Learn by moving things
Math Lab
Drag, type and step through every idea.
Long & short division
Share a number out, one digit at a time.
1 6 ) 7248 6 1
How many 6s fit in 7? 1. Write 1 on top. 1 × 6 = 6; 7 − 6 = 1. Then bring down the next digit.
Calculus
Slopes (derivatives) and areas (integrals).
- Secant slope
- 4.0000
- Derivative f′ = 2x
- 3.0000
- Rectangles (6)
- 3.6924
- Exact ∫ = [x³/3]
- 3.7917
Derivative = slope = rise ÷ run. The pink line joins two points h apart: slope = [f(x+h) − f(x)] ÷ h. Shrink h towards 0 and it becomes the amber tangent. That limit is f′(x).
Integral = area under the curve (from x = −2 here). Add up thin rectangles, width × height. More, thinner rectangles approach the exact area, F(x) − F(−2), where F′ = f.
Vectors
Arrows you can add, subtract, stretch and dot.
- a
- (3, 1) |a| = 3.16
- b
- (1, 2) |b| = 2.24
- Result
- (4, 3)
- Angle between
- 45.0°
A vector is an arrow: a length and a direction, written as components (x, y). Add by putting arrows tip to tail (add the components). Subtract by adding the reverse. Scale by multiplying each component by k. The dot product a·b = aₓbₓ + a_yb_y = |a||b|cos θ measures how much they point the same way: zero means perpendicular.
Cosine & bell curves
Waves, Gaussians and the raised-cosine bell.
- Cosine period 2π/k
- 6.283
- Bell peak 1/(σ√2π)
- 0.399
- Within ±1σ
- 68.3%
- Within ±2σ
- 95.4%
y = A cos(kx − φ). A is the height of a crest. k is how tightly the waves are packed (wavelength 2π/k). φ slides the wave sideways.
Bell curve e^(−(x−μ)²/2σ²)/(σ√2π). μ is the centre, σ the spread. The shaded middle holds 68% of the area. Total area is always 1.
Cosine bell ½(1 + cos(π(x−μ)/w))/w: one hump of a cosine lifted to sit on zero, here w = 2σ. It looks like a bell but ends sharply, so it’s used to smooth signal edges.
Matrices
Add, subtract and multiply, cell by cell.
Row 1, column 1: 1·5 + 2·7 = 19
Tap any answer cell to see how it was made. det A = -2, det B = -2.
Adding and subtracting go cell by cell: the top-left of A plus the top-left of B, and so on. Both matrices must be the same size.
Multiplying is row times column. For each answer cell, run along a row of the left matrix and down a column of the right one, multiply the pairs and add them up. Switch to B × A: the answer usually changes, so order matters.
Differential equations
Slope fields and step-by-step solvers.
- Exact y(4.0)
- 3.0055
- Euler (pink)
- 3.0004
- RK4 (green)
- 3.0055
- Euler error
- 5.1e-3
Linear first-order: y = x − 1 + Ce⁻ˣ. Every solution is pulled onto the line y = x − 1.
A differential equation tells you the slope at every point; the little dashes show it. A solution is a curve that follows the dashes. Tap the field to start somewhere else. Euler walks straight along the slope for a step of h, so it drifts. Runge–Kutta 4 samples the slope four times per step and stays on the white exact curve. Halve h and Euler’s error roughly halves, while RK4’s drops about 16×.