Interactive Math & Science Lessons

Mathematics

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.

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Chapters

Short sequences rather than single tools. Each step hands the next one something specific, so they are worth doing in order.

L1: foundations

The triangle rules, and where they come from

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.

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

L2: school

Trigonometric identities and equations

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 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

L2: school

Why everything looks straight up close

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.

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

L3: undergraduate

Fitting a line to data

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.

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

L3: undergraduate

Techniques of integration

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.

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

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Calculus & Analysis

Accumulation and Motion
With velocity t² − 4t + 3 for 4 seconds an object ends 4/3 from where it started and travels 4, three times as far. Pick a velocity and compare displacement, distance and the average, and see where the average is actually reached.
Antiderivatives
The power rule for integrals is said to fail at n = −1. At n = −0.99 the textbook antiderivative of xⁿ reads 101.40 at x = 4, and 100.00 of it is the constant 1/(n + 1); what is left is 1.3959, closing on ln 4 = 1.3863. Then type antiderivatives of your own and the page checks each one by differentiating it.
Arc Length & Solid of Revolution
Same curve, same 800 slices: the arc length comes out 0.9% wrong and the surface area 0.08%. The arc integrand spikes to about 71 at the ends, where the surface one multiplies it by zero.
Area Between Curves
x³ and x enclose an area of ½ on [−1, 1], yet the integral of their difference is 0: where the curves cross, the integral must be split. Pick two curves and watch the area and the signed integral part company.
Choosing a Technique
On an exam the integral arrives without its chapter. Name your first move (the table, substitution, parts, partial fractions or a trigonometric rewrite) and the page tells you whether it reaches the table and in how many moves, from a search it runs on the integral itself. Mixed problems from the whole chapter, some needing two techniques in a row.
Combining Two Solids
Two cylinders of radius r whose axes cross at right angles share a solid of volume 16r³/3, with no π, and a sphere drilled to leave a ring of height h keeps πh³/6 however big the sphere was. Combine two solids by intersection, union or difference and find the volume one slice at a time.
Derivative & Tangent Explorer
Drag a point along any function curve and watch the tangent line and slope update live - the geometric heart of calculus.
Double Integrals
e^(y²) has no antiderivative built from elementary functions, so ∫₀¹∫ₓ¹ e^(y²) dy dx has no formula in that order; slice the other way and it is (e − 1)/2 ≈ 0.8591. Find volumes under z = f(x, y) by slicing twice, describe the region both ways, and meet the one function here whose two orders disagree.
Double Integrals in Polar Coordinates
In polar coordinates the area element is r dr dθ. Leave out the r and the unit disc's area comes out as 2π, its circumference. Keep it and e^(−x²), which has no elementary antiderivative, gives ∫e^(−x²) dx = √π over the whole line. Integrate over sectors, a cardioid, an off-centre disc and a square, one polar cell at a time.
Euler Method Visualizer
Halving the step should halve Euler's error. At x = 8 with h = 0.2 the ratio comes out 1.731285, and dragging it to 1.9933 takes h = 0.005, which is 1600 steps across the same eight units.
Fourier Series Builder
Five harmonics overshoot a square wave to 1.182, and more never fix it: the peak converges to 1.17898. A jump makes harmonics fall as 1/k and a corner as 1/k², which is why only the jump rings.
Implicit Differentiation
The folium x³ + y³ = 6xy has three points above x = 1, with slopes 1.0409, 0.3373 and −1.3782, so its slope has to depend on y. Differentiate both sides yourself, see the formula fail where ∂F/∂y = 0, and find the origin, where it reads 0/0 because the curve crosses itself.
Integration by Parts
Integration by parts trades one integral for another, and the choice of u decides which way it goes. For x²eˣ, u = x² reaches the table in two rounds while u = eˣ climbs to x⁵eˣ/60. Watch both ladders side by side, solve ∫eˣ sin x as an equation, and practise on problems that check themselves.
Integration by Substitution
Substitution turns 2x cos(x²) into cos u, and on 0 ≤ x ≤ 2 both give −0.7568. Keep x's limits after substituting and you get 0.9093, with the wrong sign. Watch the slices of the area move from the x-axis to the u-axis, then practise on generated problems that check themselves by differentiation.
Laplace Transform Lab
Study forward and inverse Laplace transforms for common functions used in ODEs and systems.
Laplace's Equation: a Plate in Steady State
Hold one edge of a square plate at 100 °C and three at 0, and ask the temperature three-quarters of the way up: the series says 54.05, and 20,000 random walkers, counting which edge they wander to first, say 54.13 ± 0.35.
Limit Explorer
Approach a limit from both sides and classify discontinuities
Non-Elementary Integrals
Why ∫e^(−x²) dx has no formula, proved rather than assumed: run Liouville's test on any polynomial times e^(−x²), count how few such integrands have a formula, and train on integrals where the right answer may be that none exists.
Numerical Integration Lab
Double the strips on eˣ over [0, 1] and the left sum's error falls by 2, the trapezoid's by 4 and Simpson's by 16. Six decimals costs 462 trapezoids or six Simpson panels.
Optimisation: the Box and the Can
Cut squares from the corners of a 30 cm sheet and the best box holds 2,000 cm³ at a 5 cm cut; cut 4 cm and it holds 3.2% less. Solve the open box and the can of least metal step by step, watch the tangent level off, and see why a depth limit puts the best box on the edge.
Parametric Curves
(cos t, sin t) and (cos 2t, sin 2t) draw the same circle and no picture tells them apart. On the Lissajous curve at t = π the speed is 2 and the radius of curvature 0.444.
Partial Derivatives and the Gradient
Walk east and the ground rises at 3, north at 4, and the steepest heading rises at 5. A road held to half that gradient must run 60° off it rather than 45°, and travel twice as far.
Partial Fractions
Integrate rational functions by partial fractions and check any answer you type: the cover-up rule, repeated and quadratic factors, one constant for every interval between poles, and x⁵ − x − 1, whose integral is elementary though its roots have no formula.
Polar Curves
A rose r = cos(kθ) has k petals when k is odd and 2k when it is even, so cos 3θ draws 3 petals and cos 4θ draws 8. The cardioid encloses 4.7124, exactly one and a half unit circles.
Product Rule and Quotient Rule
Why the derivative of uv is u′v + uv′ and not u′v′. A rectangle whose sides grow gains two strips and a corner, and the corner is the piece that vanishes. Shrink the step and watch the strips settle on the rule, read the quotient rule off the same picture, and check your own derivatives.
Radius of Convergence
1/(1 + x²) is smooth on every real number, yet its Taylor series about 0 diverges past 1: degree 80 gives 1,121.975 at x = 1.1, where the curve is 0.4524887. Drag the centre on the complex plane and watch the poles at ±i set the radius.
Related Rates
Related rates worked by hand: a ladder sliding down a wall, a cone filling, a shadow lengthening. Differentiate the relation, put in the instant, check your answer against a rate measured from the motion, and find the height at which a uniform ladder, pulled at a steady speed without friction, leaves the wall.
Riemann Rearrangement
Re-order 1 − ½ + ⅓ − ¼ + … and it adds up to π: after 100,000 terms, 99,259 positive against 741 negative give 3.1415535. Watch the same terms walk to any target, and see why the squared series cannot be moved.
Riemann Sum Explorer
Approximate integrals with left, right, midpoint, and trapezoid sums.
Slope Field & ODE Solver
Visualize dy/dx = f(x,y) slope fields and click to add solution curves with RK4.
Taylor Series Explorer
Degree 3 gives sin 1 = 0.83333 against a true 0.84147. The number that matters is 1/120 = 0.0083333, an error bound available before the answer was, and it overshoots the real error by 2.4%.
The Chain Rule
The chain rule says the rates of linked functions multiply. Carry a small interval through each link and watch it stretch, type your own dy/dx and see which slip it is, and find out why a forgotten inner factor can pass a check at one point.
The Fundamental Theorem of Calculus
The area under e^(−t²) from 0 to 1 is 0.7468, and the area function rises there at 0.3679 per unit, the height of the curve at 1, though e^(−t²) has no formula. Drag x and watch the slope of the area match the curve everywhere.
The Shell Method
Turn the arch of sin x about the y-axis and shells give 2π² ≈ 19.7392 in one line, while washers need arcsin for the same number. Compare the two slicings on four solids and see why one is often far easier.
Trigonometric Integrals
Odd powers of sine cancel over a turn and even powers pile up, which is why ∫sin³x takes one substitution and ∫sin²x needs the half-angle identity. See ∫√(1 − x²) split into a triangle and a sector on the circle that x = sin θ draws, then practise on generated problems that check themselves by differentiation.
Volume by Slicing
Turn the region between y = x and y = x² about the x-axis and the volume is 2π/15 = 0.4189; subtract the radii instead of the areas and you get 0.1047, exactly a quarter. Slide a slice through six solids and watch the volume pile up as the area under A(x).

Linear Algebra & Geometry

2D Eigenvector Explorer
Visualize eigendirections and scaling under a 2×2 linear transform.
Angle Chase
Chase an angle across overlapping triangles, then watch the same puzzle solved by elimination. Every rule is a linear equation, and the tool shows both routes.
Area and Perimeter
A trapezium bed holds 32.00 m² behind 24.94 m of edging; the same edging as a circle holds 49.51 m². Build it half again as large and compactness stays at 0.646, because scaling cancels out of it.
Bezier Curve Explorer
Drag control points to explore Bezier curves and the de Casteljau algorithm
Circle Angle Theorems
Drag four points around one circle to connect the central-angle theorem, equal angles in the same segment, Thales' theorem, and cyclic quadrilaterals.
Compass and Straightedge Constructions
With a compass and an unmarked ruler you can build a regular polygon with 17 sides, as Gauss showed at 18, but never one with 7 sides, and no one will ever trisect a 60° angle. Watch the constructions step by step, and see the algebra that forbids the rest.
Conic Parameters Explorer
Only the parabola gets a directrix row, because only the parabola is defined by one, its vertex exactly 3 from both. An ellipse with a = 4 and b = 3 puts its foci at c = 2.645751.
Conic Sections Explorer
B² − 4AC survives every rotation of the axes, because it is −4 times the determinant of the quadratic part. On 4x² + xy + 2y² = 16 it reads −31, an ellipse tilted 13.3° with eccentricity 0.737.
Coordinate Geometry Explorer
Drag two points on a grid and read the distance, midpoint, gradient and equation of the line — with the right triangle behind the distance formula drawn in.
Gauss's Theorema Egregium
Walk the same distance from one point in every direction and measure the ring you mark. On a cylinder it is exactly 2πr, on a sphere it comes up short, on a saddle it runs long: Gauss's curvature, measured without leaving the surface.
Gimbal Lock & Quaternions
Turn a three-ring gimbal with yaw, pitch and roll, watch two rings fold together, and read the quaternion that does not care. Why 90° of pitch costs a degree of freedom, and why a 360° turn leaves the quaternion at −1.
Gram–Schmidt Orthogonalisation
Build an orthonormal basis by subtracting shadows. Reorder two lines of the algorithm and, on columns 10⁻⁸ apart, the answer moves from a right angle to 60°.
Least-Squares Geometry Explorer
Drag points, compare linear/quadratic fits, and inspect residual bars and SSE impact.
Matrix Transformation Visualiser
See how 2×2 matrices transform the plane — eigenvectors, determinant, and area scaling
Parallel Transport on a Sphere
Carry an arrow round a triangle with three right angles, never turning it, and it comes home turned 90°. Drag the corners on a sphere and watch the turn match the area inside, the angle excess, and Foucault's pendulum.
PCA 2D/3D Visualizer
Project 2D data into principal-component space and inspect explained variance and scores.
Pick's Theorem: Area by Counting Points
Draw a polygon with its corners on the dots of a grid, count the dots inside and on the edge, and you have its area exactly: 53 inside and 8 on the edge make 56, with no measuring at all. In three dimensions the two counts alone cannot give the volume.
Pyramid Geometry
The Great Pyramid's π and φ coincidences are the same coincidence, a builder's seked of 11/14. Eight times it gives 6.28571 against 2π = 6.28319, and the slant ratio 1.6186 against φ = 1.618034.
Pythagoras Lab
a² + b² = c², proved from similar triangles, and then broken on purpose. A curvature slider bends the plane into a sphere or a hyperbolic surface and the hypotenuse moves — because the theorem is not a fact about triangles, it is a fact about flatness.
Reflections and the Rigid Motions of the Plane
Reflect a shape in one mirror and then in a second that crosses it at 45°, and the result is a rotation by 90°. Two parallel mirrors 2 apart make a translation by 4. Every rigid motion of the plane is at most three reflections, and this page builds rotations, translations and glides from mirrors.
Regular Tilings Lab
Regular p-gons, q of them at every corner. One inequality decides everything: (p−2)(q−2) against 4 gives a sphere, a flat plane or a hyperbolic disk — five tilings, three tilings, or infinitely many.
Secrets of Circle & Pi
See why π emerges from circumference, area, collisions, probability, and infinite series through seven interactive experiments.
Similar Triangles Lab
Scale a triangle and watch what changes. Lengths follow the scale factor, area follows its square, and the angles do not move at all — with the big triangle drawn subdivided into the copies it really holds.
Solids Lab: Sphere, Cone & Cylinder
A sphere is two thirds of its tight cylinder by volume and again by surface, because the curved wall alone equals the whole sphere. At equal volume a cylinder costs 14.5% more surface and a cone 26.0%.
The Hyperbolic Plane
Drag a triangle's corners in the Poincaré disc and watch its angle sum shrink as it grows: no triangle in the hyperbolic plane holds more than π. Through a point off a line, a whole fan of lines never meets it.
Triangle Centres & the Euler Line
Drag a triangle and watch three of its four centres stay on one straight line, with the centroid always dividing it in the ratio 2 : 1. The incentre joins them only when two sides are equal.
Vector Calculator
Compute dot product, cross product, and angle between 2D/3D vectors

Algebra & Precalculus

Compound-Angle Formulae
sin 45° + sin 30° = 1.2071, more than any sine can be, while sin 75° = (√6 + √2)/4 ≈ 0.9659. Turn the axes through A and the four products in sin(A ± B) and cos(A ± B) appear as the legs of two right triangles, with signs, in any quadrant.
Expanding and Factorising
Brackets multiplied out as the area of a rectangle, and factorising as the same picture read backwards. One test tells you in advance whether a trinomial will factorise over the integers.
Exponential Growth & Decay Explorer
Set a rate and read the doubling time, the half-life and the amount at any moment, from the one equation that covers growth and decay alike.
Function Composition Calculator
Step-by-step function composition calculator for f(g(x)) and g(f(x)) with custom functions, equations, domain warnings, and graphs.
Gaussian Elimination
Three equations in three unknowns, solved by the same elimination as two equations in two, plus a row swap. The order is what turns a puzzle into a procedure, and one choice inside it decides whether a computer gets the right answer.
How Close a Fraction Can Get
Continued fractions and the fractions they hand back: why 355/113 pins π to seven places from a three-digit denominator, and why φ resists hardest of all.
Inequalities
−4x ≥ 12 gives x ≤ −3, and at x = −3 both sides come to 12, so the endpoint is in. Multiplying by x is refused outright, because nobody knows its sign yet.
Inverse Functions Explorer
See f(x) and f⁻¹(x) mirrored across y=x, and test when restriction is needed.
Inverse Trigonometric Functions
sin⁻¹(sin 150°) is 30°, not 150°. See why arcsin, arccos and arctan each answer from a fixed range, what that range does to a negative input, and how cos(sin⁻¹ x) comes off a right triangle with its sign already decided.
Linear Equations
Four buttons, two moves underneath. Adding is always safe; multiplying is safe for every number but zero. Multiply x = 3 by 0 and you get 0 = 0, true for every x, and x has stopped being mentioned.
Logarithm Explorer
Explore y = log_b(x) with a live base slider and compare against ln(x) and log10(x).
Logical Contradiction Lab
Three speakers open on 8 possible worlds, and a sentence about other people halves the field. One that counts its own speaker leaves 6 of 8, clearing a quarter of the board where the others clear half.
Piecewise Modeling Lab
Load the bulk discount and ten units cost 9.5 while eight cost 9.6. Between 7.916667 units and the breakpoint you pay more for less, which is what every downward jump in a price list buys.
Polynomial Division Trainer
Practice polynomial long division with explicit quotient, remainder, and step traces.
Quadratic Equation Solver
Find roots of ax² + bx + c = 0 with discriminant working and complex root support.
Radical Equations Checker
√(x − 1) = x − 2 gives 3.618, and the rejected 1.382 is an exact solution of the sign-flipped twin. The radicand test everyone is taught rejects neither candidate; only the right-hand side's sign does.
Rational Functions Explorer
Set (x − 1)/(x + 2) equal to its horizontal asymptote y = 1 and you get −1 = 2, so the curve never reaches it. Other curves cross theirs freely; a vertical asymptote is the strict one.
Rearranging Formulas
Make any letter the subject of a formula, one undo at a time. It is the same procedure as solving an equation, until the letter you want appears twice and it stops being possible at all.
Reciprocal Functions and the Pythagorean Identities
sin θ = 3/5 in quadrant II gives cos θ = −4/5, where the square root alone says +4/5. Draw tan, cot, sec and csc as lengths out to the unit circle's two tangent lines and the three Pythagorean identities turn out to be one right triangle at three sizes.
Simultaneous Equations
Two equations in two unknowns, drawn as two lines. One number decides whether they cross exactly once, and the same number says how much you should trust the answer when they do.
Sine/Cosine Wave Explorer
The wave shifts by −C/B, not −C, so at B = 2 it moves half as far as the bracket suggests. Any a cos x + b sin x collapses into one wave of amplitude √(a² + b²), phase and all.
Solving Trigonometric Equations
sin(2x − 30°) = √3/2 has four solutions between 0° and 360°, not two, because the argument turns the circle twice. Every solution in an interval, in degrees or radians, shown on the graph and the unit circle, with the general solution and quadratics in sin or cos.
Transforming Graphs: Shifts, Stretches & Reflections
(x − 3)² moves the parabola to the right, √(2x − 6) starts at x = 3 and not 6, and (2x)² is the same curve as 4x². Seven parent graphs moved by y = a·f(bx + c) + d, with every marked point worked out.
Triangle Solver
Three measurements are not always three facts: a = 7, b = 9 and A = 40° describe two triangles, of area 31.3423 and 8.5424, and nothing in the arithmetic prefers either.
Unit Circle Explorer
Tangent is named after a line: extend the radius at 60° to x = 1 and it lands at height 1.7321, which is why 90° reads +∞. From sin 45° alone, an octagon floors π at 2.8284 and a 16-gon at 3.0615.
Word Problems: from Story to Equations
Hugo is 18 years older than Kai, and in 9 years their ages will add up to 54. Write the equations yourself and the page checks them against the story, names any part you left out, and solves them step by step, across fourteen kinds of problem with new numbers every time.

Discrete Math & Sequences

Golden Ratio & Fibonacci Spiral
144/89 misses φ by 5.6461 × 10⁻⁵ and Hurwitz's floor at that denominator is 5.6459 × 10⁻⁵. Every Fibonacci convergent lands on the bound, which is what the most irrational number actually means.
Invariants
Half of the arrangements of a 3 × 3 sliding puzzle can never be reached, a chessboard with two opposite corners cut cannot be tiled, and Königsberg’s bridges cannot be walked. Play each one and see the quantity no move can change.
Lottery Math
German Lotto and EuroMillions have exactly 139,838,160 possible tickets each, by completely different routes. The number that decides a ticket is the 28.6% return rate, and it is printed on no ticket anywhere.
Markov Chain Stationary Explorer
Eight steps still leave the chain 0.08893 from its stationary distribution, which is exactly 8/13, 3/13 and 2/13 and the same from every start. PageRank is this calculation run on the whole web.
Nash Equilibrium in a Two-by-Two Game
Reward the row player for nerve, 5 up to 12, and their mixing stays at 33.3% while their payoff falls from 1.67 to 1.20. What moved was the other driver, swerving 10.0% of the time instead of 33.3%.
Proof by Induction
n² + n + 41 is prime for forty values of n in a row and then equals 41 × 41. See a proof by induction as a row of dominoes, and find the one broken step in the argument that all horses are the same colour.
Scale of the Universe Explorer
Zoom from subatomic to cosmic scales and compare sizes using powers of ten.
Sequence & Series Basics
Doubling the length of an arithmetic list does not double its sum: 8 terms give 80 and 16 give 288, a factor of 3.6. On a geometric series the fraction still missing after n terms is exactly rⁿ.
Series Convergence Explorer
After 50 terms the harmonic series reads 4.499205 and it still diverges. Reaching a sum of 10 takes about 12,000 terms and reaching 100 takes 1.5 × 10⁴³, because the growth is only logarithmic.
Sigma Summation Tool
Evaluate finite sigma notation sums and inspect the partial-sum trajectory.
The Evolution of Cooperation
One percent of defectors in a world of unconditional cooperators holds 99.2% of it by generation 21, and is extinct by 155. Cooperation comes back; the cooperators do not.
The Pentagonal Number Theorem
Multiply out (1 − x)(1 − x²)(1 − x³)… and almost everything cancels: after 200 brackets only 23 of the coefficients up to x²⁰⁰ are left, all ±1, at the pentagonal numbers. Watch Franklin’s pairing prove why, and use it to compute p(n).
The Pigeonhole Principle
Pick nine numbers from 1 to 16 and one always divides another; any six people include three mutual friends or three mutual strangers. Choose the boxes and watch the pigeonhole principle force the collision.
The Prisoner's Dilemma, and Axelrod's Tournament
One round of this game has an obvious answer that leaves both players worse off. Repeat it against eight opponents and the answer changes at round four.
The Tragedy of the Commons
One herder on a shared pasture does exactly the right thing. Add a second and the overuse begins — and the effort never exceeds twice the optimum however many you add, while the share of the value that survives falls with no floor at all.

Other tools

Adding Fractions, and the Mistake
Add two fractions properly, then add the tops and the bottoms instead and watch where that lands. The second answer is wrong for a sum and exactly right for something else.
Break-Even Point Calculator
1800 of fixed cost at a price of 6 and a variable cost of 2.4 breaks even at 500 units. Cut the price 10% and you need 600; cut the unit cost by the same 10% and break-even improves only 6.25%.
Complex Number Calculator
Multiplying by i is a quarter turn and nothing else: 3 + 2i becomes −2 + 3i, the argument moving from 33.7° to 123.7° while the modulus stays 3.606. Turned, not stretched.
Cross-Multiplication Grid
Four numbers, one cross, one rule. Solve percentages, gains, discounts and everyday ratios by looking at a picture instead of recalling a formula — including the reverse case almost everyone gets wrong.
Fermi Estimate
Six numbers you had to guess, multiplied together, and an answer that lands within a factor of two. Not luck: errors that multiply add their logarithms, and independent errors that add tend to cancel.
Goalkeeper Reaction Window
A 0.20 s reaction is 28% of the flight and costs 48% of the ground the keeper could cover. From 16 m at 80 km/h he ends 0.99 m short, and the save only arrives once the shot drops to 59.1 km/h.
Hyperbolic Functions Explorer
cosh 1 = 1.543081 is the Lorentz factor at 0.7616 c. Set β = tanh φ and relativity becomes hyperbolic trigonometry: rapidities add, so two boosts of 1 give tanh 2 = 0.9640 c.
Index Laws, and Why a⁰ Is 1
The laws of exponents on one screen, with the working shown. Put the powers of a base in a column and the value of a⁰ stops being a rule you memorise.
Negative Numbers
A number line you can move along, and the three rules about negatives read off it instead of memorised. Subtracting a negative adds because subtraction was always addition of the opposite.
Newton Method Visualizer
Iterate tangents to find roots and inspect convergence behavior.
Order of Operations, and Where It Is Not Agreed
Type an expression and watch it reduce one operation at a time. Four conventions read the same line, and they do not all return the same number.
Pendulum Phase Portrait
A 1 m pendulum released at 30° swings in 2.0410 s where the small-angle formula promises 2.0061. A 10 s period needs a release within 0.182° of vertical; 20 s needs 7.2 × 10⁻⁵ degrees.
Repeating Decimals
Type any fraction and any base, and see where the digits stop or why they never do. Whether a decimal terminates depends on the base as much as on the fraction, and the rule that decides it is exact.
Rounding: the rules, and what they cost
Six rounding rules side by side, the answer that changes when you round twice, and how far a column of rounded figures drifts from its own total.
Weighted Average Lab
Understand why averaging averages yields the wrong total when groups differ in size, and how to compute the true weighted mean.
Why Angles Are Measured in Radians
A radian is defined so that the arc equals the radius times the angle, and that one choice is why the derivative of sine is cosine. Switch the unit and watch three separate formulas all pick up the same factor of 0.017453.
Related categories: Statistics & Finance, Computing, Physics