MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Mathematics begins with sets. This course covers everything from basic definitions and membership notations to complex set algebra and De Morgan's laws. You will master cardinality, power sets, and the classification of number systems including rational, irrational, and complex numbers. The curriculum moves from simple operations like union and intersection into element-wise proofs, Cartesian products, and the mechanics of relations and functions. Set theory is the language of modern data and logic. These concepts are essential for computer programming, database management, and statistical analysis. Understanding functions and mappings allows you to model real-world dependencies in engineering, economics, and the sciences. Mastering these foundations provides the exact logical framework needed to solve complex problems in technology and research. Upon completion, you will be able to simplify set expressions and solve grouping problems using Venn diagrams and the inclusion-exclusion principle. You will know how to perform element-wise proofs and calculate set cardinalities. You will also gain the ability to evaluate composite functions and prove whether a mapping is one-to-one, onto, or bijective. This course is designed for undergraduate students and secondary school leavers entering STEM disciplines. It provides a necessary logical foundation for anyone moving into calculus, data science, or advanced mathematics. The clear, direct instruction ensures that any student can develop the systematic thinking required for professional technical roles.

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.

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.

Mathematical induction proves a statement is true for every whole number using a starting point and a logical link. It works like a chain reaction where one step confirms the next. This course explains the core principle and applies it to many types of problems. You will study standard series, fractional series, divisibility, and inequalities. The content also covers recursive sequences, matrix powers, and set theory to give you a complete understanding of the method. Programmers use this logic to check that software code works without mistakes. Engineers use it to prove that formulas for structural designs or machines are safe. This skill is vital for anyone in technology, finance, or data science because it makes logical thinking a clear, repeatable process. It removes guesswork and provides the certainty needed for professional work in any technical field. You will learn the three-step induction method to solve different kinds of mathematical proofs. You will prove formulas for series, check if expressions are divisible, and handle difficult inequalities. You will also find general formulas for repeating sequences and solve problems involving matrices and sets. By the end, you will write clear, logical proofs that are required for university mathematics examinations. This course is for university students starting science or engineering degrees and senior secondary school students preparing for higher education. It is also helpful for workers who want to improve their reasoning and problem-solving. Even those not studying mathematics will benefit from the clear way of thinking this course teaches. It provides the basic skills needed for any career that requires logic and proof.

Stop counting one by one. This course provides the mathematical tools to calculate arrangements and selections with speed. You will master the fundamental counting rules including product, addition, subtraction, and division. The curriculum covers linear and cyclic permutations, identical objects, and restricted placements where items must be together or apart. We conclude with combinations, derangements, and the calculation of shapes within geometric frameworks. Counting is the engine of probability and modern computing. These skills are vital for cryptography, software development, and statistical analysis. You will apply these methods to calculate network paths, secure passwords, and optimise logistics. In any technical career, the ability to quantify possibilities is the difference between guessing and knowing. You will solve complex arrangement problems using standard formulas and logical shortcuts. You will handle specific constraints like fixed positions or items that cannot be next to each other. You will learn to distinguish between permutations and combinations in practical scenarios. You will also calculate the number of lines, triangles, and diagonals in geometric figures using combinatorial principles. This foundation is for undergraduate students and secondary school leavers aiming for high scores in competitive exams. It is also essential for professionals in engineering, finance, and data science who require a refresher on discrete mathematics. Anyone seeking to improve their logical deduction and analytical speed will find value here.

Stop manually multiplying brackets. This course teaches the systematic expansion of binomial expressions using Pascal's triangle and the formal Binomial Theorem. We cover the fundamental proof for positive integers before moving to individual term identification and the general expansion for negative or fractional powers. You will master the mechanics of series expansion from simple squares to complex infinite series. The Binomial Theorem is a core tool in probability, statistics, and financial engineering. It allows for the approximation of complex functions and the calculation of compounding interest or risk factors. Engineers and data scientists use these principles to manage error margins and optimise algorithms. Understanding these patterns is essential for any career involving quantitative analysis or technical forecasting. You will expand binomials to any power and identify specific terms within a series without full expansion. You will demonstrate the theorem's proof and apply Pascal's triangle for rapid computation. You will also calculate expansions for non-integer indices and determine the range of validity for these series. These skills enable the simplification of advanced algebraic expressions. This curriculum is for undergraduate students and secondary school leavers entering science or technology tracks. It provides the necessary foundation for engineering, physics, and economics degrees. Professionals needing a mathematical refresher for data analysis will find the direct approach efficient. Any learner requiring precise algebraic tools for higher education will gain immediate value.

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.