List of study guides

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

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, subtraction, multiplication, and division on complex numbers.

Charlotte McCarthy

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 sketch graphs of conic sections, and is widely used in integral calculus.

Tom Coleman

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 idea of a complex number in the form \(z = a + bi\), as well as the concepts of real and imaginary parts, complex conjugates, and Argand diagrams.

Tom Coleman

Introduction to differentiation and the derivative

Calculus
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 of differentiation and the derivative is introduced from first principles, its role in explaining the behaviour of functions is explained, and derivatives of common functions are introduced and used.

Tom Coleman

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 comparing a null hypothesis, which represents the status quo, against an alternative hypothesis using sample data, a chosen significance level, and either critical values or \(p\)-values. This guide will focus on the general structure of a hypothesis test, critical values, how to choose which type of test to use, as well as when to reject, or not reject a hypothesis.

Ellie Trace

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 logarithm and learning how to change the base of a logarithm are key in mastering their functionality.

Ritwik Anand

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 discriminant \(D = b^2 - 4ac\), are essential steps in solving quadratic equations.

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 converting between radians and degrees is an important skill. This guide also explains how to find lengths of circular arcs and areas of circular sectors.

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 us to simplify problems and better understand the relationships between variables. This will guide introduce the concept and take you through how to undo common operations like addition, multiplication, and powers.

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 in almost any area of mathematics.

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 introduce you to finding if a pair of simultaneous equations in two variables has a solution or not, and then demonstrate two ways of solving a pair of simultaneous equations — the substitution method and the elimination method.

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

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 more efficiently than writing them out in full.

Zheng Chen, Zoe Gemmell

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 describe how probabilities are distributed across the possible outcomes of random events. PMFs, PDFs, and CDFs are commonly used to model probability distributions, helping to visualize and understand the behaviour of random processes. This guide will explore the role of each function, how they differ, and highlight their applications.

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 denominators through the use of the quadratic conjugate.

Maximilian Volmar

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 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 yourself with the standard trigonometric identities as they come in handy later whether that be for deriving further trigonometric identities or solving integrals involving trigonometric functions.

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 yourself with the standard trigonometric identities as they come in handy later whether that be for deriving further trigonometric identities or solving integrals involving trigonometric functions.

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. This guide uses degrees to measure angles.

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. This guide uses radians to measure angles.

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 explains the terminology and usage of the quadratic formula, as well as an explanation of why the quadratic formula works.

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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