 Applications of real functions of real variable and operations on such functions

4.6 A historical note

4.7 Instructions for self-study

Lecture 5. Real FUNCTIONS of One Real Variable: Graphs, Polynomials

5.1 Graphical representation of real functions of one real variable

5.2 How to use graphs of real functions y = f(x) of one real variable

5.3 Applications of graphs

5.4 Elementary functions: monomials (natural powers) and polynomials

5.5 Instructions for self-study

Lecture 6. Real FUNCTIONS of One Real Variable: Exponential Functions, Logarithmic Functions, Inverse Functions

6.1 Exponential functions

6.2 Logarithmic functions

6.3 Revision: Trigonometry

6.4 A historical note

6.5 Instructions of self-study

Lecture 7. Real FUNCTIONS of One Real Variable: Trigonometric Functions, Inverse Trigonometric Functions, Hyperbolic Functions

7.1 Trigonometric functions

7.2 Inverse trigonometric functions

7.3 Hyperbolic functions and their inverses

7.4 Instructions for self-study

Lecture 8. Real FUNCTIONS of One Real Variable: Sketching and Using Graphs, Simple Transformations

8.1 Sketching graphs using a table

8.2 Using graphs to find y1 given x1

8.3 Using graphs to find x1 given y1

8.4 Sketching graphs using simple transformations

8.5 Sketching graphs using several simple transformations

8.6 Sketching a parabola by simple transformations

8.7 Applications of simple transformations

8.8 Instructions for self-study

Lecture 9. Real FUNCTIONS of One Real Variable: Sketching Graphs by Simple Transformations (ctd.)

9.1 Sketching graphs using several simple transformations (ctd.)

9.2 Sketching graphs using pointwise operations

9.3 Instructions for self-study

Lecture 10. ALGEBRA: Addition of Complex Numbers, the Argand Diagram, Forms of Complex Numbers

10.1 Imaginary unity j

10.2 Applications of complex numbers: solving quadratic equations

10.3 Operations: the whole powers of j

10.4 Variables: definition of a complex number z

10.5 Operations: addition of complex numbers z1 and z2

10.6 Operations: subtraction of complex numbers

10.7 The Argand diagram and Cartesian form of a complex number

10.8 The Argand diagram and polar form of a complex number

10.9 The Cartesian-polar and polar-Cartesian transformations

10.10 The trigonometric form of a complex number

10.11 Euler’s formula

10.12 The exponential form of a complex number

10.13 Applications

10.14 A historical note

10.15 Instructions for self-study

Lecture 11. ALGEBRA: Multiplication and Division of Complex Numbers

11.1 Multiplication of a complex number z by a real number

11.2 Operations: multiplication of complex numbers z1 and z2

11.3 Operations: complex conjugate z* of a complex number z

11.4 Operations: division of complex numbers z1 and z2

11.5 Integer powers of complex numbers

11.6 The fractional powers (k-th roots) of complex numbers

11.7 Instructions for self-study

Lecture 12. ALGEBRA: Fractional Powers, Logs and Loci of Complex Numbers

12.1 The fractional powers (whole roots k) of complex numbers (ctd.)

12.2 Logs of complex numbers

12.3 Loci on the Argand diagram

12.4 Applications

12.5 Instructions for self-study

IV. SUMMARIES

Algebra Summary

Functions Summary

Order of Operations Summary

Quadratics Summary

Trigonometry Summary

Complex Numbers

Decision Tree For Solving Simple Equations

Sketching Graphs by Simple Transformations

V. GLOSSARY

VI. STUDY SKILLS

VII. TEACHING METHODOLOGY (FAQs)

VIII. SOLUTIONS TO EXERCISES FOR SELF-STUDY (WORKED EXAMPLES)

Solutions to Exercises in Lecture 4

Solutions to Exercises in Lecture 5

Solutions to Exercises in Lecture 6

Solutions to Exercises in Lecture 7

Solutions to Exercises in Lecture 8

Solutions to Exercises in Lecture 9

Solutions to Exercises in Lecture 10

Solutions to Exercises in Lecture 11

Solutions to Exercises in Lecture 12

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