Interactive Math & Science Lessons
Fix a few rules and the consequences stop being up to you. What follows is rarely what you would have guessed, and the tools here exist to show you which consequence you are stuck with.
Short sequences rather than single tools. Each step hands the next one something specific, so they are worth doing in order.
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
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.
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
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.
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
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
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 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.
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.
L3: undergraduate
MATHThree tools that turn a picture of scattered points into a number you can defend. 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.
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.
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.
