Algebra: Basic to Advanced

The course the rest of the series rests on

Algebra is the load-bearing subject. Number theory uses its manipulation on every line, combinatorics borrows its generating functions, probability depends on its series work, and the linear algebra at the end is the prerequisite for anything quantitative that follows.

This course runs from factorisation and quadratics through polynomials, functional equations and inequalities to groups, rings, fields and linear algebra.

Fluency is not the same as knowledge

Parents often ask whether a child who is good at school algebra needs the basic stage. The answer usually turns on speed rather than knowledge.

School algebra tests whether a taught method has been learned. Competition and university algebra assume manipulation is automatic and put the difficulty somewhere else entirely. A student who is reliably correct but deliberate finds the intermediate units heavy going — not because the ideas are beyond them, but because every line of working costs attention they need for the actual problem.

For such students the basic stage is short. It is rarely wasted.

Functional equations are learnable

These have a reputation for requiring inspiration, and that reputation is mostly wrong.

What they require is a repertoire: substitute particular values, test whether the function must be injective or surjective, look for fixed points, exploit symmetry, guess a form and verify it. Cauchy's functional equation and its pathological solutions are worth knowing for their own sake. Once a student has these moves and has used each a dozen times, most competition functional equations reduce to trying them in a sensible order.

The students who find these impossible are almost always the ones who have never been shown that a method exists.

Inequalities, with an order of attack

Inequalities recur constantly and students approach them without a plan, trying whatever they remember until something works.

We teach AM-GM, Cauchy-Schwarz, rearrangement, Chebyshev's sum inequality, power mean and Jensen as a toolkit with a stated order: what to try first, how to recognise the shape of a problem that suits Cauchy-Schwarz rather than AM-GM, when the real content is convexity. That ordering is most of the skill, and it is teachable in a way that "be clever" is not.

Polynomials, properly

The remainder and factor theorems, roots and multiplicity, Vieta's formulas, symmetric functions of roots, the rational root theorem, complex roots in conjugate pairs.

Vieta in particular is undervalued in school teaching: a great many problems that look as though they need the roots explicitly only need symmetric functions of them, and a student who sees that saves themselves an afternoon.

Why abstraction arrives last

Groups, rings and fields are placed at the end deliberately. They are not needed for most competitions, and introducing them early makes them look like arbitrary rules.

Taken after a student has worked modular arithmetic, roots of unity and polynomial structure, they land differently: the abstraction explains things the student has already noticed. Lagrange's theorem makes sense of orders in number theory; field structure explains which equations are solvable. Students heading for an undergraduate mathematics degree should do this stage; others can stop before it without loss.

The bridge into linear algebra

The final unit — vector spaces, span, independence, basis and dimension, linear maps and matrices, determinants, eigenvalues and the rank-nullity theorem — is the single most useful thing in the course for anyone continuing into a quantitative degree.

It is also the prerequisite for the Advanced Maths for Research course, where eigen theory and convex optimisation start immediately and assume all of it.

Who this suits

Students from Class 9 upward through undergraduate study. The course rebuilds the foundations before extending them, so a student who is shaky can start here; a strong student moves through the basic stage quickly.

Classes are live and online in groups of at most eight, with written work corrected individually.