# Pure Maths – fundamental principles and core tools

We can think of **Pure Maths** as the fundamental principles and reasoning at the heart of any mathematics we do. It is the collection of underlying core tools required to explain and develop mathematical knowledge. This is as opposed to Applied Maths where we take knowledge of the pure mathematical subjects and apply it in real-life situations. For example, the study of functions, their features and their graphs, such as quadratics, is Pure Mathematics. However, using quadratics to model the trajectory of a projectile is Applied Mathematics.

Pure Maths is thousands of years old. It was the ancient Greeks who first began to study maths as an organised science. Pythagoras, Euclid and Archimedes are just a few of the earliest known mathematicians – see the Mathematics Timeline.

In the AS Maths and A-Level Maths qualifications, Pure Maths makes up two-thirds of the content and hence the examinations. Click here to see more information on AS Maths exams or here for more A-Level Maths exam information. Also, see below for the topics studied in Pure Maths in AS and A-Level Mathematics.

## AS Pure Mathematics

**Proof:**

**Algebra & Functions:**

- completing the square
- cubics
- curve sketching
- discriminant
- indices
- inequalities
- polynomials
- quadratics
- simultaneous equations
- surds
- transformations

**Coordinate Geometry:**

**Sequences & Series:**

**Trigonometry:**

- non-right angled triangles: the sine and cosine rule, area of a triangle
- trigonometric equations
- trigonometric graphs
- trigonometric identities

**Exponentials & Logarithms:**

- exponential & logarithmic graphs
- logs, their rules and solving log equations
- growth & decay
- differentiating e to the kx

**Differentiation:**

- differentiation by first principles and differentiating polynomials
- increasing & decreasing functions
- stationary points
- tangents & normals
- differentiating e to the kx

**Integration:**

**Numerical Methods:**

- (not covered at AS Level)

**Vectors:**

## A2 Pure Mathematics

Proof:

- the modulus of a function
- partial fractions
- inverse and composite functions
- compound transformations

- radians, arc length & area of a sector
- small angle approximations,
- reciprocal trigonometric functions
- inverse trigonometric functions
- double & compound angle formulae

- concavity, convexity & inflection points
- derivatives of trigonometric functions
- product, quotient & chain rule
- parametric & implicit differentiation
- differentiating exp/log functions
- differential equations & rates

- further integration (exp/log/ trig functions etc)
- integrating with trigonometric identities
- integration by substitution
- integration by parts
- integrate using partial fractions
- solving differential equations

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