Library

This course explores Einstein's special relativity, from its two postulates to mass-energy equivalence. You will learn to read Minkowski spacetime diagrams and to resolve the famous twin paradox.

Discover the foundations of mathematical reasoning: logical connectives, truth tables, quantifiers and set operations. You will also learn the main methods of proof, the essential groundwork before tackling proof by induction.

Get as close to the hardware as possible: from elementary logic gates to the adder, from a one-bit memory to the execution cycle of a simplified processor. Each step is illustrated by a diagram.

How can you draw reliable conclusions about an entire population from a mere sample? Estimators, confidence intervals, hypothesis tests and p-values, illustrated with curves.

The idea that changed science: every signal can be broken down into a sum of sinusoids. Fourier series and the Fourier transform, spectrum, sampling and Shannon's theorem, illustrated with diagrams.

What is "information," and how far can a file be compressed? From Shannon entropy to Huffman coding and LZ77, the art of saying the same thing with fewer bits.

How can a message arrive intact despite bits damaged along the way? From the simple parity bit to the Hamming code that corrects itself, by way of CRC and the idea behind QR codes.

How bowl-shaped ligands and palladium ions build themselves into nanometer-sized cages. A real case of supramolecular chemistry, illustrated with figures from the original paper.

Is work a quantum observable? By including the controller in the description, a well-defined work operator emerges. A course based on a recent research paper.

The most famous of all sequences: its tiny rule, the golden ratio hidden behind it, and why computing it reveals the whole of dynamic programming and fast exponentiation.

Solving several equations at once: what the solutions mean geometrically, how Gaussian elimination always finds them, and why a linear system never has anything other than zero, one, or infinitely many solutions.

Why polynomials, functions and matrices are "vectors": closure under linear combination, the subspace test, and the notions of independent family, basis and dimension.

Transformations that respect addition and scalar multiplication: why each one reduces to a matrix, what the kernel and image are, and how the rank-nullity theorem links what is crushed to what is reached.

The directions a matrix does not turn but only stretches: how to find eigenvalues and eigenvectors, when a matrix reduces to a diagonal one, and why this solves powers, recurrence sequences and dynamical systems.

A number whose square is -1: the algebraic form a + ib, computing with the conjugate and the modulus, the representation in the plane, and solving quadratics with a negative discriminant.

Locating a complex number by its distance and its angle: trigonometric form, the notation e^(iθ) that turns multiplication into rotation, de Moivre's and Euler's formulas, and the n-th roots that draw regular polygons.

Why every nonconstant complex polynomial has a root: factorization, conjugate roots, and the proof via the minimum of the modulus.

Modelling a random experiment before computing: sample space and events, probability distributions, the union formula and complementary events, the law of large numbers, and why the drawing protocol changes the answer.

Counting without enumerating: the multiplication principle, permutations, arrangements and combinations, Pascal's triangle and the binomial theorem, and the classic configurations — anagrams, grid paths, simultaneous draws.

Attaching a number to each outcome: the distribution of a random variable, expectation as a long-run average, variance and standard deviation as measures of risk, and their uses in insurance, finance and quality control.

Repeating a two-outcome trial and counting successes: the Bernoulli scheme, the formula P(X=k) = C(n,k)p^k(1-p)^(n-k), expectation np and variance np(1-p), and the threshold argument underlying statistical tests.

The bell curve and why it is everywhere: probability density, the parameters μ and σ, the 68-95-99.7 rule, the standardised variable, the central limit theorem, and the situations where the model fails.

Summarising a data series without betraying it: mean, median and mode, quartiles and box plots, range, interquartile range and standard deviation, and the reading traps — truncated axes, misleading means, Simpson's paradox.

The elementary bricks of the integers: the sieve of Eratosthenes and testing up to the square root, the fundamental theorem of arithmetic, Euclid's proof of infinitude, thinning out, and modern primality tests.