Operations with Real Numbers - Mathematics (Undergraduate Foundation)

Master the essential algebraic structures required for rigorous undergraduate mathematical study. This course moves rapidly beyond basic arithmetic to establish foundations in manipulating the real number system and solving diverse equation types. We examine the mechanics of polynomials, linear and quadratic equations, simultaneous systems, and the precise handling of inequalities. The curriculum progresses to advanced algebraic techniques, including the decomposition of rational expressions into partial fractions and the comprehensive application of laws governing indices, logarithms, and surds to solve complex exponential equations. Algebraic fluency is non-negotiable in quantitative professional fields. The ability to model relationships using equations and inequalities is fundamental to engineering design, economic forecasting, and algorithmic development in computer science. Mastery of indices and logarithms is essential for analysing exponential growth or decay patterns in scientific data and financial markets. This course equips you with the computational fortitude required to handle complex data and execute rigorous analytical tasks in STEM and business sectors without error. On completion, you will possess the skills to correctly classify real numbers and solve linear, quadratic, and higher-degree polynomial equations with precision. You will demonstrate competence in utilising elimination techniques for simultaneous systems and decomposing algebraic fractions into partial fractions for integration or further analysis. Furthermore, you will command the laws of indices, logarithms, and surds, applying them effectively to simplify radical expressions and solve equations where the unknown variable appears as an exponent. This course is targeted at students entering undergraduate foundation programmes requiring immediate algebraic proficiency, particularly in science, engineering, and economics. It also serves as an intensive refresher for professionals returning to academia or shifting into data-driven roles demanding strong mathematical literacy. Prior exposure to standard secondary school algebra is assumed; focus is placed strictly on mastery and application of these core concepts.

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

6 Lessons

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

Problem Walkthroughs

3 Lessons

3. Equations in One Unknown

This chapter governs the transition from arithmetic to formal algebraic manipulation. You will master the mechanics of isolating variables and maintaining equality across transformations, providing the essential precision required for solving the multi-variable systems and complex rational expressions encountered later in this course. You will master: identifying linear equation structures; executing systematic variable isolation; performing cross-multiplication for fractional expressions; and verifying solutions through substitution.

Concept Overviews

1 Lesson

Problem Walkthroughs

5 Lessons

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 methods for combined linear and quadratic systems; identifying points of intersection between straight lines and curves; solving equations involving fractional components; and verifying coordinate solution sets.

Concept Overviews

1 Lesson

Problem Walkthroughs

8 Lessons

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 algebraic manipulation; solving linear and compound inequalities; expressing solution sets through interval notation; and representing valid regions on the real number line.

Concept Overviews

2 Lessons

Problem Walkthroughs

6 Lessons

6. Partial Fractions

This chapter governs the inverse operation of fraction addition by decomposing complex rational expressions into simpler components. Mastering this technique is a prerequisite for advanced calculus, specifically integration, and for simplifying complex transfer functions in control engineering and signal processing systems. You will master: identifying proper and improper rational functions; decomposing expressions with distinct linear factors; handling repeated linear denominators; and resolving expressions containing irreducible quadratic factors.

Concept Overviews

2 Lessons

Problem Walkthroughs

4 Lessons

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

Problem Walkthroughs

2 Lessons

8. Logarithms

This chapter governs the inverse relationship between powers and exponents. You will master the logical structure of logarithmic transformations to handle nonlinear data scales, establishing the computational proficiency required for solving complex exponential equations and analysing growth models in engineering and financial sectors. You will master: converting between index and logarithmic forms; applying the product, quotient, and power laws of logarithms; executing the change of base formula; and resolving algebraic equations involving unknown exponents.

Concept Overviews

3 Lessons

Problem Walkthroughs

2 Lessons

9. Surds

This chapter governs the manipulation of irrational roots and the maintenance of exact numerical precision. You will master the arithmetic of radical expressions to avoid approximation errors, establishing the computational accuracy required for rigorous geometric analysis, trigonometric identities, and advanced engineering derivations. You will master: simplifying radical expressions; executing fundamental operations with like and unlike surds; rationalising denominators using conjugate pairs; and solving equations involving radical terms.

Concept Overviews

3 Lessons

Problem Walkthroughs

2 Lessons

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 the essential analytical tools for modelling exponential growth, decay constants, and complex scaling in engineering and financial data sets. You will master: equating bases for direct solution; applying logarithmic transformations to isolate exponents; reducing equations to quadratic forms; and executing change of base operations for mixed-base equations.

Concept Overviews

1 Lesson

Problem Walkthroughs

2 Lessons