List of study guides

summary
Our study guides, written by students for students on a specific area of mathematics or statistics, can be found here.

Arithmetic on algebraic fractions
Algebra

Algebraic fractions extend numerical fractions to expressions that include variables. This guide explains how to add, subtract, multiply and divide algebraic fractions…

Donald Campbell

Arithmetic on complex numbers
Arithmetic
Algebra
Complex numbers

Doing arithmetic using complex numbers is a core concept in mathematics as it allows you to perform calculations in the complex plane. This guide covers addition…

Charlotte McCarthy

Arithmetic with fractions
Arithmetic

Arithmetic with fractions involves combining, comparing, and scaling parts of a whole through addition, subtraction, multiplication, and division. Mastering these operations…

Donald Campbell

Completing the square
Algebra
Solving equations
Key skills

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…

Tom Coleman

Conditional probability
Probability
Statistics

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
Probability
Statistics

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…

Tom Coleman

Factorization
Algebra
Solving equations
Key skills

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 algebraic fractions
Algebra

Algebraic fractions extend the idea of numerical fractions to expressions that include variables. This guide explains what algebraic fractions are and why certain values are…

Donald Campbell

Introduction to complex numbers
Key skills
Algebra
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…

Tom Coleman

Introduction to confidence intervals
Statistics

In statistics, to estimate an unknown parameter you can construct a confidence interval. This is the range of values you expect the true estimate to fall between if you were…

Millie Harris

Introduction to differentiation and the derivative
Calculus
Differentiation

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…

Tom Coleman

Introduction to fractions
Arithmetic

Fractions are a fundamental concept used to represent parts of a whole. They provide a precise way to describe values that are not whole numbers, helping you to understand…

Donald Campbell

Introduction to hypothesis testing
Statistics
Probability

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
Calculus

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…

Tom Coleman

Introduction to linear regression
Statistics

Linear regression is a statistical model which uses one or more variables to predict the behaviour of another. It is widely used in fields such as business, economics and…

Flora Green

Introduction to logarithms
Key skills
Algebra

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 matrices
Linear mathematics
Matrices

Matrices are rectangular arrays of mathematical objects with entries arranged in rows and columns. Matrices are a very useful concept within mathematics, you’ll see them…

Jessica Taberner

Introduction to partial differentiation
Calculus

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
Statistics
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
Algebra
Solving 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…

Tom Coleman

Introduction to radians
Geometry
Trigonometry

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…

Ifan Howells-Baines, Mark Toner

Introduction to rearranging equations
Key skills
Algebra

Being able to rearrange equations is a crucial skill in mathematics as it can help you to simplify problems, solve equations, and better understand the relationships between…

Shanelle Advani, Ciara Cormican

Introduction to sigma notation
Arithmetic
Algebra
Key skills

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…

Tom Coleman, Ifan Howells-Baines, Mark Toner

Introduction to solving simultaneous equations
Algebra
Solving 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
Geometry
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
Probability
Statistics

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
Key skills
Algebra

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

Matrix multiplication
Linear mathematics
Matrices

Matrices are rectangular arrays of mathematical objects with entries arranged in rows and columns. Matrices are a very useful concept within mathematics and statistics.…

Jessica Taberner

Matrix multiplication with special matrices
Linear mathematics
Matrices

Matrices are rectangular arrays of mathematical objects with entries arranged in rows and columns. Matrices are a very useful concept within mathematics and statistics.…

Tom Coleman

Multivariate chain rule
Calculus

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
Statistics

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
Algebra
Key skills

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
Solving equations
Algebra

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.

Krish Chaudhary, Tom Coleman, Ellie Gurini, Mark Toner

Solving exponential equations
Solving equations
Algebra

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

Ritwik Anand, Zheng Chen, Zoe Gemmell

The chain rule
Calculus
Differentiation

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

Tom Coleman

The product rule
Calculus
Differentiation

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…

Tom Coleman

The quotient rule
Calculus
Differentiation

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

Tom Coleman

The scalar product
Geometry
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)
Geometry
Trigonometry

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…

Shanelle Advani, Krish Chaudhary, Dzhemma Ruseva

Trigonometric identities (radians)
Geometry
Trigonometry

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…

Shanelle Advani, Krish Chaudhary, Dzhemma Ruseva

Trigonometry (degrees)
Geometry
Trigonometry

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

Ciara Cormican, Ellie Gurini, Dzhemma Ruseva

Trigonometry (radians)
Geometry
Trigonometry

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

Ciara Cormican, Ellie Gurini, Dzhemma Ruseva

Using the quadratic formula
Solving equations
Algebra

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…

Tom Coleman

Vector addition and scalar multiplication
Geometry
Vectors
Linear mathematics

Understanding and utilising vector addition and scalar multiplication are essential when studying vectors, particularly when it comes to solving equations involving vectors.

Renee Knapp, Kin Wang Pang
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