Operations with Real Numbers - Mathematics (Undergraduate Foundation)

Mathematics requires absolute precision. This course provides a rigorous grounding in real number operations, from integers and rational numbers to complex systems and algebraic laws. You will learn to handle polynomials, solve equations ranging from linear to biquadratic, and navigate simultaneous systems. The syllabus covers inequalities, sign tables, partial fractions, indices, logarithms, and the systematic simplification of surds. These tools are essential for success in engineering, accounting, and the sciences. Clear mathematical thinking allows you to model financial risks, calculate structural loads, and write efficient computer code. Proficiency in these operations ensures accuracy in any career that relies on quantitative data and logical deduction. You will gain the ability to classify number systems, apply algebraic theorems, and solve complex equations in one or more unknowns. You will acquire the skills to manipulate inequalities, resolve algebraic fractions, and simplify expressions using the laws of indices and logarithms. The course provides the technical competence required to handle surds and find roots of compound expressions effectively. This training is built for undergraduate foundation students and secondary school leavers entering university. It provides a necessary bridge for anyone needing to strengthen their mathematical base before advanced study. Even those in non-technical roles will find value in the disciplined analytical approach required to master these foundational concepts.

34 hrs

Enrolment valid for 12 months

This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.

MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the foundational mathematical structures essential for success in quantitative undergraduate degrees and professional technical roles. This comprehensive learning track delivers a rigorous treatment of algebra and trigonometry, moving rapidly from fundamental set theory and real number operations to advanced topics including matrix algebra, complex numbers, and analytical trigonometry. You will establish the critical problem-solving framework required for advanced study in calculus, engineering mechanics, and data science. This programme is primarily designed for first-year university students in STEM disciplines requiring strong analytical bases, particularly engineering, physics, computer science, and economics. It also serves as an intensive, high-level refresher for professionals returning to academia or shifting into data-driven roles demanding precise numerical literacy and logical structuring. Prior competence in standard secondary school mathematics is assumed; focus is placed strictly on mastery and application of core definitions. Upon completion, you will possess the skills to construct rigorous logical arguments using set theory and mathematical induction, model complex relationships with functions and matrices, and analyze periodic systems using advanced trigonometry. You will demonstrate competence in solving diverse equation types, from quadratics to linear systems, and manipulating complex numbers in engineering applications. This track prepares you directly for the mathematical demands of second-year university studies and technical professional certification exams.

Course Chapters

1. Introduction

This chapter establishes the rigorous classification and fundamental operations of the real number system. You will master the structural hierarchy of numbers and basic algebraic manipulations, providing the essential precision required for solving complex equation systems and inequalities in subsequent chapters. You will master: classifying real numbers into rational and irrational subsets; applying order of operations to complex expressions; identifying properties of real number subsets; and executing fundamental algebraic simplifications.

Concept Overviews

9 Lessons

2:34:40

Problem Walkthroughs

3 Lessons

19:50

2. Polynomials

This chapter governs the manipulation and factorisation of higher-degree algebraic expressions. You will master the structural logic of polynomial functions, establishing the essential skills required for advanced partial fraction decomposition and the analysis of non-linear systems in engineering and physics. You will master: executing polynomial long division and synthetic division; applying the Remainder and Factor Theorems; factorising cubic and higher-order expressions; and identifying roots of polynomial equations.

Concept Overviews

4 Lessons

1:22:14

Problem Walkthroughs

6 Lessons

52:56

3. Equations in One Unknown

This chapter establishes the rigorous mechanical framework for resolving equations of varying degrees and structures. Mastering these transformation techniques is essential for isolating variables in engineering models and scientific formulas with absolute logical consistency. You will master the systematic isolation of unknowns in linear and quadratic forms; the resolution of rational and radical equations; and the reduction of cubic, biquadratic, and symmetric expressions into solvable factors.

Concept Overviews

1 Lesson

45:13

Problem Walkthroughs

7 Lessons

1:54:00

4. Simultaneous Equations

This chapter governs the resolution of mixed systems where linear and non-linear dependencies intersect. You will master the algebraic techniques required to solve for variables across multiple constraints, providing the essential logic for intersecting geometric paths and complex equilibrium states in engineering and economics. You will master: executing substitution and elimination for dual linear systems; identifying points of intersection between straight lines and curves; resolving homogeneous quadratic systems through variable substitution; and simplifying complex rational and product constraints to determine precise coordinate solution sets.

Concept Overviews

1 Lesson

28:51

Problem Walkthroughs

7 Lessons

2:32:54

5. Inequalities

This chapter governs the algebra of non-equality and region-based constraints. You will master the rigorous handling of directionality in mathematical relationships, establishing the essential logic required for economic optimisation, engineering tolerance analysis, and defining valid data ranges in computational algorithms. You will master: executing sign-reversal rules during manipulation; solving linear, quadratic, and rational inequalities; using interval notation; and shading valid regions.

Concept Overviews

8 Lessons

3:10:18

Problem Walkthroughs

14 Lessons

3:22:50

6. Partial Fractions

This chapter covers the inverse operation of fraction addition by breaking down complex rational expressions into simpler parts. Mastery of this decomposition is a prerequisite for advanced calculus and engineering system analysis. You will master identifying proper and improper functions; decomposing expressions with distinct or repeated linear factors; resolving irreducible quadratic denominators; and applying polynomial long division to handle high-degree numerators.

Concept Overviews

3 Lessons

1:05:47

Problem Walkthroughs

6 Lessons

1:34:33

7. Indices

This chapter governs the mechanics of exponential notation and the foundational laws for manipulating powers. You will master these operations to establish the technical proficiency required for solving complex exponential equations and performing the logarithmic transformations essential to scientific and financial data analysis. You will master: applying the product, quotient, and power laws; evaluating zero, negative, and fractional indices; and simplifying complex exponential expressions through systematic reduction.

Concept Overviews

3 Lessons

31:31

Problem Walkthroughs

2 Lessons

11:55

8. Logarithms

This chapter establishes the inverse relationship between powers and exponents to handle nonlinear data. Mastery of logarithmic transformations is essential for solving complex exponential equations and modelling growth patterns in engineering and financial sectors. You will master: converting between index and logarithmic forms; applying the product, quotient, and power laws; executing the change of base formula; and solving algebraic equations involving unknown exponents or bases.

Concept Overviews

6 Lessons

1:15:07

Problem Walkthroughs

6 Lessons

45:52

9. Surds

This chapter covers surds, which are roots that do not have whole number answers. You must learn to use them to keep your calculations exact, as using decimals in engineering and science leads to errors. You will master simplifying and performing arithmetic on surds; rationalising denominators using conjugates; finding square roots of compound surds; and solving algebraic equations involving radical terms.

Concept Overviews

8 Lessons

2:30:09

Problem Walkthroughs

9 Lessons

1:18:50

10. Equations with Unknown Index

This chapter governs the synthesis of indices and logarithms to solve equations where variables appear as exponents. You will master the algebraic techniques required to isolate and resolve unknown indices, providing essential analytical tools for modelling exponential growth, decay constants, and complex scaling in engineering and financial data. You will master: equating bases for direct solution; applying logarithmic transformations; reducing equations to quadratic forms; and solving non-linear simultaneous systems involving mixed bases.

Concept Overviews

1 Lesson

21:07

Problem Walkthroughs

6 Lessons

1:01:58