Pure Mathematics

The Further Pure Mathematics core extends A-Level Mathematics into the abstract and powerful territory required for university STEM study. These topics — complex numbers, matrices, advanced calculus, differential equations, and vector geometry — form the mathematical language of engineering, physics, and computer science.

Topics Covered

Complex Numbers

• Argand diagrams — geometric representation of $z = a + bi$
• Modulus-argument form — $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$
• De Moivre’s theorem — $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$; proof by induction
• Roots of unity — solving $z^n = 1$; geometric interpretation as regular polygons
• Euler’s formula — $e^{i\theta} = \cos\theta + i\sin\theta$; applications to trigonometric identities

Matrices

• Matrix arithmetic — addition, multiplication, properties (non-commutativity of multiplication)
• Determinants — $2\times 2$ and $3\times 3$; geometric interpretation as scaling factor
• Inverses — formula for $2\times 2$; row operations for $3\times 3$
• Linear transformations — rotation, reflection, enlargement, shears in 2D and 3D
• Eigenvalues and eigenvectors — characteristic equation $\det(A - \lambda I) = 0$; diagonalisation (AQA)

Further Algebra

• Roots of polynomial equations — relationships between roots and coefficients ($\alpha + \beta + \gamma$, $\alpha\beta + \alpha\gamma + \beta\gamma$, $\alpha\beta\gamma$)
• Substitution of roots — finding polynomials whose roots are transformations of the originals
• Partial fractions with irreducible quadratics — decomposition including $\frac{Ax + B}{x^2 + px + q}$ terms

Further Calculus

• Inverse trigonometric integration — $\int \frac{1}{\sqrt{1-x^2}}\,dx$, $\int \frac{1}{1+x^2}\,dx$
• Integration by parts (repeated) — reduction formulae
• Volumes of revolution — $V = \pi \int y^2\,dx$; parametric form
• Parametric differentiation — $\frac{dy}{dx} = \frac{\dot{y}}{\dot{x}}$

Polar Coordinates

• Conversion — $x = r\cos\theta$, $y = r\sin\theta$, $r^2 = x^2 + y^2$
• Sketching polar curves — cardioids, limacons, roses
• Area enclosed — $A = \frac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta$

Hyperbolic Functions

• Definitions — $\sinh x = \frac{e^x - e^{-x}}{2}$, $\cosh x = \frac{e^x + e^{-x}}{2}$, $\tanh x = \frac{\sinh x}{\cosh x}$
• Identities — $\cosh^2 x - \sinh^2 x = 1$; Osborne’s rule
• Calculus — derivatives and integrals of hyperbolic functions; inverse hyperbolic functions

Differential Equations

• First-order — separable variables, integrating factor method
• Second-order linear — constant coefficients; complementary function + particular integral
• Boundary conditions — determining arbitrary constants from initial/boundary values

Maclaurin and Taylor Series

• Maclaurin series — $f(x) = f(0) + xf'(0) + \frac{x^2}{2!}f''(0) + \cdots$
• Standard expansions — $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $(1+x)^n$
• Convergence — radius of convergence; when series are valid approximations

Vectors in 3D

• Scalar product — $\mathbf{a}\cdot\mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$; finding angles
• Vector product — $\mathbf{a}\times\mathbf{b}$; magnitude and direction; applications
• Lines and planes — vector equations, Cartesian equations, intersections, distances

Study Tips

1. Learn complex number forms fluently — convert between Cartesian, modulus-argument, and exponential form without hesitation.
2. Practise matrix operations — matrix multiplication and finding inverses are procedural skills that must be fast and accurate.
3. Derive, don’t memorise — know where the Maclaurin series comes from, why De Moivre’s theorem holds, and how reduction formulae work.
4. Sketch polar curves by plotting key points ($\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$) rather than trying to visualise the entire curve at once.
5. Practise differential equations systematically — identify the type (separable, linear, second-order), apply the correct method, then use boundary conditions.

How to Use These Notes

Follow the sidebar order. Each page provides rigorous definitions, proofs, worked examples, and exam-style problems. Complex numbers and matrices are foundational — master them before moving to calculus and differential equations.