List of study guides

Completing the square is an important technique to learn for dealing with quadratic expressions in a variety of places. It can help you solve quadratic equations, accurately…

Conditional probability

Conditional probability describes how the likelihood of an event changes when new information is introduced. It is a key concept for understanding relationships between…

Sophie Chowgule

Expected value, variance, standard deviation

This guide introduces expected values, variance, and standard deviation. These are key ideas in the concept of random variables, and they give information about the…

Factorization

Factorization is an important concept in mathematics, especially in algebra. It involves breaking down expressions into smaller parts called factors which, when multiplied…

Millie Pike

Introduction to complex numbers

Complex numbers

Complex numbers are an extremely important concept in mathematics, as you can express a solution to any algebraic equation using complex numbers. This guide introduces the…

Introduction to differentiation and the derivative

The idea of differentiation is everywhere in modern mathematics and in the sciences as it is related to the rate of change of an object or process. In this guide, the idea…

Introduction to hypothesis testing

A hypothesis test is a statistical process used to determine whether there is enough evidence in a sample to support or reject a claim about a population. It involves…

Ellie Trace

Introduction to integration

As the reverse process of differentiation, integration is a key area of mathematics. It has uses in almost all places that requires calculus, such as in the sciences, social…

Introduction to logarithms

Using logarithms to help understand large numbers is a fundamental skill when working in mathematics. Understanding the laws of logarithms, working with the natural…

Ritwik Anand

Introduction to partial differentiation

Partial differentiation is a key technique used in calculus for dealing with functions of multiple variables. It focuses on how a function changes with respect to one…

Donald Campbell

Introduction to probability

Probability theory is an important branch of mathematics, which is foundational to statistics and is used in quantum mechanics. This guide on an introduction to probability…

Michelle Arnetta

Introduction to quadratic equations

Solving quadratic equations of the form \(ax^2 + bx + c\) is a core skill in mathematics. Identifying variables and coefficients of quadratic equations, as well as finding the…

Introduction to radians

Ifan Howells-Baines, Mark Toner

Radians are an important concept to understand in mathematics. They are used instead of degrees in areas like calculus and trigonometry, and so becoming comfortable…

Introduction to rearranging equations

Shanelle Advani, Ciara Cormican

Being able to rearrange equations is a crucial skill in mathematics as it can help us to simplify problems and better understand the relationships between variables. This…

Introduction to sigma notation

Arithmetic

Tom Coleman, Ifan Howells-Baines, Mark Toner

Sigma notation is used to express many additions at once. Understanding what this notation is, how it works, and how to manipulate them is a valuable skill to learn for use…

Introduction to solving simultaneous equations

Solving simultaneous equations is a crucial skill in algebra. Understanding this concept allows you to find the common solution to a set of two equations. This guide will…

Ollie Brooke

Introduction to vectors

Vectors

Linear mathematics

Quantities expressing both a magnitude and a direction are called vectors. Vectors are used in various disciplines and have a wide range of real-life applications.

Akshat Srivastava

Law of total probability and Bayes’ theorem

The law of total probability and Bayes’ theorem build on the fundamentals of conditional probability. They are used to calculate probabilities across different scenarios and…

Sophie Chowgule

Laws of indices

In mathematics, a knowledge of indices is important for an understanding of algebraic processes. The laws of indices enable expressions involving powers to be manipulated…

Zheng Chen, Zoe Gemmell

Multivariate chain rule

The multivariate chain rule is used in calculus to differentiate a function when its variables depend on other variables. It shows how the change in one variable affects the…

Donald Campbell

PMFs, PDFs, and CDFs

Probability mass functions (PMFs), probability density functions (PDFs), and cumulative distribution functions (CDFs) are fundamental concepts in statistics. These functions…

Sophie Chowgule

Rationalizing the denominator

Rationalizing the denominator is a technique for simplifying fractions involving square roots in the denominator. This study guide also explains how to rationalize…

Maximilian Volmar

Solving equations involving logarithms

Krish Chaudhary, Tom Coleman, Ellie Gurini, Mark Toner

This guide discusses the idea of rearranging and solving equations involving logarithms, which is a key skill in both mathematics and many areas of science.

Solving exponential equations

Ritwik Anand, Zheng Chen, Zoe Gemmell

This guide applies the laws of indices to solve equations involving powers; a key skill in mathematics and many areas of science.

The chain rule

The chain rule is one of three central techniques of differentiation, allowing you to differentiate any composition of two differentiable functions. This guide introduces…

The product rule

The product rule is one of three central techniques of differentiation, allowing you to differentiate any product of two differentiable functions. This guide introduces the…

The quotient rule

The quotient rule is one of three central techniques of differentiation, allowing you to differentiate any quotient of two differentiable functions. This guide introduces…

The scalar product

Vectors

Linear mathematics

The scalar product is an important concept in the theory of vectors, with many geometric and algebraic applications.

Isabella Lewis

Trigonometric identities (degrees)

Shanelle Advani, Krish Chaudhary, Dzhemma Ruseva

Trigonometric identities are equations expressed in terms of trigonometric functions that hold true for all values of the variable defined. It is important to familiarize…

Trigonometric identities (radians)

Shanelle Advani, Krish Chaudhary, Dzhemma Ruseva

Trigonometric identities are equations expressed in terms of trigonometric functions that hold true for all values of the variable defined. It is important to familiarize…

Trigonometry (degrees)

Ciara Cormican, Ellie Gurini, Dzhemma Ruseva

Trigonometric functions are key concepts which will appear frequently throughout your mathematical studies. They have uses in geometry, calculus, mechanics, and much more.…

Trigonometry (radians)

Ciara Cormican, Ellie Gurini, Dzhemma Ruseva

Trigonometric functions are key concepts which will appear frequently throughout your mathematical studies. They have uses in geometry, calculus, mechanics, and much more.…

Using the quadratic formula

Solving quadratic equations of the form \(ax^2 + bx + c\) is a core skill in mathematics. A guaranteed method to solve these is use of the quadratic formula. This guide…

Vector addition and scalar multiplication