Mathematics
0 · What mathematics is 0 Start 1 · Counting: from notches to bits 1 Counting 2 · Zero and minus 2 Zero and minus 3 · Prime numbers 3 Primes 4 · Divisibility and Euclid's algorithm 4 Divisibility 5 · Fractions 5 Fractions 6 · Root two and the real numbers 6 Root two 7 · Letters for numbers 7 Letters 8 · Equations: the art of balance 8 Equations 9 · Functions and graphs 9 Functions 10 · Quadratic equations 10 Quadratics 11 · Systems and inequalities 11 Systems 12 · Powers and logarithms 12 Powers and logs 13 · Sequences and induction 13 Sequences 14 · Polynomials 14 Polynomials 15 · Complex numbers 15 Complex numbers 16 · Elements: axioms and constructions 16 Euclid 17 · The triangle 17 Triangle 18 · Pythagoras' theorem 18 Pythagoras 19 · The circle and π 19 Circle and π 20 · Area and volume 20 Area and volume 21 · Trigonometry 21 Trigonometry 22 · Vectors and coordinates 22 Vectors 23 · Symmetry and transformations 23 Symmetry 24 · Conic sections 24 Conics 25 · Limits 25 Limits 26 · The derivative 26 Derivative 27 · Extrema and optimization 27 Extrema 28 · The integral 28 Integral 29 · e and the exponential 29 The number e 30 · Infinite series 30 Series 31 · Differential equations 31 Diff. equations 32 · Several variables 32 Several variables 33 · Fourier series 33 Fourier 34 · Complex analysis 34 Complex analysis 35 · Matrices as transformations 35 Matrices 36 · Gaussian elimination 36 Elimination 37 · Vector spaces 37 Vector spaces 38 · Eigenvectors 38 Eigenvectors 39 · Orthogonality, least squares and SVD 39 Least squares, SVD 40 · Groups 40 Groups 41 · Modular arithmetic and ciphers 41 Modular and ciphers 42 · Rings, fields and codes 42 Rings and fields 43 · Galois theory 43 Galois theory 44 · Zeta and elliptic curves 44 Zeta and curves 45 · Combinatorics 45 Combinatorics 46 · Graphs 46 Graphs 47 · Probability 47 Probability 48 · Random variables 48 Random variables 49 · Statistics 49 Statistics 50 · Markov chains and information 50 Markov chains 51 · Logic and sets 51 Logic and sets 52 · Infinities 52 Infinities 53 · Real analysis 53 Real analysis 54 · Measure and the Lebesgue integral 54 Measure 55 · Functional analysis 55 Functional an. 56 · Gödel, Turing and the limits of proof 56 Gödel and Turing 57 · Topology 57 Topology 58 · Non-Euclidean geometry 58 Non-Euclidean 59 · Chaos and fractals 59 Chaos and fractals 60 · The frontier 60 The frontier I II III IV V VI VII IX VIII
\+ − ⤢
Drag to move the map
Open the full map
Drag the map with a finger; pinch or use the buttons to zoom
Mathematics is the queen of the sciences, and arithmetic the queen of mathematics.
Gauss, as remembered by Sartorius von Waltershausen
1. 1
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 → 2. 2Can 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 → 3. 3Why 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 → 4. 4Are 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 → 5. 5How 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 → 6. 6Why 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 → 7. 7Can 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 → 8. 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 →
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 
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 
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 
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 
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 
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 
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 
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 - Quadratic equation
- Linear equation
- System of linear equations
- Higher-degree equations and factoring
- Inequalities by the interval method
- Derivative
- Integral
- Limit
- Curve sketching
- Fraction arithmetic
- GCD, LCM and Euclid's algorithm
- Prime factorisation
- Solving a triangle
- Matrices: determinant, inverse, rank, eigenvalues
- Trigonometric equation
- Exponential and logarithmic equations
- Percentages, compound interest, loans
- Permutations and combinations
- Number bases
- Complex numbers
- Congruences
- Progressions
- Probability: Bayes and Bernoulli trials
- Differential equation
- 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
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.
1. 1
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. 2. 2Live diagrams
The widgets answer “what happens if…”. Drag, change, break things — nothing can be spoiled. 3. 3Formulas, piece by piece
Every important formula is taken apart: where each piece comes from, an example with numbers, and where the formula stops working. 4. 4Practice 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
Chapters
Introduction
I Numbers
1. 1 Counting 2. 2 Zero and minus 3. 3 Primes 4. 4 Divisibility 5. 5 Fractions 6. 6 Root two
II Algebra
1. 7 Letters 2. 8 Equations 3. 9 Functions and graphs 4. 10 Quadratic equations 5. 11 Systems and inequalities 6. 12 Powers and logarithms 7. 13 Sequences 8. 14 Polynomials 9. 15 Complex numbers
III Geometry
1. 16 Euclid's Elements 2. 17 The triangle 3. 18 Pythagoras' theorem 4. 19 The circle and π 5. 20 Area and volume 6. 21 Trigonometry 7. 22 Vectors 8. 23 Symmetry 9. 24 Conic sections
IV Calculus
1. 25 Limits 2. 26 The derivative 3. 27 Extrema 4. 28 The integral 5. 29 The number e 6. 30 Series 7. 31 Differential equations 8. 32 Several variables 9. 33 Fourier series 10. 34 Complex analysis
V Linear algebra
1. 35 Matrices 2. 36 Gaussian elimination 3. 37 Vector spaces 4. 38 Eigenvectors 5. 39 Least squares and SVD
VI Structures
1. 40 Groups 2. 41 Remainders and ciphers 3. 42 Rings and fields 4. 43 Galois theory 5. 44 Zeta and curves
VII Chance & data
1. 45 Combinatorics 2. 46 Graphs 3. 47 Probability 4. 48 Random variables 5. 49 Statistics 6. 50 Markov and information
VIII Foundations
1. 51 Logic and sets 2. 52 Infinities 3. 53 Rigorous analysis 4. 54 Measure and integral 5. 55 Functional analysis 6. 56 Gödel and Turing
IX Horizons
1. 57 Topology 2. 58 Non-Euclidean geometries 3. 59 Chaos and fractals 4. 60 The frontier