From counting to university

Mathematics

The Queen of the Sciences

Sixty-one chapters on nine lines, from notches on a bone to Gödel’s theorems. The formulas here can be touched, and the interchanges show where an idea from one part works in another.

Start at the beginning What’s inside Algorithm catalogue

Mathematics is the queen of the sciences, and arithmetic the queen of mathematics.

Gauss, as remembered by Sartorius von Waltershausen

Seven big questions

The course starts with questions school rarely answers. Each of them gets its answer here.

Why does minus times minus make plus?

Everyone memorises the rule of signs at school. But why should a debt times a debt turn into a profit — is that a convention, or could it not be otherwise?
Answered in chapter 2 · Zero and minus
Can you cut a disc into pieces and reassemble them into a square?

Same area, different shapes. Scissors won’t do it, and neither will a compass and straightedge. Then in 1990 it turned out you can — if the pieces are ones no artist could ever draw.
Answered in chapter 54 · Measure and integral
Why is there no formula for the roots of a quintic?

The Babylonians could solve quadratics; cubics and quartics fell in the sixteenth century. Then nearly three hundred years of nothing — and not for lack of trying.
Answered in chapter 43 · Galois theory
Are there more whole numbers or points on a line segment?

Both are infinite, so it seems there is nothing to compare. But infinities come in different sizes, and the proof takes a few lines.
Answered in chapter 52 · Infinities
How does the cipher that protects your bank work?

You and your bank agree on a secret key in full view of the whole internet, and an eavesdropper still learns nothing. How is that even possible?
Answered in chapter 41 · Remainders and ciphers
Why can’t the weather be forecast a month ahead?

The equations of moving air are known and computers are fast. Yet beyond a couple of weeks forecasts give up — and it isn’t the forecasters’ fault.
Answered in chapter 59 · Chaos and fractals
Can a statement be true but unprovable?

It seems that in mathematics everything true gets proved sooner or later. In 1931 Kurt Gödel showed otherwise — and no new axioms can fix it.
Answered in chapter 56 · Gödel and Turing
The answers are hidden in the chapters. The first comes as early as chapter 2, where a sceptic from the 1600s argues with the rule of signs.
Start at the beginning

Chapters by part

The course reads in order, but you can open any chapter directly: each one says up front what it builds on.

Introduction

Why mathematicians prove things, and how the course works

Chapter 0 · Ages 11–15

0 What mathematics is A rope round the Earth, a sunflower, a dragon made of a paper strip and an ant that suddenly starts building a highway. Five short stories about how mathematics surprises, why it needs proofs and which seven questions lie ahead. Ages 11–15 30 min

Numbers

From notches on a bone to the real numbers

Chapters 1–6 · Ages 11–15

1 Counting: from notches to bits How do you write a number too big for your fingers or for notches on a stick? A walk through the halls of a museum of counting, from a notched bone to binary code, to see how people learned to pack any number into a few signs. Ages 11–15 40 min

2 Zero and minus What is 3 − 5? A seventeenth-century sceptic is sure it’s nothing, and he calls “minus times minus” nonsense. Let’s argue with him by the rules: every sign rule will have to be defended. Ages 11–15 40 min

3 Prime numbers Some numbers can’t be broken into factors. Everything else is built from them by multiplication, there are infinitely many of them, and nobody can say in advance where the next one will turn up. Ages 11–15 40 min

4 Divisibility and Euclid's algorithm Puzzles about a calendar, a long number, tiles, jugs and coins, and one answer to all of them: the greatest common divisor, and a way to find it that is more than two thousand years old. Ages 11–15 40 min

5 Fractions How to share seven pies fairly among three people, why you flip the fraction when you divide, and where the endless merry-go-round of the digits 142857 in one seventh comes from. Ages 11–15 40 min

6 Root two and the real numbers The case of the square’s diagonal: three independent expert reports prove that its length can’t be written as a fraction. We look at how the Babylonians lived with it, what continued fractions are, and where the real numbers come from. Ages 11–15 45 min

Algebra

Letters, equations, functions — and numbers that “don’t exist”

Chapters 7–15 · Ages 11–15 → Ages 16–18

7 Letters for numbers A magician guesses the number you thought of and does mental arithmetic faster than a calculator. We’ll expose his tricks one by one, and along the way learn to write down a rule that holds for all numbers at once. Ages 11–15 40 min

8 Equations: the art of balance An equation is a pair of scales in balance with a box of unknown weight on it. We’ll learn to take things off the pans without making the scales tip, and find out where the word “algebra” itself comes from. Ages 11–15 40 min

9 Functions and graphs Sometimes the answer to a question isn’t a number but a rule. Let’s play with machines that turn some numbers into others: learn to crack them from a few throws, see them whole in a single picture, tune them, chain them into a production line and run them in reverse. Ages 11–15 40 min

10 Quadratic equations The problem the last chapter ended with is nearly four thousand years old. Let’s excavate it layer by layer, from a clay tablet to the quadratic formula, and see what the formula really says. Ages 11–15 40 min

11 Systems of equations and inequalities When there are several conditions, one equation isn’t enough. Let’s open a small toy workshop: find prices from two invoices, work an ancient Chinese counting board, run into the limits of the storeroom and find out why the most profitable plan always sits in a corner. Ages 11–15 45 min

12 Powers and logarithms A quantity that is multiplied at every step sooner or later outgrows any quantity that is added to, and usually sooner than you’d think. Let’s hold a race of growth rates, then find a tool that turns multiplication into addition and tames astronomical numbers. Ages 11–15 40 min

13 Sequences and induction How to add a hundred numbers in a minute, why an infinite sum can be finite, and how to prove a statement for every n at once: by standing dominoes in a row and pushing the first one. Ages 11–15 45 min

14 Polynomials In 1535, in Venice, two mathematicians exchanged thirty problems, and every problem set by one of them was a cubic equation. Let’s follow that duel to the end, and along the way gather the weapons used ever since to solve equations of high degree. Ages 16–18 50 min

15 Complex numbers For almost three hundred years a number whose square is −1 was thought impossible. Let’s follow how it turned from a trick into a symbol, from a symbol into a point, and from a point into a rotation, and what happened to equations after that. Ages 16–18 55 min

Geometry

From Euclid’s axioms to the orbits of planets

Chapters 16–24 · Ages 11–15 → Ages 16–18

16 The Elements: axioms and constructions Around 300 BC a book was written in Alexandria that teaches you to tell ‘looks right in the drawing’ from ‘always true’. Its rules are compass and straightedge, and you can play by them. Ages 11–15 45 min

17 The triangle A laboratory with a single instrument: a movable triangle. We drag the vertices, notice what doesn’t change, and prove why, from the congruence tests to the Euler line. Ages 11–15 60 min

18 Pythagoras' theorem The most famous theorem in geometry and its gallery: five halls, each with its own proof: Chinese, Euclid’s, by similarity, a president’s, and a puzzle without formulas. On the way out: whole-number triples, distances and Fermat’s riddle. Ages 11–15 50 min

19 The circle and the number π A four-thousand-year chase after one number, from a rope round a barrel to supercomputers. Along the way, the theorems of the circle: the angle in a semicircle, the inscribed angle, the tangent, crossing chords and the area of a disc. And to finish, the trap of the chords: 1, 2, 4, 8, 16, and then 31. Ages 11–15 70 min

20 Area and volume A workshop with scissors: we cut figures, rearrange the pieces and find out what area is. Any polygon can be re-cut into any other of the same area, but a disc and a cube won’t give in to scissors. Ages 11–15 55 min

21 Trigonometry A surveying expedition: from a pole’s shadow to the width of a river, the height of a mountain and the size of the whole Earth. Along the way angles turn into numbers (sine, cosine, tangent) and the circle unwinds into a wave. Ages 16–18 60 min

22 Vectors and coordinates The engine pulls the boat one way, the current another, the wind blows a third. The navigator adds all this up as arrows, and calculates with arrows as confidently as with numbers. Ages 16–18 45 min

23 Symmetry and transformations An ornament workshop: we draw a motif, and mirrors, rotations and shifts turn it into a rosette, a frieze and a wallpaper. Along the way it turns out that there are only four ways to move the plane without distortion, and seventeen ways to repeat a pattern over the whole plane. Ages 16–18 55 min

24 Conic sections Cut a cone with a plane and you get the curves along which planets, comets and interstellar visitors fly. One night at the observatory: from Apollonius’ drawings to satellite dishes and the Earth’s orbit. Ages 16–18 50 min

Calculus

The infinitely small: speed, area, growth and waves

Chapters 25–34 · Ages 16–18 → Years 1–2

25 Limits Speed at an instant looks like zero divided by zero. For a century and a half mathematicians used it without being able to explain what it was. The explanation turned out to be a game against a stubborn opponent: let’s learn its rules. Ages 16–18 50 min

26 The derivative What does a speedometer really show? Let’s get into a car with a dashcam, turn average speed into instantaneous speed, and learn to find the slope of any smooth graph at any of its points. Ages 16–18 50 min

27 Extrema and optimization An inventors’ contest: a box from a sheet, a speed camera on a highway, a lifeguard on a beach, a tin can and the square root of two on a calculator with no square-root key. Each task has infinitely many options, and the derivative finds the best. Ages 16–18 50 min

28 The integral The biography of one idea: cut a figure into thin strips and add them up. It was born in Syracuse, grew up in the age of indivisibles, met the tangent in the hands of Newton and Leibniz, and got a rigorous definition from Riemann. Ages 16–18 55 min

29 The number e and the exponential The same number, 2.71828…, has been spotted in a bank, on a graph, under a hyperbola, in radioactive decay, on a circle, in a cloakroom and at a bride’s choosing of a husband. Let’s log every sighting, like a naturalist in a field diary, and find out whether it’s one number or several look-alikes. Ages 16–18 55 min

30 Series An infinite sum can be added up, but it has to be handled with care: one converges, another runs off to infinity, a third changes its answer if you rearrange the terms. Let’s walk through a cabinet of curiosities of infinite sums, from a stack of books over an abyss to a polynomial that passes itself off as the sine. Years 1–2 55 min

31 Differential equations The laws of nature rarely say how much of something there will be; they say how fast it changes. Welcome to a lab where processes are grown from such laws: cooling coffee, a spring, wolves and hares, an epidemic. And we’ll find out why a uniquely determined future isn’t the same as a predictable one. Years 1–2 55 min

32 Functions of several variables The height on a mountainside depends on two coordinates at once. Let’s take a map, a magnifier and a compass and find out which way such a function grows fastest, where its peaks and passes are, and why the area under the Gaussian bell is the square root of π. Years 1–2 60 min

33 Fourier series The sound of a violin, the temperature in a ring and an outline drawn with a finger can all be taken apart into sine waves and put back together. We’ll find out how to get a curve’s harmonics, why a step always keeps a ringing, and how the same trick compresses photos and music. Years 1–2 55 min

34 Complex analysis A function of a complex variable can’t be drawn as a graph, but it can be hung on the wall as a painting. Let’s walk round an exhibition of such paintings and find out why the series for 1/(1 + x²) converges only for |x| < 1, and why every polynomial has a root. Years 1–2 75 min

Linear algebra

Matrices as transformations of space and data

Chapters 35–39 · Years 1–2

35 Matrices as transformations Four numbers in two columns can rotate the plane, stretch it, skew it, reflect it in a mirror or squash it flat onto a line. Let’s learn to read a matrix as the blueprint of a deformation: chain deformations together, undo them, and tell from a single number how much area changes. Years 1–2 50 min

36 Gaussian elimination How to find a lost planet from a few observations: turn a system of many equations into a staircase and read the answer from the bottom up. The steps show at once whether the system has one solution, infinitely many or none. Years 1–2 55 min

37 Vector spaces Arrows, polynomials, sound signals and even magic squares are added and multiplied by numbers by the same rules. Eight axioms, and everything we know about arrows starts working in all these worlds at once. Years 1–2 60 min

38 Eigenvectors A matrix usually turns vectors, but many matrices have directions they only stretch. Let’s find them, and it will turn out that the Fibonacci numbers, the Google search engine and a tilted ellipse all work the same way. Years 1–2 50 min

39 Orthogonality, least squares and SVD A data lab: the measurements won’t lie on a line, a cloud of points is stretched on a slant, a photograph takes up too much space. One idea copes with all three problems: drop a perpendicular. Years 1–2 60 min

Structures

Groups, rings and fields — and the ciphers the internet runs on

Chapters 40–44 · Years 1–2 → Year 3 and up

40 Groups A cardboard square, a deck of cards, the fifteen puzzle and the Rubik’s cube: four puzzles behind which stands a single structure of four rules. Let’s assemble it and find out why 14 and 15 can’t be swapped in the fifteen puzzle, and how many times R U has to be repeated for the cube to come back solved. Years 1–2 55 min

41 Modular arithmetic and ciphers How do you agree on a secret when everyone can hear your conversation? A cipher-maker’s workshop: Caesar’s wheel, key exchange, RSA, and the arithmetic of remainders on which each of these locks rests. Years 1–2 55 min

42 Rings, fields and codes How do you correct an error in a message when you can’t ask for it again? An engineer’s logbook: from triple repetition and Hamming’s three circles to Galois fields and the Reed–Solomon codes hidden in every QR code. Years 1–2 55 min

43 Galois theory Why there’s no formula for the quintic, why an angle can’t be trisected with compass and straightedge, and why a regular seventeen-sided polygon can be constructed. All three questions are answered by one idea: the symmetries of the roots. Its author, Évariste Galois, died at twenty. Year 3 and up 75 min

44 The zeta function and elliptic curves How accurate is Gauss’s formula for the number of primes? The answer is hidden in the zeros of one function of a complex variable, and where all those zeros lie nobody knows. Next door are cubic curves whose points add up like numbers: they proved Fermat’s Last Theorem and they guard your bank. Year 3 and up 70 min

Chance & data

Counting, probability, statistics and information

Chapters 45–50 · Ages 11–15 → Ages 16–18 → Years 1–2

45 Combinatorics A board-game evening: dice, cards, counters, hats and guests. We’ll learn to count possibilities without listing them, and see that in a large enough crowd some order appears all by itself. Ages 11–15 45 min

46 Graphs A walk round town: bridges, crossroads, roads and maps. Euler reduced a city to nothing but points and the lines between them, and solved the bridge problem. The same trick is used today to plan routes, draw up timetables and colour maps. Ages 16–18 45 min

47 Probability Intuition on trial: seven cases of faulty reasoning about chances, at the gaming table, in a TV studio, in a doctor’s surgery and in a courtroom. We’ll learn to compute probabilities and to recognise the traps that caught gamblers, professors and juries. Ages 16–18 50 min

48 Random variables A laboratory of chance: we throw dice and needles, drop balls onto pegs, count kicks from horses. Each law is first observed in an experiment and then proved, and we see why the average of many random numbers is hardly random at all. Years 1–2 60 min

49 Statistics Five newspaper headlines, five investigations. An average salary, a poll with a three per cent margin of error, a ‘proven’ drug, chocolate and Nobel prizes, university admissions: we learn to judge the whole from a sample without fooling ourselves. Years 1–2 55 min

50 Markov chains and information The letters of a text depend on the ones before them, the weather on yesterday’s. We’ll count this dependence as Markov did in Eugene Onegin, and measure how much information a letter carries, as Shannon did. Years 1–2 50 min

Foundations

Logic, infinities and the limits of proof

Chapters 51–56 · Years 1–2 → Year 3 and up

51 Logic and sets For fifty chapters we’ve said ‘for every’, ‘there exists’, ‘therefore’ and ‘set’ without looking at the grammar. We’ll learn it on an island where some inhabitants always tell the truth and others always lie. Years 1–2 55 min

52 Infinities Galileo’s three characters argue about which is larger: the natural numbers or the points of a segment. On the way they meet a hotel that always has a free room, Cantor’s diagonal and a tower of infinities with no top floor. Years 1–2 55 min

53 Rigorous analysis A continuous function without a single tangent, a function that can’t be drawn, a limit of continuous functions with a jump. A bestiary of the monsters that made analysis rigorous, and the tools that tamed them. Years 1–2 75 min

54 Measure and the Lebesgue integral How much do all the fractions in a segment weigh? The answer ‘nothing’ forced length to be defined afresh, and led to an integral that can handle the Dirichlet function, to sets that can’t be measured, to a ball from which two are assembled, and to a circle cut into a square. Year 3 and up 75 min

55 Functional analysis A function is a point of a space with infinitely many dimensions. It has lengths, angles and projections, Pythagoras’ theorem with infinitely many legs gives π²/6 once again, and the fixed point of a contraction draws a fern. But some familiar things break at this border. Year 3 and up 60 min

56 Gödel, Turing and the limits of proof A puzzle with four rules, a number that talks about itself, and a program that can’t be written. In 1931 Gödel proved that arithmetic has true but unprovable statements, and in 1936 Turing proved that some questions can’t be settled by any algorithm. Years 1–2 65 min

Horizons

Topology, non-Euclidean worlds, chaos and open problems

Chapters 57–60 · Years 1–2 → Year 3 and up

57 Topology Geometry in which shapes can be kneaded. Here a mug can’t be told from a doughnut, a strip of paper can have one side, a knot provably can’t be untied, and at any moment there’s a place on Earth where no wind blows. Years 1–2 55 min

58 Non-Euclidean geometries A travel diary through three worlds. In the plane the angles of a triangle add up to exactly 180°, on a sphere to more, and in Lobachevsky’s world to less, and there infinitely many parallels pass through a point. Years 1–2 60 min

59 Chaos and fractals A one-line formula, Lorenz’s three equations, a game of dice, and behaviour that no computer can predict. A lab notebook of nine experiments: why the weather can’t be forecast a month ahead, and where a coastline gets a fractional dimension. Years 1–2 60 min

60 The frontier The course ends where the answers end. A map of the unknown: the Millennium Problems, the Collatz conjecture, P versus NP; how mathematics is done today, together, with computers, Lean and AI; and a return to the seven questions we started with. Year 3 and up 55 min

Problem solvers

Type in your own problem — a solver works it out step by step and explains why each step is allowed.

Also in the course

Practice Endless problems: fresh ones every time, any correct form of the answer accepted, worked solutions on request. Topics Chapters by topic, level and part — say, everything about probability, or everything useful for exams. Glossary Every term in the course with a short definition and a link to the chapter that introduces it.

How the course works

A wall, then a ladder
A chapter opens with a problem nothing so far can solve, and builds a tool for it. It ends at a new wall — where the next chapter begins.
Live diagrams
The widgets answer “what happens if…”. Drag, change, break things — nothing can be spoiled.
Formulas, piece by piece
Every important formula is taken apart: where each piece comes from, an example with numbers, and where the formula stops working.
Practice and solvers
Practice sets hand out problems for as long as you like. Solvers take your own problem and solve it step by step, with reasons.

Four levels

Ages 11–15 School Arithmetic and curiosity are enough
Ages 16–18 High school Builds on school algebra and geometry
Years 1–2 University Like the first years at MIT, Cambridge or Moscow State
Year 3 and up Beyond For readers who enjoy abstraction