Queen of the Sciences RU

Queen of the Sciences

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 Five points on a circle cut it into 16 pieces; six points give 31, although your hand wants to write 32. The course begins with this trap: why examples are never enough in mathematics, what replaces them, 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 14RU Polynomials Ages 16–18 50 min 15RU Complex numbers Ages 16–18 55 min 16RU Elements: axioms and constructions Ages 11–15 45 min 18RU Pythagoras' theorem Ages 11–15 50 min 19RU The circle and π Ages 11–15 55 min 20RU Area and volume Ages 11–15 55 min 21RU Trigonometry Ages 16–18 60 min 24RU Conic sections Ages 16–18 50 min 25RU Limits Ages 16–18 50 min 26RU The derivative Ages 16–18 50 min 28RU The integral Ages 16–18 55 min 29RU e and the exponential Ages 16–18 55 min 30RU Infinite series Years 1–2 55 min 33RU Fourier series Years 1–2 55 min 36RU Gaussian elimination Years 1–2 55 min 40RU Groups Years 1–2 55 min 43RU Galois theory Year 3 and up 75 min 44RU Zeta and elliptic curves Year 3 and up 70 min 50RU Markov chains and information Years 1–2 50 min 52RU Infinities Years 1–2 55 min 53RU Real analysis Years 1–2 75 min 55RU Functional analysis Year 3 and up 60 min 56RU Gödel, Turing and the limits of proof Years 1–2 65 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
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