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1,585 cards. Each one is a 40-second story with a five-minute read behind it. Page 11 of 27.

601. Wittgenstein gets credit for the truth table, but Peirce sketched one decades earlierLudwig Wittgenstein is usually credited with inventing the truth table, in the Tractatus he finished in 1918. Yet research by Irving Anellis points to Charles Sanders Peirce drawi…
602. The word tuple was carved out of double, triple and quintupleMathematicians took the family of words single, double, triple, quadruple, quintuple and so on, and abstracted the ending into n-tuple, a finite ordered list of any length n. Yet …
603. The first University of Chicago went bankrupt before Rockefeller money built a secondBaptist educators incorporated a University of Chicago in 1857, but fire, a financial panic and debt ruined it, and creditors foreclosed in 1886. A new university took the name in…
604. PageRank ranks web pages by imagining a surfer clicking links at randomGoogle's founding algorithm pictures an aimless web user who follows links, and every so often gets bored and jumps somewhere random. The share of time that surfer spends on each …
605. Peter Scholze became Germany's youngest full professor at 24, then won a Fields MedalPeter Scholze finished a Bonn bachelor's degree in three semesters and a master's in two more. Shortly after his doctorate in 2012, the university made him a full professor at 24,…
606. William Rowan Hamilton carved quaternions into a Dublin bridge in 1843Walking beside Dublin's Royal Canal on 16 October 1843, William Rowan Hamilton suddenly saw how to multiply points in space, and scratched the rule into Brougham Bridge with a poc…
607. Recursion defines things in terms of themselves, and programmers love joking about itSearch Google in English for recursion and it asks whether you meant recursion. In some editions of the classic C programming book, the index entry for recursion lists its own pag…
608. A circle is not a polygon with infinitely many sides, geometry insistsAdd sides to a regular polygon and it looks ever more like a circle. A shape with 10,000 equal sides has interior angles of 179.964 degrees. Yet the angle can never reach exactly …
609. Rotate a phone two ways in different order and it ends up facing differentlyTurn a phone about one axis and then another, and it lands in one position; swap the order and it lands somewhere else. Rotation matrices, the grids of numbers that computers use …
610. The spectral theorem says when a tangled matrix can be simplified to pure stretchingMany matrices scramble space in complicated ways, but a special class can be rewritten, in the right coordinates, as nothing more than stretching along perpendicular directions. T…
611. Oxford and Cambridge women sailed to Dublin to collect degrees their colleges withheldBetween 1904 and 1907, women who had studied at Oxford and Cambridge but been refused degrees crossed the Irish Sea to Trinity College Dublin, which granted them one. They were ni…
612. Lie algebras capture a whole symmetry group from its tiniest motionsTurn a book a quarter-turn forward and then a quarter-turn sideways, and it ends up differently than if you reverse the order. Rotations do not commute. A Lie algebra measures exa…
613. Sophus Lie chased a grand dream for equations and built modern physics' toolkitSophus Lie wanted to do for differential equations what Galois had done for polynomial ones: sort them all by their symmetries. That dream never fully came true. But the continuou…
614. The simplest rule in logic may break when Hobbes wrote HamletIf P implies Q, and P holds, then Q holds. Modus ponens is about as basic as reasoning gets, and logicians have leaned on it since antiquity. Yet the philosopher Vann McGee built …
615. Monoids: the humble structure behind text strings, logic gates and surfacesGluing words end to end, adding whole numbers and combining true-or-false values with AND all share the same bare skeleton. Each has a way of combining two things where grouping d…
616. Any rotation in the plane is really two mirror flips in disguiseStand between two mirrors set at an angle, and reflecting twice turns out to be the same as rotating by double that angle. That fact sits at the heart of the orthogonal group, the…
617. Jean-Pierre Serre won the Fields Medal at 27 and still holds the recordIn 1954 the French mathematician Jean-Pierre Serre received the Fields Medal at 27, the youngest winner then and still today. Almost half a century later, in 2003, he became the f…
618. A linear map's kernel is everything it sends to zeroEvery linear transformation has a blind spot: the collection of inputs it crushes down to zero. Mathematicians call this the kernel, or null space. Knowing its size tells you exac…
619. Khan Academy started with a finance worker tutoring his cousin onlineIn 2004 Sal Khan, then working in finance, began helping a young cousin with maths over the internet. Other relatives wanted lessons too, so he recorded short videos on his home c…
620. Set unions and least common multiples share one hidden mathematical shapeMerging two sets, finding the least common multiple of two numbers and combining two logical conditions look like unrelated chores. Order theorists see the same pattern in each: a…
621. Least common multiples predict when gears and planets line up againMark a line across two meshing gears and set them turning. How long until the mark lines up again? The answer is the least common multiple of their tooth counts. The same arithmet…
622. A lost asteroid made least squares famous, and started a priority fightOn 1 January 1801 Giuseppe Piazzi spotted the asteroid Ceres, tracked it for 40 days, then lost it in the Sun's glare. Astronomers needed to know where it would reappear. The only…
623. Mathematicians' favourite shorthand 'iff' dates only from 1955The four-letter word 'iff', meaning 'if and only if', first appeared in print in John L. Kelley's General Topology in 1955. Paul Halmos usually gets the credit for inventing it, t…
624. Inner products let mathematicians measure angles in spaces nobody can pictureSchool geometry measures length and angle with rulers and protractors. An inner product space captures the same ideas algebraically, through a single operation that takes two vect…
625. A matrix can be undone only if it never squashes space flatThink of a matrix as a machine that moves every point in space. If it keeps distinct points apart and still reaches everywhere, another matrix can reverse the move exactly. If it …
626. The Jacobian tells you how a smooth map stretches and twists spaceWarp a photograph smoothly and each tiny patch gets stretched, rotated or sheared. The Jacobian matrix, named after the German mathematician Carl Gustav Jacob Jacobi, records exac…
627. Euclid's shortcut for finding the greatest common divisor still runs inside computersWhat is the largest square tile that will cover a 24 by 60 floor with no cutting? The answer, 12, is the greatest common divisor of the two lengths. Listing every divisor gets slo…
628. Grigori Perelman solved the Poincaré conjecture and turned down a million dollarsIn 2002 and 2003 a reclusive Russian geometer posted proofs of a problem that had resisted mathematicians for a century. Then Grigori Perelman refused the Fields Medal, saying he …
629. The Hodge conjecture, a million-dollar question about shapes you cannot pictureSome geometric spaces have too many dimensions to imagine, yet mathematicians still want to know how many holes they have. The Hodge conjecture bets that this information can be r…
630. Ideals began as imaginary numbers invented to rescue prime factorisationIn some number systems, whole numbers can be broken into primes in more than one way, which wrecks a lot of arithmetic. Ernst Kummer patched the hole by imagining missing factors …
631. Why flipping a triangle twice is the same as turning itPick up a cardboard equilateral triangle, flip it over along one line of symmetry, then flip it along another. It lands exactly as if you had simply rotated it by 120 degrees. Tha…
632. Seventeen particles, 61 variations: the Standard Model's list of basic ingredientsAtoms were named for being uncuttable, and they turned out not to be. Today's Standard Model lists seventeen particles that, as far as anyone can tell, have no smaller parts: twel…
633. How to add two points on an elliptic curve with a rulerOn an elliptic curve you can add points using nothing but straight lines. Draw a line through two points, find where it hits the curve a third time, and flip that point across the…
634. Euclidean space began as a model of the real world and became pure algebraAncient Greek geometers wanted a model of the space we live in. Euclid's great step was to prove every property of it from a handful of starting assumptions. Over two thousand yea…
635. The wedge product turns area and volume into algebra with a signTake two arrows on a flat page and they outline a parallelogram. Hermann Grassmann's wedge product multiplies the two arrows and gets that area directly, with a plus or minus sign…
636. Gram–Schmidt straightens a tilted set of arrows into perpendicular onesHand the Gram–Schmidt process a set of arrows pointing in awkward, overlapping directions, and it returns a tidy set at right angles to each other, spanning exactly the same space…
637. A 1,500-year-old Chinese puzzle now speeds up giant computer calculationsAn old Chinese manual poses a riddle: count some objects in threes and two are left over, in fives and three remain, in sevens and two remain. How many are there? The answer, 23, …
638. André Weil did his career-making mathematics inside a military prisonIn early 1940 the French mathematician André Weil sat in a military prison in Rouen, charged with failing to report for army duty. Between February and May he finished the work th…
639. A teacher can tell seats equal students without counting eitherStudents file into a classroom and sit down. The teacher glances around: nobody is standing and no seat is empty. Without counting anything, she knows there are exactly as many st…
640. Expanding a bracket and choosing a team give the same numbersMultiply out (1 + x) to the fourth power and the coefficients come out as 1, 4, 6, 4, 1. Ask how many ways there are to pick two items from four, and the answer is also 6. That co…
641. A Mafia boss was caught partly by a code Julius Caesar usedIn April 2006 police in Sicily captured the fugitive Mafia boss Bernardo Provenzano, helped by the fact that some of his notes were written in a clumsy variant of the Caesar ciphe…
642. A Q-Q plot reveals at a glance whether two distributions share a shapeHistograms can hide the difference between a bell curve and something with dangerously heavy tails. Sort two sets of numbers, plot them against each other, and see whether the dot…
643. R began as a teaching tool in Auckland and became a data science workhorseTwo statistics professors in New Zealand wanted a better way to teach their introductory course. They named their language R, partly after their own first initials, Ross and Rober…
644. Flip a coin, step left or right, and discover why gamblers always go brokePut a marker at zero, flip a fair coin, and move one step right for heads or left for tails. Keep going forever and the marker will revisit every point on the line infinitely ofte…
645. Drawing the best straight line through data explains regression toward the meanMeasure something once, then measure it again, and the second result will on average sit closer to the typical value than the first did. That pull, known as regression toward the …
646. Skewness measures lopsided data, and the textbook rule about it often failsMany statistics courses teach that in right-skewed data the mean sits to the right of the median. A 2005 journal article pointed out that this rule of thumb fails with surprising …
647. Affine transformations keep straight lines straight and parallel lines parallelStretch, slide, rotate, reflect or shear a drawing, and lengths and angles may change, but straight lines stay straight and parallel lines never meet. That guarantee defines an af…
648. The Pareto distribution explains the 80:20 rule and why averages can explodeVilfredo Pareto noticed that a small slice of society held most of its wealth. The mathematical curve named after him now describes everything from insurance claims to natural phe…
649. Pearson's correlation is really the cosine of an angle between two data vectorsThe most widely used measure of how two things move together carries Karl Pearson's name, but a French scientist published its formula decades earlier. Underneath the algebra sits…
650. Percentiles quietly set internet bills, speed limits and children's growth chartsYour internet provider may ignore your heaviest five per cent of usage when it bills you, and the speed limit on your road may have been set by watching how fast most drivers alre…
651. Psychometrics tries to measure things nobody can see, like intelligence and introversionThe word psychometry was coined in 1842 by an American who used it for supposed paranormal powers. Today it names a serious and argumentative science: working out how to put numbe…
652. Mean squared error scores guesses by squaring every missWhenever a model predicts a number, somebody has to decide how wrong it was. The most common answer is to take every miss, square it, and average the results. That simple recipe, …
653. MIDI succeeded because its inventors gave it awayIn the early 1980s a Roland keyboard could not talk to a Yamaha or an Oberheim; each maker guarded its own connection scheme. Roland's president and a Californian synth builder de…
654. Minimax means planning for the worst your opponent can doSuppose one of your options could pay 5 but might cost you 100, while another guarantees at least 2. Minimax thinking picks the safe one. The rule, born in game theory, asks a pes…
655. Norbert Wiener earned a Harvard doctorate at 19 and then founded cyberneticsNorbert Wiener finished high school at 11, collected a mathematics degree at 14 and a Harvard PhD at 19. He went on to argue that intelligent behaviour comes down to feedback loop…
656. One hot oven can wreck the average temperature of a roomMeasure ten objects in a room: nine sit between 20 and 25 degrees Celsius, and one is an oven at 175. The median stays sensibly between 20 and 25, but the mean jumps to somewhere …
657. The Mann-Whitney test is a tournament between two groups of numbersWant to know whether one group tends to score higher than another without assuming a bell curve? Pair every value in the first group against every value in the second, award a poi…
658. The doubling bet that inspired a branch of probabilityEighteenth-century French gamblers loved a system called the martingale: double your stake after every loss, and the first win recovers everything plus a profit. Given unlimited m…
659. KL divergence measures the cost of believing the wrong modelSuppose you design a code for messages assuming one pattern of letter frequencies, but the real messages follow another. On average you will waste some bits per symbol. That waste…
660. Kurtosis measures a distribution's outliers, not how pointy its peak isMany textbooks describe kurtosis as peakedness, the sharpness of a curve's summit. Statisticians now agree that description is wrong. Kurtosis is about the tails: how prone a dist…

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