Library

Why does ice float? Why does boiling water remain at 100 °C? And what really happens at the molecular level when matter changes state?

The number that remained ‘indecipherable’ for three centuries, and the method that cracked it. A lesson on repetition, the enemy of secrecy.

How have people been hiding messages for 2,000 years? Caesar’s cipher, substitution ciphers, and why they all eventually fail. No prior knowledge required.

Proving that a file hasn’t been altered, storing a password without knowing it, signing a document: what encryption doesn’t do.

How can you agree on a secret without ever having met? The idea that made online shopping possible, and the way RSA works.

Congruences, modular inverses and fast exponentiation: the mathematical tools underpinning RSA and Diffie-Hellman.

Move from intuition to rigour with the epsilon-N definition, convergence theorems and techniques for calculating the limits of sequences.

Learn how to determine whether a sequence is increasing, decreasing or constant using rigorous methods illustrated with examples.

Discover geometric sequences, which grow by multiplication and can be used to model growth, decline and compound interest.

Discover arithmetic sequences: definition, explicit formula, direction of variation and calculation of sums, with examples of increasing complexity.

Find out how to model a list of numbers that changes step by step, and get to grips with arithmetic and geometric sequences.

Understand limited expansions using the Taylor–Young formula and use them to resolve indeterminacies and study the local behaviour of curves.

Learn how to solve first- and second-order differential equations, from separable variables to the resonance of forced systems.

Master integration by parts, from its proof to cyclic integrals and recurrence relations, by correctly choosing the functions to be differentiated and integrated.

Gradually explore integral calculus, from antiderivatives to calculating areas under curves, to master a key pillar of analysis.

Learn how to find a function from its derivative, using common antiderivatives, composite functions and the link to integration.

Learn how to compare logarithms, powers and exponential functions in terms of their behaviour as they approach infinity, in order to resolve indeterminate forms.

Discover the natural logarithm and master its properties, its derivative and its limits to solve equations and inequalities.

Discover the exponential function, which is equal to its own derivative, and learn how to work with its algebraic properties and its asymptotic behaviour.

Understand the continuity of a real-valued function and the Intermediate Value Theorem, including its applications to finding zeros and solving equations.

Covers limits of functions, indeterminate forms and notable limits, with an introduction to the rigorous definition and the link to continuity.

Learn how to use the derivative to analyse a function, determine its extrema, draw tangents and solve optimisation problems.

Use the derivative to determine when a function is increasing, decreasing or reaching an extremum, and then draw up a complete table of variations.

Discover the derivative, the key formulas for calculating it, and its applications to the study of tangents and variations.