Numbers and algebra
L1: foundations
MATHFour tools that build one lesson: a summary number is a lossy compression, and how you compress determines what the summary is honest about. Each step hands the next one a specific limitation of single-number summaries.
L1: foundations
MATHThree tools and one argument: some numbers cannot be written out in full, the best short version of one can be searched for rather than guessed, and cutting a number short has a cost you can measure.
L1: foundations
MATHSix tools that build equation-solving in the order the ideas actually depend on each other, starting from what a minus sign does and ending with a machine method for three unknowns at once. Every step hands the next one a specific thing: a move, a restriction on a move, or a test for whether a move was legal.
L2: school
MATHThree tools, one habit: when quantities differ by factors rather than amounts, stop counting and start counting exponents. The first builds the ruler, the second walks it, and the third puts a number on it with no data at all.
L2: school
MATHFour tools, and each one exists because the tool before it runs out. Factorising works when the roots are tidy, the formula works when they are not, division works past degree two, and a rational function is what two polynomials do when one sits over the other.
L2: school
MATHThree pages about applying one function after another. Shifting and stretching a graph is already that, with the order fixed by the brackets. Composition in general shows that the order changes the answer. An inverse is the function that, composed with f, gives back x, and its graph is the reflection of f's in the line y = x.
Geometry and trigonometry
L1: foundations
MATHSix tools that build the standard triangle and circle results in the order they actually depend on each other. Nothing here is asserted: every rule arrives as a consequence of the one before it, and the route ends by putting the first one back on a grid, where it turns into the distance formula.
L2: school
MATHAn identity holds at every angle where both sides are defined, which is what lets you trade the function a question gives you for the one it asks about. The chapter starts with the reciprocal functions and the three Pythagorean identities, then the inverse functions a calculator answers with, the equations they start, the compound-angle and double-angle formulae and the harmonic form, and ends with proving identities, one line at a time.
L2: school
MATHThree tools, and the point of the sequence is that they are not three subjects. A sine wave is what uniform circular motion looks like from the side, and a swinging pendulum is what it looks like when gravity supplies the turning.
Calculus
L2: school
MATHFive tools on one idea: a smooth curve, magnified far enough, is indistinguishable from a straight line. That single fact is what a derivative measures, what a Taylor series extends, and what a gradient does in more than one direction. The path starts with the angle measure that makes the arithmetic come out clean and ends on a hillside where the steepest slope beats every direction you actually measured.
L2: school
MATHA derivative is a rate, and each rule of differentiation says how rates combine: through a product, through a function inside another, through an equation nobody solved for y. The last two pages put the rules to work on the questions a course actually asks, how fast something changes at one instant and which choice is best.
L3: undergraduate
MATHThe sign of f′ says where a graph rises and falls, and the sign of f″ which way it bends. The chapter starts with the theorem that makes those readings safe, the mean value theorem, and is building towards the first and second derivative tests and reading f from a graph of f′.
L3: undergraduate
MATHAn infinite sum is a limit of finite ones, and the limit depends on things no finite sum cares about: the order of the terms, and where the nearest singularity sits, even when it is off the real line.
L3: undergraduate
MATHNewton's method, Simpson's rule and Euler's method all run on the same engine: inside a small enough interval, treat the curve as straight. Each one therefore has a failure that is not a bug, and the useful knowledge is which failure belongs to which method. Every step here hands the next a named error and the rate at which it shrinks.
L3: undergraduate
MATHDifferentiation is a procedure and integration is a search, so the trainers in this chapter grade an answer one way: differentiate it and compare it with the integrand. Substitution and parts are the chain and product rules read backwards, partial fractions covers the one large family where an elementary answer always exists, and the last page proves that for e⁻ˣ² none does.
L3: undergraduate
MATHAn integral is an amount piled up: area, distance, volume. This chapter starts with the theorem that ties piling up to rates of change, and builds from there to areas between curves, motion, and volumes found by slicing.
L3: undergraduate
MATHA differential equation says how fast something is changing, never what it is. The slope field draws that statement and shows a whole family of solutions, one through each starting point. The exponential is the family you can solve in a line, and the Laplace transform turns its rate into a point in the s-plane, where the ending can be read without solving anything.
Linear algebra and data
L3: undergraduate
MATHSeven tools that turn a picture of scattered points into a number you can defend, then ask how the number is found, how much of it to believe, what happens when nobody tells you the answer — and finally how to grade a prediction that was only ever a probability. The route runs through geometry rather than statistics, because the geometry is what makes the statistics honest.
L3: undergraduate
MATHOne question, asked four times with less and less to go on: what does this matrix actually do to space? The columns answer it outright. The eigenvectors answer it for the directions that survive. The SVD answers it when there are none, and PCA points the same machinery at a cloud of measurements.
Proof, counting and games
L2: school
MATHThe birthday problem looks like a party trick and is actually the standard tool for reasoning about collisions. The second step is where computer scientists meet it again, under a different name.
L2: school
MATHA proof is a reason that covers every case, including the ones nobody will ever check. These pages build that reason four ways: by induction, by counting things into boxes, by a quantity no move can change, and by pairing things off. The first begins with a formula that gives a prime forty times running and then gives 41 × 41.
L3: undergraduate
MATHFour games and one operation: find the move that answers whatever the other player does. It settles the first game on its own, takes a whole square to locate in the second, and carries a price tag in the third. In the fourth nobody computes it at all and the population arrives at the answer anyway.
Money and investing
L2: school
MATHFour tools that build one idea: money is worth different amounts at different times, and a repeated payment can be collapsed into a single formula. Each step here hands the next one something specific.
Bio, Geo, Chem
L2: school
GEOOn a curved surface the rules proved on paper stop holding, and the amount by which each one fails measures the curvature from inside. A straight route changes its compass heading, an arrow carried without turning comes home turned, a ring comes out the wrong length, and in the hyperbolic plane no triangle can hold more than π.
Waves, light and quanta
L3: undergraduate
PHYThree equations built on one operator, the Laplacian, that differ only in what time does. Heat forgets, the sharpest detail first. A wave carries a disturbance away, and on a sheet never quite finishes arriving. Laplace's equation has no time in it at all, and it is where heat forgets its way to: every point the average of its neighbours.
An antiderivative with no formula, and an area of exactly ln 2
The rules of the integration chapter get nowhere with (x − 1)/ln x. The area under it from 0 to 1 is still exactly ln 2, and the route there is three lines long once you give the integral a dial to turn.
Every rule you memorised in algebra is about the same one move
There are two moves in school algebra: add something to both sides, or multiply both sides by something. Adding has never needed a warning, and every rule you had to memorise is about the other one.
The fraction for π that nothing smaller has beaten in 1,500 years
Zu Chongzhi wrote down 355/113 in the fifth century. It is still the best fraction for π that anyone can reach with three digits over three, and the reason is a single large number sitting in π’s continued fraction.
Two countries measured the same border and disagreed by 227 kilometres
Spain put its border with Portugal at 1,214 km. Portugal put the same border at 987 km. Both had measured carefully; they had used different rulers, and a shorter ruler always finds more coast.
π is in the bell curve, and there is no circle anywhere in it
Drop needles on floorboards and you can measure π. Collide two blocks and you can count it. Neither has a circle anywhere in it. π is really about rotation and the sum of two squares; circles are just where most of us meet it first.
You typed one digit. The answer came back with six.
A calculator cannot be more precise than what you fed it, and none of them say so. Here is what the extra digits actually mean, why a reciprocal is the worst place to forget it, and why this site prints them anyway.
Halving the step size should halve the error. The tool says 1.73.
"First order" is a promise about the limit, not about the step size you are running. The gap between the promise and the measurement is itself first order, which is why an error estimate built on the promise is wrong in a predictable direction.
Newton's method can double its mistake every step, and the answer was 0.01 away
The standard way to solve an equation numerically converges so fast it feels like cheating. Give it the cube root and it walks away from the answer, doubling the distance every step, forever.
Primes thin out at a rate you can name
The density of primes near a number n is about one in ln n. That one fact tells you how many primes lie below any bound, why they never run out, and how long a computer must search to find a 1024-bit one.
Twice as tall is four times the paint, and eight times the concrete
Lengths scale as k, areas as k² and volumes as k³. Getting this wrong is probably the most expensive arithmetic mistake ordinary people make.
The rule that was the ceiling of arithmetic for a thousand years
Cross-multiplication is usually dismissed as a procedural trick. It was written down in 628 CE, carried across three continents, and was the end of the mathematics syllabus for most people who ever received one.
Every football has twelve pentagons, and it could not have any other number
Any closed cage of pentagons and hexagons with three faces at each corner has exactly twelve pentagons. Always twelve, whatever size the ball, and the proof never counts a single one.
Pythagoras is not a fact about triangles
a² + b² = c² fails on a sphere and fails the other way in hyperbolic space. What the theorem actually asserts is that the surface you are standing on is flat.
