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Mathematics

85 lessons across 6 eras, from c. 2000 BC to today. Tap any question to read its lesson.

  1. c. 2000 BC – c. AD 400

    Era 1: Ancient measurement

    1. The Rhind Papyrus and area in EgyptWhen the Nile floods, how is a field found again?Completed
    2. Plimpton 322 and base-sixty numbersWhy does an hour have sixty minutes?Completed
    3. Measuring a pyramid by its shadowCan a pyramid's height be measured without touching it?Completed
    4. Pythagoras and the irrational crisisWhy can no fraction name the square root of two?Completed
    5. Zeno's paradoxesWhy can't the fast runner catch the tortoise?Completed
    6. Euclid's ElementsCan all of geometry come from five rules?Completed
    7. Euclid and the infinity of primesDo the prime numbers ever run out?Completed
    8. Aristarchus measures the Moon and the SunHow far away is the Sun?Completed
    9. Archimedes squeezes the circleHow long, exactly, is a circle's circumference?Completed
    10. Archimedes and the method of exhaustionHow do you find the area of a curved shape?Completed
    11. Eratosthenes measures the EarthCan a shadow tell the size of the planet?Completed
    12. Apollonius and conic sectionsIf you slice a cone, how many different curves can you get?Completed
    13. Hipparchus and the first table of chordsHow is an angle in the sky turned into a number?Completed
    14. Diophantus and the unknownWhat do we gain by putting a letter in place of a number?Completed
  2. 200 BC – AD 1500

    Era 2: The birth of algebra — China, India, Islam

    1. The Nine Chapters and equations in ChinaCan mathematics be a handbook?Completed
    2. Liu Hui and the 3,072-sided polygonHow many digits of pi can be chased down?Completed
    3. The Chinese remainder theoremCan soldiers be counted without counting them one by one?Completed
    4. The Maya zero and the Long CountHow many times was zero invented?Completed
    5. Brahmagupta and the rules of zeroCan nothing be a number?Completed
    6. Indian numerals and place valueWhy is no new symbol needed after 9?Completed
    7. Al-Khwarizmi's al-JabrHow do you get the unknown on its own?Completed
    8. Al-Karaji breaks algebra away from geometryCan algebra stand on its own?Completed
    9. Al-Biruni measures the Earth from a mountainCan a planet be measured from a single hilltop?Completed
    10. Omar Khayyam draws the cubicCan an equation be solved by drawing?Completed
    11. Al-Tusi sets trigonometry apartHow did the triangle come down from the sky to the ground?Completed
    12. Madhava and the Kerala seriesCan an infinite sum give a finite number?Completed
    13. Ulugh Beg and the Samarkand tablesHow precisely can the sky be measured with the naked eye?Completed
    14. Fibonacci's Liber AbaciHow did Indian numerals reach Europe?Completed
  3. 1500 – 1680

    Era 3: Symbol and space

    1. Cardano, Tartaglia and the cubic feudWhy fight a duel over a formula?Completed
    2. Bombelli and imaginary numbersIs the square root of a negative number any use?Completed
    3. Viète writes with lettersWhat changed when letters took the place of numbers?Completed
    4. Napier and the logarithmCan multiplication be turned into addition?Completed
    5. Kepler and the volume of a wine barrelHow do you find the volume of a curved vessel?Completed
    6. Cavalieri and indivisiblesCan an area be built from infinitely thin strips?Completed
    7. Descartes's coordinate planeCan geometry be turned into numbers?Completed
    8. Fermat and the top of a curveHow do you find the highest point of a curve?Completed
    9. Fermat's little theoremDo prime numbers have a hidden rule?Completed
    10. The Pascal–Fermat lettersHow do you split the stakes of an unfinished game?Completed
  4. 1660 – 1800

    Era 4: The infinitely small

    1. Newton and instantaneous speedIs there such a thing as speed in a single instant?Completed
    2. Leibniz's notation and the priority disputeWhat happens when two people find the same idea?Completed
    3. The fundamental theorem of calculusWhy are the derivative and the integral inverses of each other?Completed
    4. Berkeley's objectionIs the infinitely small something real?Completed
    5. Bernoulli and the brachistochroneIsn't the fastest way down a straight line?Completed
    6. Euler and the number eWhat happens if interest is added continuously?Completed
    7. The seven bridges of KönigsbergCan you cross every bridge exactly once?Completed
    8. Euler's identityWhat are e, i, π, 1 and 0 doing on the same line?Completed
    9. Taylor seriesCan every curve be turned into a polynomial?Completed
    10. Bayes and inverse probabilityHow much should new evidence change a belief?Completed
  5. 1800 – 1900

    Era 5: Structure and abstraction

    1. Gauss and the regular 17-gonWhich polygons can be drawn with compass and straightedge?Completed
    2. Gauss and clock arithmeticWhy does adding 5 hours to 10 o'clock give 3?Completed
    3. Gauss and the normal distributionHow is measurement error tamed?Completed
    4. Fourier takes the wave apartIs every wave a sum of simple waves?Completed
    5. Cauchy, Weierstrass and making the limit preciseWhat exactly does "infinitely small" mean?Completed
    6. Dedekind and the definition of a real numberAre there gaps in the number line?Completed
    7. Babbage, Lovelace and the first programDoes a machine only do arithmetic?Completed
    8. Cayley, matrices and linear algebraCan a transformation be written as a table of numbers?Completed
    9. The Riemann hypothesisAre the prime numbers scattered at random?Completed
    10. Abel and the unsolvable equationWhy is there no formula for the fifth degree?Completed
    11. Galois and symmetry groupsWhat decides whether an equation can be solved?Completed
    12. Lobachevsky and parallel linesCan parallel lines meet?Completed
    13. Hamilton and quaternionsCan multiplication be defined in three dimensions?Completed
    14. Riemann and curved spaceCan space itself be curved?Completed
    15. The Möbius strip and the birth of topologyCan a surface have only one side?Completed
    16. Cantor and the sizes of infinityAre some infinities bigger than others?Completed
    17. Boole and the algebra of logicCan true and false be calculated?Completed
    18. Poincaré and the three-body problemCan the future be calculated?Completed
  6. 1900 – present

    Era 6: Uncertainty and the machine

    1. Hilbert's 23 problemsCan the future of mathematics be put on a list?Completed
    2. Russell's paradoxDoes the set of all sets that don't contain themselves contain itself?Completed
    3. Noether's theoremWhy does a symmetry give rise to a conservation law?Completed
    4. Ramanujan's notebooksWhere do the formulas come from?Completed
    5. Gödel's incompleteness theoremCan every true statement be proved?CompletedHands-on version →
    6. The Turing machineWhat can a machine do, and what can't it?Completed
    7. Enigma and the BombeHow is a cipher broken?Completed
    8. Shannon and information theoryCan information be measured?Completed
    9. Von Neumann and game theoryHow do you play knowing your opponent is thinking too?Completed
    10. Nash equilibriumWhat happens if everyone chases their own interest?Completed
    11. The Monte Carlo methodCan randomness do calculations?Completed
    12. Markov chainsCan the present state alone decide the future?Completed
    13. Fisher and the design of experimentsHow do you show a result is not a coincidence?Completed
    14. Perelman and the Poincaré conjectureHow else can a sphere be recognised?Completed
    15. Lorenz and the butterfly effectHow much can a small difference grow?Completed
    16. Mandelbrot and fractalsHow many kilometres long is a coastline?Completed
    17. The four colour theoremCan a computer's proof be trusted?Completed
    18. Fermat's Last Theorem and WilesHow was a problem unsolved for 350 years finally solved?Completed
    19. P and NPIs every problem that is easy to check also easy to solve?Completed