Mathematics RU

Mathematics

History

Every chapter of the course by topic, level and part. You can pick several topics at once — say, “Puzzles” and “Probability”.

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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
Mathematics

Chapters

Introduction

  1. 0 What mathematics is

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