Queen of the Sciences RU

An interactive maths textbook

Queen of the Sciencesfrom counting to university

Sixty-one chapters, from notches on a bone to Gödel’s theorems. Here you can touch the formulas — drag points, change numbers and watch what breaks. Every chapter opens with a problem the old tools can’t crack.

The English edition is being translated: 14 of 61 chapters are ready. The others, marked RU, open in the Russian original for now.

Start at the beginning Course map
  • 61chapters in 10 parts
  • 24step-by-step problem solvers
  • 4levels, from middle school to year 3
Mathematics is the queen of the sciences, and arithmetic the queen of mathematics. Gauss, as remembered by Sartorius von Waltershausen

The map of mathematics

Coloured lines are the parts of the course, stations are chapters. At each interchange one part hands over to the next, and dotted threads join chapters where an idea from one part is at work in another.

Course map: lines are parts, stations are chapters Chapter 0 · What mathematics is · Ages 11–15 0 Start Chapter 1 · Counting: from notches to bits · Ages 11–15 1 Counting Chapter 2 · Zero and minus · Ages 11–15 2 Zero and minus Chapter 3 · Prime numbers · Ages 11–15 3 Primes Chapter 4 · Divisibility and Euclid's algorithm · Ages 11–15 4 Divisibility Chapter 5 · Fractions · Ages 11–15 5 Fractions Chapter 6 · Root two and the real numbers · Ages 11–15 6 Root two Chapter 7 · Letters for numbers · Ages 11–15 7 Letters Chapter 8 · Equations: the art of balance · Ages 11–15 8 Equations Chapter 9 · Functions and graphs · Ages 11–15 9 Functions Chapter 10 · Quadratic equations · Ages 11–15 10 Quadratics Chapter 11 · Systems of equations and inequalities · Ages 11–15 11 Systems Chapter 12 · Powers and logarithms · Ages 11–15 12 Powers and logs Chapter 13 · Sequences and induction · Ages 11–15 13 Sequences Chapter 14 · Polynomials · Ages 16–18 14 Polynomials Chapter 15 · Complex numbers · Ages 16–18 15 Complex numbers Chapter 16 · Elements: axioms and constructions · Ages 11–15 16 Euclid Chapter 17 · The triangle · Ages 11–15 17 Triangle Chapter 18 · Pythagoras' theorem · Ages 11–15 18 Pythagoras Chapter 19 · The circle and π · Ages 11–15 19 Circle and π Chapter 20 · Area and volume · Ages 11–15 20 Area and volume Chapter 21 · Trigonometry · Ages 16–18 21 Trigonometry Chapter 22 · Vectors and coordinates · Ages 16–18 22 Vectors Chapter 23 · Symmetry and transformations · Ages 16–18 23 Symmetry Chapter 24 · Conic sections · Ages 16–18 24 Conics Chapter 25 · Limits · Ages 16–18 25 Limits Chapter 26 · The derivative · Ages 16–18 26 Derivative Chapter 27 · Extrema and optimization · Ages 16–18 27 Extrema Chapter 28 · The integral · Ages 16–18 28 Integral Chapter 29 · e and the exponential · Ages 16–18 29 The number e Chapter 30 · Infinite series · Years 1–2 30 Series Chapter 31 · Differential equations · Years 1–2 31 Differential equations Chapter 32 · Several variables · Years 1–2 32 Several variables Chapter 33 · Fourier series · Years 1–2 33 Fourier Chapter 34 · Complex analysis · Years 1–2 34 Complex analysis Chapter 35 · Matrices as transformations · Years 1–2 35 Matrices Chapter 36 · Gaussian elimination · Years 1–2 36 Elimination Chapter 37 · Vector spaces · Years 1–2 37 Vector spaces Chapter 38 · Eigenvectors · Years 1–2 38 Eigenvectors Chapter 39 · Orthogonality, least squares and SVD · Years 1–2 39 Least squares, SVD Chapter 40 · Groups · Years 1–2 40 Groups Chapter 41 · Modular arithmetic and ciphers · Years 1–2 41 Modular and ciphers Chapter 42 · Rings, fields and codes · Years 1–2 42 Rings and fields Chapter 43 · Galois theory · Year 3 and up 43 Galois theory Chapter 44 · Zeta and elliptic curves · Year 3 and up 44 Zeta and curves Chapter 45 · Combinatorics · Ages 11–15 45 Combinatorics Chapter 46 · Graphs · Ages 16–18 46 Graphs Chapter 47 · Probability · Ages 16–18 47 Probability Chapter 48 · Random variables · Years 1–2 48 Random variables Chapter 49 · Statistics · Years 1–2 49 Statistics Chapter 50 · Markov chains and information · Years 1–2 50 Markov chains Chapter 51 · Logic and sets · Years 1–2 51 Logic and sets Chapter 52 · Infinities · Years 1–2 52 Infinities Chapter 53 · Real analysis · Years 1–2 53 Real analysis Chapter 54 · Measure and the Lebesgue integral · Year 3 and up 54 Measure Chapter 55 · Functional analysis · Year 3 and up 55 Functional analysis Chapter 56 · Gödel, Turing and the limits of proof · Years 1–2 56 Gödel and Turing Chapter 57 · Topology · Years 1–2 57 Topology Chapter 58 · Non-Euclidean geometry · Years 1–2 58 Non-Euclidean geometry Chapter 59 · Chaos and fractals · Years 1–2 59 Chaos and fractals Chapter 60 · The frontier · Year 3 and up 60 The frontier Course map: lines are parts, stations are chapters Chapter 0 · What mathematics is · Ages 11–15 0 Start Chapter 1 · Counting: from notches to bits · Ages 11–15 1 Counting Chapter 2 · Zero and minus · Ages 11–15 2 Zero and minus Chapter 3 · Prime numbers · Ages 11–15 3 Primes Chapter 4 · Divisibility and Euclid's algorithm · Ages 11–15 4 Divisibility Chapter 5 · Fractions · Ages 11–15 5 Fractions Chapter 6 · Root two and the real numbers · Ages 11–15 6 Root two Chapter 7 · Letters for numbers · Ages 11–15 7 Letters Chapter 8 · Equations: the art of balance · Ages 11–15 8 Equations Chapter 9 · Functions and graphs · Ages 11–15 9 Functions Chapter 10 · Quadratic equations · Ages 11–15 10 Quadratics Chapter 11 · Systems of equations and inequalities · Ages 11–15 11 Systems Chapter 12 · Powers and logarithms · Ages 11–15 12 Powers and logs Chapter 13 · Sequences and induction · Ages 11–15 13 Sequences Chapter 14 · Polynomials · Ages 16–18 14 Polynomials Chapter 15 · Complex numbers · Ages 16–18 15 Complex numbers Chapter 16 · Elements: axioms and constructions · Ages 11–15 16 Euclid Chapter 17 · The triangle · Ages 11–15 17 Triangle Chapter 18 · Pythagoras' theorem · Ages 11–15 18 Pythagoras Chapter 19 · The circle and π · Ages 11–15 19 Circle and π Chapter 20 · Area and volume · Ages 11–15 20 Area and volume Chapter 21 · Trigonometry · Ages 16–18 21 Trigonometry Chapter 22 · Vectors and coordinates · Ages 16–18 22 Vectors Chapter 23 · Symmetry and transformations · Ages 16–18 23 Symmetry Chapter 24 · Conic sections · Ages 16–18 24 Conics Chapter 25 · Limits · Ages 16–18 25 Limits Chapter 26 · The derivative · Ages 16–18 26 Derivative Chapter 27 · Extrema and optimization · Ages 16–18 27 Extrema Chapter 28 · The integral · Ages 16–18 28 Integral Chapter 29 · e and the exponential · Ages 16–18 29 The number e Chapter 30 · Infinite series · Years 1–2 30 Series Chapter 31 · Differential equations · Years 1–2 31 Differentialequations Chapter 32 · Several variables · Years 1–2 32 Several variables Chapter 33 · Fourier series · Years 1–2 33 Fourier Chapter 34 · Complex analysis · Years 1–2 34 Complex analysis Chapter 35 · Matrices as transformations · Years 1–2 35 Matrices Chapter 36 · Gaussian elimination · Years 1–2 36 Elimination Chapter 37 · Vector spaces · Years 1–2 37 Vector spaces Chapter 38 · Eigenvectors · Years 1–2 38 Eigenvectors Chapter 39 · Orthogonality, least squares and SVD · Years 1–2 39 Least squares, SVD Chapter 40 · Groups · Years 1–2 40 Groups Chapter 41 · Modular arithmetic and ciphers · Years 1–2 41 Modular and ciphers Chapter 42 · Rings, fields and codes · Years 1–2 42 Rings and fields Chapter 43 · Galois theory · Year 3 and up 43 Galois theory Chapter 44 · Zeta and elliptic curves · Year 3 and up 44 Zeta and curves Chapter 45 · Combinatorics · Ages 11–15 45 Combinatorics Chapter 46 · Graphs · Ages 16–18 46 Graphs Chapter 47 · Probability · Ages 16–18 47 Probability Chapter 48 · Random variables · Years 1–2 48 Random variables Chapter 49 · Statistics · Years 1–2 49 Statistics Chapter 50 · Markov chains and information · Years 1–2 50 Markov chains Chapter 51 · Logic and sets · Years 1–2 51 Logic and sets Chapter 52 · Infinities · Years 1–2 52 Infinities Chapter 53 · Real analysis · Years 1–2 53 Real analysis Chapter 54 · Measure and the Lebesgue integral · Year 3 and up 54 Measure Chapter 55 · Functional analysis · Year 3 and up 55 Functional analysis Chapter 56 · Gödel, Turing and the limits of proof · Years 1–2 56 Gödel and Turing Chapter 57 · Topology · Years 1–2 57 Topology Chapter 58 · Non-Euclidean geometry · Years 1–2 58 Non-Euclideangeometry Chapter 59 · Chaos and fractals · Years 1–2 59 Chaos and fractals Chapter 60 · The frontier · Year 3 and up 60 The frontier
  • chapter
  • read
  • change to the next line
  • an idea from one chapter at work in another

Seven big questions

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

  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. 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

  3. 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

  4. 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

  5. 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 · Modular and ciphers

  6. 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

  7. 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

  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

Chapters by part

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

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
14RU Polynomials Ages 16–18 50 min
15RU Complex numbers Ages 16–18 55 min

Horizons

Topology, non-Euclidean worlds, chaos and open problems

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

57RU Topology Years 1–2 55 min
58RU Non-Euclidean geometry Years 1–2 60 min
59RU Chaos and fractals Years 1–2 60 min
60RU The frontier 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.

All solvers

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

  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. Live diagrams

    The widgets answer “what happens if…”. Drag, change, break things — nothing can be spoiled.

  3. 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.

  4. 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
Queen of the Sciences

Chapters

Introduction

  1. 0 Start

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
  2. 17 Triangle
  3. 18 Pythagoras
  4. 19 Circle and π
  5. 20 Area and volume
  6. 21 Trigonometry
  7. 22 Vectors
  8. 23 Symmetry
  9. 24 Conics

IV Calculus

  1. 25 Limits
  2. 26 Derivative
  3. 27 Extrema
  4. 28 Integral
  5. 29 The number e
  6. 30 Series
  7. 31 Differential equations
  8. 32 Several variables
  9. 33 Fourier
  10. 34 Complex analysis

V Linear algebra

  1. 35 Matrices
  2. 36 Elimination
  3. 37 Vector spaces
  4. 38 Eigenvectors
  5. 39 Least squares, SVD

VI Structures

  1. 40 Groups
  2. 41 Modular 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 chains

VIII Foundations

  1. 51 Logic and sets
  2. 52 Infinities
  3. 53 Real analysis
  4. 54 Measure
  5. 55 Functional analysis
  6. 56 Gödel and Turing

IX Horizons

  1. 57 Topology
  2. 58 Non-Euclidean geometry
  3. 59 Chaos and fractals
  4. 60 The frontier