Master the bedrock of modern mathematics. This course provides a rigorous introduction to set theory, the fundamental language used to structure nearly every mathematical discipline. We move quickly from defining basic sets and elements to complex operations including unions, intersections, and complements. You will analyze relationships between collections using Venn diagrams, master set-builder notation, and examine the properties of power sets and Cartesian products, establishing the ground rules for all subsequent mathematical study. Mathematical rigour is essential for advanced problem-solving. Set theory is not abstract trivia; it is the critical organizational framework underpinning computer science database structures, logical reasoning systems in artificial intelligence, and complex data stratification in statistical analysis. Fluency in set operations allows you to define precise categories, manage data relationships efficiently, and construct watertight logical arguments required in professional technical environments. Upon completion, you will possess the skills to define sets accurately using various notations and execute operations on finite and infinite collections with precision. You will demonstrate competence in proving fundamental set identities, manipulating Venn diagrams to solve practical problems, and formalising relationships between distinct data groups. Furthermore, you will establish the foundational understanding of relations and functions necessary for progression into advanced calculus and abstract algebra. This course is targeted at students entering undergraduate foundation programmes requiring strong quantitative proficiency, particularly in mathematics, computer science, and engineering. It also serves as an intensive structural refresher for professionals shifting into data-architecture or analytical roles demanding strong logical literacy. Prior exposure to standard secondary school mathematics is assumed; focus is placed strictly on mastery and application of rigorous definitions.

Master the architecture of mathematical dependencies. This course provides a rigorous foundation in relations, mappings, and functions, which are the essential tools used to model connections between abstract mathematical sets. We progress rapidly from defining basic binary relations and analyzing properties such as reflexivity, symmetry, and transitivity, to the precise mechanics of functions. You will examine the strict rules governing inputs, outputs, domains, and codomains that differentiate a functional mapping from a general relation. Functional literacy is non-negotiable in quantitative professional fields. Computer scientists rely heavily on the concept of mappings for algorithm design, database structuring, and functional programming paradigms. Simultaneously, engineers, economists, and data analysts use functions as the primary method to model complex real-world systems, define variables, and predict outcomes based on specific inputs. This framework is indispensable for anyone required to analyze causal relationships or build predictive models professionally. Upon completion, you will possess the skills to rigorously define binary relations and identify equivalence properties within sets. You will accurately distinguish mappings from general relations, determine domains, codomains, and ranges with precision. Crucially, you will acquire the competence to classify functions structurally as injective (one-to-one), surjective (onto), or bijective, and perform essential operations including function composition and the determination of inverse mappings. This course is designed for students entering undergraduate foundation programmes requiring strong analytical bases, particularly in mathematics, computer science, and engineering disciplines. It also acts as an intensive, high-level refresher for professionals returning to academia or shifting into technical roles that demand precise logical structuring. Prior exposure to elementary algebra and basic set notation is assumed; focus is placed strictly on the mastery and application of these core definitions.

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.

Master the algebraic structures governing ordered lists of numbers and their discrete summations. This course moves rapidly beyond elementary pattern recognition to establish a rigorous foundation in the mechanics of real sequences and series. We investigate the precise definitions defining arithmetic and geometric progressions, establish formal notation for general terms, and execute the manipulation of finite and infinite series using standard sigma notation. The curriculum is focused strictly on the structural and algebraic properties required for advanced mathematical analysis. Solid competence in manipulating discrete numerical patterns is essential for quantitative professional fields. Financial analysts rely on the algebra of geometric series for modelling compound interest, loan repayments, and annuities. Computer scientists utilise sequence structures to define discrete data sets and analyse iterative loops. This framework is indispensable for modelling growth, decay, or discrete accumulation processes in business and STEM sectors with absolute precision. Upon completion, you will possess the skills to accurately identify and define sequences through their general term and recursive formulae. You will demonstrate competence in calculating the nth term and the sum of the first n terms for both arithmetic and finite geometric series. Furthermore, you will command the conditions for convergence in infinite geometric series and acquire the ability to manipulate sigma notation fluently, expanding and simplifying discrete summations efficiently. This course is designed for students entering undergraduate foundation programmes requiring strong algebraic bases, particularly in mathematics, finance, and computer science. It also serves as an intensive refresher for professionals returning to academia or shifting into roles demanding precise numerical structuring. Prior exposure to standard secondary school algebra is assumed; focus is placed strictly on the mastery and application of these core discrete definitions.

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.

Do you want to learn how to work with numbers that go beyond the real line? Do you want to understand the concepts of imaginary unit, conjugate, modulus, argument, and polar and exponential forms of complex numbers? Do you want to master the skills of performing algebraic and geometric operations on complex numbers using different methods and tools? If you answered yes to any of these questions, then this course is for you! In this course, you will learn how to: - Define and classify complex numbers and their real and imaginary parts - Perform addition, subtraction, multiplication, and division of complex numbers using the standard form a + bi - Find the conjugate, modulus, and argument of a complex number and use them to compare and simplify complex numbers - Represent complex numbers on the Argand plane and visualize their geometric properties and transformations - Convert complex numbers from rectangular to polar and exponential forms and vice versa - Use De-Moivre's theorem and Euler's formula to find the powers and roots of complex numbers in polar and exponential forms - Use complex numbers to define and manipulate trigonometric and hyperbolic functions and their inverses - Use complex numbers to define and manipulate logarithmic functions and their properties - Use complex numbers to graph and solve equations of circles, lines, and other curves on the complex plane This course is suitable for anyone who wants to learn or review the basics of complex numbers and their applications. It is especially useful for students and professionals in engineering, physics, computer science, cryptography, and other related fields. By the end of this course, you will have a solid understanding of complex numbers and their operations. You will also be able to apply the knowledge and skills you gain to real-world problems and challenges that involve complex numbers. Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

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.

Do you want to learn how to work with matrices and their properties, operations, and applications? Do you want to understand the concepts of determinants, eigenvalues, eigenvectors, diagonalization, quadratic and canonical forms? Do you want to master the skills of solving systems of linear equations, finding inverses, and computing matrix functions using different methods and tools? If you answered yes to any of these questions, then this course is for you! This course covers the fundamentals of matrix theory and its applications in mathematics and science. You will learn how to: - Define and classify matrices and their special types, such as symmetric, orthogonal, diagonal, and identity matrices - Perform matrix addition, subtraction, multiplication, and scalar multiplication using the algebraic properties of matrices - Find the transpose, conjugate, and adjoint of a matrix and use them to simplify matrix operations and expressions - Perform elementary row and column transformations on matrices and use them to find the row echelon form, reduced row echelon form, rank, and nullity of a matrix - Find the minors, cofactors, and determinants of matrices and use them to calculate the area, volume, and orientation of geometrical figures - Find the inverse of a matrix using the adjoint method or the row operations method and use it to solve systems of linear equations - Find the eigenvalues and eigenvectors of a matrix using the characteristic polynomial and the Cayley-Hamilton theorem and use them to analyze the behavior and stability of dynamical systems - Diagonalize a matrix using the eigenvalues and eigenvectors and use it to compute matrix functions, such as polynomials, exponentials, sines, and cosines of matrices - Find the quadratic and canonical forms of an equation using the coefficient matrix and the transformation matrix and use them to classify and graph conics and quadrics - Use computer software, such as MS-Excel, MATLAB, and Python, to perform matrix operations and computations efficiently and accurately This course is suitable for anyone who wants to learn or review the basics of matrix theory and its applications. It is especially useful for students and professionals in algebra, calculus, differential equations, linear programming, optimization, cryptography, computer graphics, data science, machine learning, and other related fields. By the end of this course, you will have a firm grasp of the theory and applications of matrices and determinants. You will also be able to apply the knowledge and skills you gain to solve real-world problems and challenges that involve matrices. Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.