Science

1,604 cards. Each one is a 40-second story with a five-minute read behind it. Page 27 of 27.

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…
Vectors began as carriers before they became arrowsA Euclidean vector has length and direction—an arrow that "carries" one point to another. Astronomers coined the Latin word for carrier; nineteenth-century fights over quaternions…
A field is a number system where you can always divideAdd, subtract, multiply, and divide—except by zero—and demand the same laws the rationals obey. That axiom package is a field: the stage for linear algebra's scalars, Galois theor…
The fundamental group counts loops up to stretchingFix a base point and gather every path that leaves and returns. Treat two loops as the same when a continuous deformation turns one into the other without tearing. Concatenate the…
Every complex polynomial hides at least one complex rootThe fundamental theorem of algebra says a non-constant polynomial with complex coefficients always has a complex root—and counted with multiplicity, exactly as many roots as its d…
Galois linked equation-solving to the symmetry of rootsÉvariste Galois showed that whether a polynomial can be solved by radicals depends on the structure of its root-permutation group. That bridge between fields and groups also expla…
Gaussian elimination turns matrices into staircases of zerosRow operations—swap two rows, scale a row, or add a multiple of one row to another—systematically clear entries beneath pivots. Named for Carl Friedrich Gauss, the algorithm solve…
A group is a closed club with identity and undo buttonsTake a set and a way to combine any two members into a third still in the set. Demand associativity, an identity element, and an inverse for every element. That short rulebook—ill…
Representations turn abstract groups into matrices you can computeA group representation is a homomorphism into invertible linear maps on a vector space—so each group element becomes a matrix and composition becomes multiplication. Hard group qu…
Group theory is the mathematics of reversible symmetryRings, fields, and vector spaces are groups wearing extra structure. From crystal lattices to public-key cryptography, tracking what you can compose and undo has become a central …
Homological algebra mines chain complexes for hidden invariantsString modules or spaces into a sequence of maps whose consecutive composites vanish. The failure of exactness—cycles taken modulo boundaries—is homology. That bookkeeping, born f…
Isomorphism means same shape, different costumeAn isomorphism is a reversible map that preserves structure end to end. If one exists, the two objects share every property that depends only on that structure—so mathematicians o…
Linear algebra is the mathematics of flat approximationsLines, planes, rotations, and first-order models of curved systems all live inside vector spaces. From ancient Chinese counting rods through to modern matrix algorithms, linear al…
Linear equations are named for the lines they drawPut an equation in the form of a linear polynomial set to zero, and its solutions in two variables sketch a straight line in the plane. That geometric picture—not the algebra alon…
A matrix is a rectangle that eats vectors and spits mapsStack numbers in rows and columns and you can add them, multiply them, and—when the array is square—ask about determinants and eigenvalues. That humble rectangle became the workho…
Modules are vector spaces that forgot their scalars were a fieldReplace the field of scalars by a ring and you get a module: still an abelian group with distributive scaling, but bases may vanish and ranks may misbehave. Every abelian group is…
Quadratic equations were solved four thousand years agoAn equation with a squared unknown—standard form ax² + bx + c = 0 with a nonzero—has at most two roots. Count complex numbers and double roots carefully, and there are always two.…
Representation theory makes abstract algebra act on vectorsDescribe group elements, algebra elements, or Lie brackets as linear maps—often as concrete matrices you can multiply by hand—and hard structural questions collapse into ordinary …
Rings keep addition fair and let multiplication misbehaveA ring has two operations modeled on the integers: addition forms an abelian group, multiplication is associative and distributive, and (in the modern default) a multiplicative id…
Ring theory studies integer-like worlds far from ZOnce addition and multiplication obey ring axioms, the structure theory begins: modules as representations, special classes from group rings to enveloping algebras, and properties…
Maths proves you can cut one ball into pieces and build two identical balls.In 1924 two mathematicians proved that a solid ball can be split into just a handful of pieces and reassembled, by moving and rotating alone, into two balls exactly the same size …
Put just 23 people in a room and a shared birthday becomes likely.Ask people how many strangers you need for a better-than-even chance that two share a birthday, and most guess hundreds or thousands. The real answer is 23. The trick is that you …
Chill atoms almost to absolute zero and they start acting as one.In 1924 Satyendra Nath Bose and Albert Einstein predicted a strange state of matter: cool certain atoms enough and they all drop into the same quantum state, behaving like a singl…
A jittering speck under a microscope finally proved atoms are real.In 1827 the botanist Robert Brown saw tiny particles from pollen dancing restlessly in water, and nobody could explain why. In 1905 Einstein did: they are being jostled by invisib…
The man who said continents move was right, and was laughed at for decades.In 1912 Alfred Wegener argued that the continents had once been joined in a single landmass and had drifted apart. Most geologists rejected it: nobody could say what force could p…
Fire single electrons at two slits and they still make a wave pattern.Send particles one at a time through two narrow slits and each lands as a single dot. Yet as the dots build up, they form the striped pattern of waves interfering. Try to catch wh…
A note in a book margin took mathematicians 358 years to prove.Around 1637 Pierre de Fermat scribbled in the margin of a book that he had a marvellous proof, too large to fit, that one simple-looking equation has no whole-number solutions. No…
Compasses would have pointed south for much of Earth's history.Earth's magnetic north and south poles swap places, at random, every few hundred thousand years on average. The last full flip was about 780,000 years ago. We know because volcani…
Old church windows are not thicker at the bottom because glass flows.It is a favourite classroom fact: glass is really a slow liquid, and medieval windows have sagged over the centuries. It is not true. Glass is a solid, and calculations suggest me…
Helium was found on the Sun 27 years before anyone found it on Earth.In 1868, astronomers studying a solar eclipse spotted a mysterious yellow line in sunlight that matched no known element. Norman Lockyer decided it was something new and named it …
If ice sank, Earth's lakes would freeze solid from the bottom up.Almost every substance is densest as a solid. Water is a rebel: it is densest at about 4°C, and ice is lighter still. So ice floats, forms an insulating lid on lakes and seas, and…
In 1986 a quiet lake in Cameroon exhaled a deadly cloud of gas.Lake Nyos looked peaceful, but its deep water was loaded with dissolved carbon dioxide, like a sealed bottle of fizzy drink. One August night it overturned and released a vast inv…
Water lasts longer on a very hot pan than on a warm one.Flick water onto a pan just above boiling and it hisses away at once. Make the pan much hotter, around 193°C, and the drops bead up and skate about for far longer. They are hoveri…
A tiny imaginary doorkeeper nearly broke the laws of physics.In 1867 James Clerk Maxwell imagined a being that sorts fast molecules from slow ones through a tiny door, making one side of a box hot and the other cold without doing any work. …
Can hot water freeze faster than cold? Physicists still disagree.In 1963 a Tanzanian schoolboy, Erasto Mpemba, noticed that his hot ice cream mixture froze before a cooler one. A visiting physicist doubted him, tested it and was surprised. Six …
A glass teardrop that shrugs off a hammer but explodes if you snip its tail.Drip molten glass into cold water and you get a tadpole-shaped bead. Hit its round head with a hammer and it survives. Nick its thin tail and the whole thing bursts into powder in…
Nuclear tests almost doubled the radiocarbon in the air we breathe.Radiocarbon dating works because every living thing absorbs a trace of radioactive carbon, then stops when it dies. The clock needs careful correcting, though. Burning coal dilute…
Schrödinger's famous cat was meant as a joke at quantum theory's expense.Everyone knows the cat in the box that is both alive and dead until you look. Few know that Erwin Schrödinger invented it in 1935 to mock that idea, not to promote it. His absurd …
Earth may once have frozen over, all the way to the equator.About 700 million years ago, glaciers seem to have reached sea level in the tropics. Many geologists think the whole planet froze into a 'snowball' for millions of years, and that…
Pure water can stay liquid far below zero, down to about −48°C.Water does not freeze just because it hits 0°C. Ice needs a seed to start growing on, such as a speck of dust or a scratch. Take the seeds away and water stays liquid far below fr…
The power plants in your cells were once free-living bacteria.Mitochondria, which power our cells, and chloroplasts, which let plants photosynthesise, have their own DNA and divide on their own schedule, like bacteria. That is because they o…
The half-millimetre animal that survived the vacuum of space.Tardigrades, or 'water bears', can dry out, curl into a husk called a tun and wait. In 2007 scientists sent some into orbit and exposed them to open space. Many came back, rehydra…