Intellect PlusAlgebra
259 cards. Each one is a 40-second story with a five-minute read behind it. Page 4 of 5.
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…
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 …
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,…- 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…
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…
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 …
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 …
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…- 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…
- 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…
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…
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 …
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…
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…
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…
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…
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…
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…
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…
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…
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…
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…- 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 …
- 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…
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…- 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 …
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…
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 …
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…- 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…
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…
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…- 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…
- 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…
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, …
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…
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…
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…
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…
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…
Sheaves track local data that must agree on overlapsIn mathematics a sheaf systematically attaches algebraic or set-like data to open patches of a space so that local pieces glue when they match on overlaps. Sheaf theory studies th…
Vector spaces are playgrounds closed under scale and addA vector space—or linear space—is a set of vectors you can add and stretch by scalars while staying inside the set. The idea generalizes arrows for force and velocity into pure st…
Abelian groups are named for a Norwegian teen prodigyWhen order does not matter under a group operation, mathematicians call the group abelian. Integers under addition are the everyday model. Camille Jordan borrowed the name from Ni…
Abstract algebra flipped how textbooks teach structureBefore the 1800s, algebra meant polynomials. Then scattered facts from number theory, geometry, and equation-solving hardened into axiom systems for groups, rings, and fields—almo…
Algebra began as bone-setting, then became equation craftThe Arabic word behind algebra once named a surgical fix for broken bones. In the ninth century a Persian mathematician turned that word into a method for completing and balancing…- Polynomial zeros became geometry's modern languageAlgebraic geometry turns systems of polynomial equations into shapes—curves, surfaces, and higher varieties—then studies those shapes with commutative algebra. Twentieth-century a…
- An algebraic structure is a set that obeys a short lawbookTake a nonempty set, add operations such as addition or multiplication, and insist on a finite list of identities. That package—the carrier set, the operations, and the axioms—is …
Topology borrowed algebra to count holes in spaceAlgebraic topology attaches groups to shapes so that continuous deformation becomes algebra. Homotopy, homology, and cohomology turn questions about loops and cavities into calcul…- A variety is a polynomial zero set with geometric teethClassical algebraic varieties are solution sets of polynomial systems over the reals or complexes. Hilbert's Nullstellensatz then locks those shapes to ideals in polynomial rings,…
- Parentheses can vanish—until floating-point math arrivesIf a binary operation is associative, regrouping never changes the answer, so long as the operands stay in order. Real addition and multiplication obey that rule. Computer floatin…
Boolean algebra turned true and false into circuit mathGeorge Boole built an algebra whose variables are truth values, not ordinary numbers. Decades later Claude Shannon noticed the same rules describe switching circuits—so the logic …- Ideal theory grew up and renamed itself commutative algebraCommutative algebra studies rings where multiplication swaps freely, plus their ideals and modules. Algebraic geometry and number theory both lean on it—so much that once-geometri…
- Order did not need a name until algebra got strangeFor centuries people quietly used the fact that three plus four equals four plus three. Only in the nineteenth century, as weird new algebraic structures appeared, did mathematici…
Commutative rings are where multiplication finally swapsA ring already adds like an abelian group and multiplies with distributivity. Call it commutative when ab equals ba. That single extra law unlocks ideals that are automatically tw…- Cross products make a third arrow from two in spaceIn oriented three-dimensional space, two independent vectors determine a third perpendicular to both, with length equal to the parallelogram they span. Swap the inputs and the res…
- Determinants scale volume and decide invertibilityOne number built from a square matrix tells you whether the matrix is invertible, how a linear map stretches volumes, and—via Cramer's rule—how to solve linear systems. Similar ma…
- Distributivity is why you can factor and expand at allMultiplying a sum means multiplying each piece, then adding—the move behind mental arithmetic, long multiplication, and factoring. Rings and fields build that law into their axiom…
- The dot product turns two arrows into one numberAdd the products of matching coordinates, or multiply lengths by the cosine of the angle between them—in Cartesian coordinates those two recipes agree. That single scalar rebuilds…
- Eigenvectors are the arrows a map only stretchesA linear transformation can rotate, shear, or stretch the space around it. Its eigenvectors are the special nonzero directions that refuse to turn—they only scale in place, possib…
School algebra is arithmetic with letters for unknownsElementary algebra keeps the usual number operations but lets symbols stand for unspecified values. That small move turns one numerical example into a general relationship—and sti…