This course establishes the foundational language of mathematics. It covers the core principles of set theory and applies them directly to the structure of the real number system. Mastery of this material is non-negotiable for further study in any quantitative field. The principles of set theory are the structural basis for formal logic, database theory, and computer programming. Understanding number systems as structured sets is the prerequisite for all higher mathematics, including algebra and calculus. This course provides the essential logical framework for these disciplines. By the end of this course, you will be able to use formal notation to define sets and their relationships, perform standard set operations such as union and intersection, use Venn diagrams to analyse set identities, and classify numbers within the real number system. This course is designed for first-year university students of mathematics, computer science, and engineering. It is also required foundational material for any professional in data analysis, software development, or other technical fields requiring rigorous logical thought.

This course provides a complete treatise of mathematical progressions. It formally defines sequences and series, with a specific focus on Arithmetic Progressions (APs) and Geometric Progressions (GPs). This material is the foundation for analysing patterns of growth, decay, and summation. These principles are foundational to finance, computer science, and physics. The course provides the tools to calculate compound interest, analyse algorithm complexity, and model physical phenomena. Mastery is required for calculus and financial mathematics. By the end of this course, you will be able to differentiate between sequences and series, apply all standard formulas for the nth term and sum of Arithmetic and Geometric Progressions, calculate the sum to infinity for a convergent GP, and solve problems involving arithmetic and geometric means. This course is designed for first-year university students in mathematics, economics, finance, and engineering. It is a mandatory prerequisite for the study of calculus and provides an essential foundation for financial modelling and algorithm analysis.

This course provides a rigorous, in-depth analysis of quadratic equations. It covers all methods for solving quadratics, determining the nature of their roots using the discriminant, and constructing equations from given roots. A full command of this topic is a cornerstone of algebra. Quadratic models are essential in the physical and financial sciences. They are used in physics to describe projectile motion, in engineering to design parabolic structures like antennas, and in finance for profit optimisation. This is the mathematics of trajectories and optimisation. By the end of this course, you will be able to solve any quadratic equation by factorisation, completing the square, or the quadratic formula, use the discriminant to determine the nature of the roots, apply the sum and product of roots to solve problems, and construct a quadratic equation from a given set of roots. This course is critical for first-year university students in mathematics, physics, engineering, and economics. It provides the necessary algebraic foundation for calculus, mechanics, and any field involving the optimisation of non-linear models.

This course covers two critical topics in discrete mathematics: the principle of mathematical induction as a formal proof technique, and the binomial theorem for expanding powers of binomials. The course structure directly links the proof method to one of its most important applications in algebra. These topics are the foundation of algorithm verification, probability theory, and statistics. Mathematical induction is a core technique for proving algorithm correctness and establishing formulas for series. The binomial theorem is a powerful tool used directly in statistics for the binomial distribution and in financial mathematics. By the end of this course, you will be able to apply the principle of mathematical induction to prove mathematical statements, solve complex counting problems using permutations and combinations, and use the binomial theorem to expand any binomial expression for a positive integer exponent. This course is for first-year university students and is a fundamental prerequisite for computer science, statistics, and probability theory. It provides essential skills for competitive mathematics and advanced algebraic studies.

This course provides a complete introduction to the complex number system, from the definition of the imaginary unit to the Argand diagram and polar form. It then covers De Moivre’s theorem and its application to finding the nth roots of unity. This material extends algebra into two dimensions and provides essential tools for advanced mathematics. Complex numbers are an essential tool in engineering and the physical sciences. They are used extensively in electrical engineering to analyse AC circuits, in signal processing for Fourier transforms, in quantum mechanics to describe wave functions, and in fluid dynamics. This course provides the practical foundation for these applications. By the end of this course, you will be able to perform all arithmetic operations on complex numbers, represent them on an Argand diagram, convert between Cartesian and polar forms, apply De Moivre’s theorem to find powers of complex numbers, and calculate the nth roots of unity. This course is designed for first-year university students of mathematics, physics, and engineering. It is a critical prerequisite for the study of electrical circuits, signal processing, quantum mechanics, and any field that uses wave analysis or advanced vector mathematics.

This course provides a complete guide to trigonometry, from the principles of circular measure to functions of any angle. It covers the unit circle definitions and culminates in a full treatment of the addition and factor formulae. This is the mathematical language of cycles, waves, and angles. A command of trigonometry is essential for any field that models periodic or geometric systems. It is used extensively in physics to analyse waves and oscillations, in engineering for structural analysis and signal processing, and in computer graphics for rotations and 3D modelling. By the end of this course, you will be able to convert between degrees and radians, define the six trigonometric functions for any angle using the unit circle, and apply the addition and factor formulae to manipulate and simplify trigonometric expressions and solve equations. This course is a mandatory prerequisite for any student of mathematics, physics, or engineering. It provides the necessary foundation for the study of calculus, complex numbers, and vector analysis.