Interactive Math & Science Lessons

Four 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.

1. 01Cross-multiplication
2. 02Weighted average
3. 03Mean vs median
4. 04Spread and deviation

Six 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.

1. 01Area and Perimeter
2. 02Similar triangles
3. 03Pythagoras
4. 04Solving any triangle
5. 05Circle angle theorems
6. 06The unit circle
7. 07Coordinate geometry

Three 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.

1. 01Adding fractions
2. 02Repeating decimals
3. 03Rational approximation
4. 04Rounding

Six 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.

1. 01Negative numbers
2. 02Linear equations
3. 03Inequalities
4. 04Rearranging formulas
5. 05Two equations at once
6. 06Three or more at once

Four 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.

1. 01Sequences & series
2. 02Simple vs compound
3. 03Annuity future value
4. 04Loan payment
5. 05Inflation Adjuster

Three 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.

1. 01Logarithms
2. 02Scale of the universe
3. 03Fermi estimates

The 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.

1. 01Modular Arithmetic Clock
2. 02Birthday paradox
3. 03Hash tables

Four 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.

1. 01Expanding and factorising
2. 02Quadratic solver
3. 03Complex Number Calculator
4. 04Polynomial division
5. 05Rational functions

An 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 and the compound-angle formulae, and is building towards the double-angle and harmonic forms and proving identities.

1. 01Pythagorean identities
2. 02Inverse trig functions
3. 03Trig equations
4. 04Compound angles

Three 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.

1. 01Sine & cosine waves
2. 02Circular motion
3. 03Pendulum & SHM
4. 04Fourier Series Builder

Five 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.

1. 01Radians
2. 02Limits
3. 03The derivative
4. 04Riemann Sum Explorer
5. 05Taylor series
6. 06More than one direction

A 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.

1. 01Proof by induction
2. 02The pigeonhole principle
3. 03Invariants
4. 04The pentagonal number theorem

A 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.

1. 01Product rule
2. 02Chain rule
3. 03Implicit differentiation
4. 04Related rates
5. 05Optimisation

Seven 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.

1. 01Vectors
2. 02Least-squares geometry
3. 03Linear regression
4. 04Gradient descent
5. 05Overfitting
6. 06k-means
7. 07Calibration

One 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.

1. 01Matrix transforms
2. 02Eigenvectors
3. 03Singular value decomposition
4. 04Principal components

Four 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.

1. 01Prisoner's dilemma
2. 02Nash equilibrium
3. 03Tragedy of the commons
4. 04Evolution of cooperation

An 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.

1. 01Series convergence
2. 02Riemann rearrangement
3. 03Radius of convergence

Newton'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.

1. 01Newton's method
2. 02Integrating numerically
3. 03Euler's method

Differentiation 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.

1. 01Antiderivatives
2. 02Substitution
3. 03Integration by parts
4. 04Partial fractions
5. 05Trigonometric integrals
6. 06Choosing a technique
7. 07No formula at all

An 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.

1. 01The fundamental theorem
2. 02Area between curves
3. 03Accumulation and motion
4. 04Volume by slicing
5. 05The shell method
6. 06Combining two solids
7. 07Double integrals
8. 08Polar double integrals

Ohm’s law, as its own page says, describes one resistor at a steady temperature, on direct current, with nothing else in the loop. Each page after it puts back one of the things that sentence left out: a second component, the wire, the battery, a circuit that will not reduce, and a voltage that changes.

1. 01Ohm’s law
2. 02Series and parallel
3. 03Resistivity
4. 04Internal resistance
5. 05Kirchhoff’s laws
6. 06AC and RMS

Two tools and one shortcut: when all you want is speeds and heights, you can skip the forces entirely and let energy keep the books.

1. 01Free fall
2. 02Roller coaster
3. 03Momentum & Collision Lab

A lens is a boundary that somebody curved, and no page on this path introduces physics the first one does not already contain. What changes is how much is asked of it: one surface, then an ideal lens, then a piece of glass.

1. 01Snell's law
2. 02Ray-tracing a lens
3. 03Real lenses

Five machines and one bargain. None of them creates energy, and every one of them lets you choose how to spend it: a smaller force over a longer path, or the reverse. Each step hands the next a specific piece of that bargain, and by the end you can read any machine by asking two questions about it.

1. 01Lever classes
2. 02Wheel and axle
3. 03Pulley systems
4. 04Inclined plane, screw and wedge
5. 05Gear trains
6. 06Screw Thread

A capacitor has no resistance and limits the current anyway, and what it sets is a time rather than a number of ohms. The second page shows that the time belongs to any system with one store and one drain. The third adds a second store, and the time becomes a frequency.

1. 01RC circuit
2. 02Step response
3. 03RLC resonance

Three tools, and the third one breaks the first two. Constant acceleration has a closed-form answer, so the first two steps are algebra. Air resistance is not constant, and from there the only honest method is to integrate it step by step.

1. 01SUVAT
2. 02Projectile motion
3. 03Drag and wind

Two tools built on a single result. Water leaves a hole at exactly the speed a stone reaches falling the same depth, and everything else follows from where that speed is spent: sideways it is a projectile, downwards it is the tank emptying itself.

1. 01Three holes in a tank
2. 02The tank emptying

One restoring force, followed as far as it will go. A mass on a spring oscillates, a row of them passes the oscillation along, two of those meeting decide whether to add or cancel, and a boundary at each end throws away every frequency that does not fit.

1. 01Hooke's law
2. 02Resonance
3. 03Interference
4. 04Standing waves

Ray optics says light goes straight and an edge casts a sharp shadow. Each page here draws that answer as a ghost beside the wave answer, and what stands in the light's way changes from page to page: two surfaces, one slit, a round aperture, and finally an opaque disc whose shadow is bright at its centre.

1. 01Thin films
2. 02Single slit
3. 03The diffraction limit
4. 04The spot of Arago

The path a light ray takes, a sliding bead follows and a thrown ball flies is the one along which a small change makes no first-order change in a total: time for the first two, action for the ball. Every page lets you bend the path and watch the total barely move, and two of them find a path whose total is stationary without being least.

1. 01Fermat's principle
2. 02Mirages
3. 03The brachistochrone
4. 04Least action

A spinning body carries a vector, its angular momentum, and nearly everything surprising about rotation is that vector refusing to change. Pull your arms in on a turntable and you spin faster; a wheel turns sideways instead of falling; a racket flips; a pendulum's plane stays put while the floor turns under it.

1. 01Angular momentum
2. 02The gyroscope
3. 03The tennis racket theorem
4. 04Coriolis and Foucault

Ohm's law describes one kind of part. The five here store energy in a voltage or a current, conduct in one direction, amplify, and amplify precisely, and each is unreliable on its own: a transistor's current gain β varies threefold between samples of the same type. The circuit around a component is what makes it predictable, and every page shows how.

1. 01Capacitor
2. 02Inductor
3. 03Diode
4. 04Transistor amplifier
5. 05Op-amp

Light arrives in packets, and the same constant h then gives every particle a wavelength. The rest follows from taking that wave seriously: made to fit a box, it has a lowest energy that is not zero; it leaks through walls it cannot climb; and squeezed into a small space it spreads over many wavenumbers, which is the uncertainty principle.

1. 01The photoelectric effect
2. 02Matter waves
3. 03Particle in a box
4. 04Quantum tunnelling
5. 05Uncertainty and wave packets

Seven pages on one question: which part of a structure is actually carrying the weight, and in what form. It starts inside a single beam, where the middle does almost nothing, and ends with a truss that has more bars than equations and still says exactly when it will fall.

1. 01Beam deflection
2. 02Shaft torsion
3. 03Mohr’s circle
4. 04Truss analysis
5. 05Euler buckling
6. 06Arches and cables
7. 07Redundant bars

Three 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.

1. 01The heat equation
2. 02Seasons underground
3. 03The wave equation
4. 04Laplace's equation

From a gate to a running program. A NAND gate is enough to build every circuit, an adder made of gates does subtraction too, and the Little Man Computer puts that adder beside a hundred mailboxes: the program sits in the same memory as its numbers, and the machine runs whatever its program counter lands on.

1. 01Logic gates
2. 02Binary arithmetic
3. 03The Little Man Computer
4. 04Memory and bytes
5. 05The call stack
6. 06Cache locality

Each cipher here hides the letters and leaves something else showing. A substitution leaves the letter counts, a keyword cipher leaves them in columns, and a one-time pad used twice leaves the spaces. Only a random key as long as the message shows nothing at all, and then the hard part becomes getting that key to the other side, which the last two pages do in public.

1. 01Substitution cipher
2. 02Vigenère cipher
3. 03One-time pad
4. 04Diffie–Hellman
5. 05RSA

Every structure here answers one question fast and pays for it somewhere else. A Bloom filter keeps no keys at all and can still say definitely not; a trie answers the prefix questions a hash table cannot; a heap always knows its smallest key and little else. The last page asks which page to throw out of memory, where the best answer needs the future.

1. 01Bloom filter
2. 02Trie
3. 03Binary heap
4. 04Page replacement

A video encoder is not one algorithm but eight stages in a fixed order, and the order is not arbitrary: each stage exists because the one before it made its job possible. This tool models one of them. The chain below links to the other seven.

1. 01Sampling
2. 02Chroma subsampling
3. 03GOP structure
4. 04Motion estimation
5. 05Quantization
6. 06Entropy coding
7. 07Rate control
8. 08Size budget

Two halves of the same problem, in the order a studio meets them. First, a key has leaked and the licence must keep working for everyone else — which turns out to cost a header that does not grow with the size of the system at all. Then the harder half: knowing whose key it was, from a copy that several subscribers built together specifically so that it names nobody.

1. 01Device revocation
2. 02Forensic watermarking

Two tools that open up a large language model at the moment it chooses a word. The second literally runs on the first: the candidates it ranks are entries from the vocabulary the first one builds.

1. 01Tokenizer
2. 02Token Vectors and Similarity
3. 03Temperature & sampling

A stopwatch measures your laptop. Counting operations measures the algorithm. Two tools built on that one habit.

1. 01Big-O growth
2. 02Sorting Algorithm Race
3. 03Recursion Tree Explorer
4. 04Pathfinding

A signal is a sum of spinning arrows, and every practical way of reading them off costs something the textbook formula leaves out. A finite window smears a tone that does not fit it across every bin; a shorter window buys time and pays in pitch; an average is a filter with a shape of its own; and the last page is the reason any of it is affordable.

1. 01Discrete Fourier transform
2. 02Spectrogram
3. 03Moving average as a filter
4. 04Fast Fourier transform

An algorithm is right for a reason that can be stated, and wrong for one a small test can find. Each page sets a correct version beside a plausible wrong one: fourteen of the sixteen ways to write binary search fail, greedy change-giving loses on old British coins and never on US cents, Kruskal and Prim agree for a reason, and a code that corrects one flipped bit makes two flips worse.

1. 01Binary search
2. 02Coin change
3. 03Minimum spanning tree
4. 04Hamming code

Four encoders you could run on paper, ordered by what each one can see that the last one could not. The fourth compresses nothing whatsoever, and it is the reason the first one works on text.

1. 01Run-length
2. 02Huffman
3. 03LZ77
4. 04Burrows-Wheeler

One exponential, read three ways. Forwards it tells you how much is left; backwards it tells you how old something is; and applied to DNA rather than to nuclei it needs a rock to set it by.

1. 01Radioactive decay
2. 02Radiometric dating
3. 03Molecular clock

On 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.

1. 01Great circle routes
2. 02Parallel transport
3. 03Theorema egregium
4. 04The hyperbolic plane

The heat a reaction gives out does not decide whether it runs; ice melts by taking heat in. These four pages build the bookkeeping that does decide: enthalpy, which adds like equations; standard free energy, which is an equilibrium constant in other units; the difference between moving K and moving Q; and a battery that reads that difference as a voltage.

1. 01Hess's law
2. 02Gibbs free energy
3. 03Le Chatelier's principle
4. 04The Nernst equation

A gas in a mixture acts at its own partial pressure, which sets a diver's depth limit. A liquid boils where its vapour pressure meets the pressure above it. Two liquids that mix badly stop a still at about 95.6% ethanol by mass, dissolved particles make one kelvin of frost worth thirteen bar of osmotic pressure, and a salt dissolves only until its ions reach a fixed product.

1. 01Partial pressure
2. 02Vapour pressure and boiling
3. 03Fractional distillation
4. 04Colligative properties
5. 05Solubility product

Every population here is held back by something, and the something gets less obliging at each step: first the crowding it causes itself, then a fixed catch taken off the top, then a predator that is a population in its own right, and finally running out of people left to infect.

1. 01Logistic growth
2. 02Harvesting
3. 03Predator and prey
4. 04Epidemics

A forecast is a model run forward from a starting state nobody knows exactly, so forecasters run it many times and let the disagreement say how far to trust it. The second page opens the model and finds a way to fail that has nothing to do with chaos. The third goes back to Lorenz's equations of 1963, where the run's position is soon lost and the shape it draws never is.

1. 01Ensemble forecasting
2. 02Weather on a grid
3. 03The Lorenz attractor

Five tools that build one argument: DNA is a real storage medium and a real computer, and both run into limits you can calculate rather than guess at. Each step hands the next a specific number.

1. 01DNA data storage
2. 02DNA computing scale
3. 03Levinthal’s paradox
4. 04Strand displacement gate
5. 05Energy per operation

Six tools and one distribution. Everything here is the binomial in a different limit, or the binomial read backwards: count the successes, let the count get large, and watch a bell curve arrive that nobody put there.

1. 01Binomial
2. 02Poisson
3. 03Normal distribution
4. 04Central limit theorem
5. 05Z-score
6. 06Confidence interval

Four tools asking one question with progressively less to work with: two groups, then any number of them, then counts with no average in them at all. The last tool asks it before the data exist.

1. 01t-test
2. 02ANOVA
3. 03Chi-square
4. 04Sample size

Nobody objects to the arithmetic in Bayes' theorem. They object to the prior — where the first number came from, and who chose it. These three tools take that question in the order it gets harder: a prior you can look up, a prior the apparatus hands you, and a prior nobody has.

1. 01Bayes' theorem
2. 02Bayes' billiard
3. 03Biosignatures

A p-value answers one question about one planned test. Real studies run twenty tests, look at the data as it comes in, and publish the results that came out significant, and each of those changes what p < 0.05 is worth. Every page here takes a procedure that sounds harmless and works out exactly what it costs.

1. 01Multiple comparisons
2. 02Optional stopping
3. 03The winner’s curse
4. 04False findings

An expected value is a promise about the average over a great many players, and a gambler is one player with one bankroll. A game can be fair, or even in your favour, and still ruin almost everyone who plays it. Each page draws the average as a dashed ghost and lets you watch what happens to the players who make it up.

1. 01Gambler's ruin
2. 02The doubling system
3. 03The St Petersburg paradox
4. 04The Kelly criterion

Six tools on the same 98 years of returns, each handing the next a specific quantity. The sequence answers one question in six steps: what is actually risky about an investment, and how much of what you are told about it can be checked?

1. 01Where the risk is
2. 02Drawdown and drag
3. 03Order matters
4. 04Rebalancing
5. 05How sure can you be
6. 06The frontier, held

Nothing about a star can be worked out from how bright it looks until you know how far away it is. That is the order these four go in, and it is the order astronomy went in — each rung measured against the one below it, all the way from a surveyor’s triangle to the edge of the observable universe.

1. 01Parallax distance
2. 02Solar luminosity
3. 03Stellar magnitude
4. 04The distance ladder

An orbit is fixed by its size and its shape, and almost everything else follows. These five tools go from the ellipse to the fuel: where a spacecraft is, how fast it is going there, how long the trip takes, which trip is cheapest, and what that costs in propellant.

1. 01Kepler orbits
2. 02Vis-viva
3. 03Orbital period
4. 04Hohmann transfer
5. 05The rocket equation