It begins with counting and never really stops. Numbers and algebra is the study of quantity and structure — of the whole numbers and the hidden patterns inside them, and of the symbolic machinery that lets us reason about all numbers at once. Algebra is arithmetic with the specific numbers rubbed out and letters left in their place, and that one move unlocks astonishing power.

The areas to explore

Begin at the beginning with arithmetic — counting, place value, fractions and the four operations. Zoom in on the whole numbers themselves in number theory, the study of primes, divisibility and Diophantine puzzles. Learn to work with the unknown in algebra — equations, expressions and graphs. Then meet the two great generalisations: linear algebra, the maths of vectors, matrices and transformations that powers graphics and machine learning, and abstract algebra, which distils the very idea of structure into groups, rings and fields.