Elementary formulas next to figures you can drag. Open a construction, then keep it on the board or ask in chat.
Measuring angles
Why a turn is 2π, not only 360°. Signed angles, one radian as an arc equal to the radius, then a triangle that already sums to π. Drag the live board.
Quadrilaterals: area and perimeter
Square, rectangle, parallelogram, and rhombus as one area formula with extra freezes. Drag a vertex on the live board, then keep the figure or ask in chat.
Triangles: area and perimeter
One triangle area ½ah = ½ab sin γ, then freeze a right angle or three equal sides. Drag a vertex on the live board, then keep the figure or ask in chat.
Circles, sectors, and ellipses: area and circumference
Circle, sector, and ellipse as one radius, then a wedge, then a stretch. Drag a point on the live board, then keep the figure or ask in chat.
Regular n-gons: area and perimeter
One regular n-gon from the circumradius r: n isosceles triangles at the centre, then freeze n at 3, 4, 5, or 6. Drag a vertex on the live board.
Prism, cylinder, box, and ball
Five solids as base area times height, then a ball that is not an extrusion. Drag a point on the 3D board, then keep the figure or ask in chat.
Pyramid, cone, and frustum volume
A pyramid has one-third the volume of the prism with the same base and height. Drag a point on the solid, and a cone or a frustum follows the same cut.
Spherical cap, torus, and ellipsoid
Volume of a spherical cap, zone, torus, barrel, and ellipsoid, plus the ellipsoid surface from Legendre's formula. Drag a point on the board.
Volumes and surface areas of regular polyhedra
Five regular polyhedra from one edge a: Euler c−e+f=2, then freeze the face at each vertex. Drag an edge on the 3D board.
Lines in the plane
Cartesian coordinates name a point in the plane. The distance to a second point is a line you can drag, including a line with no slope.
Circles in the plane
A circle is every point at one fixed distance from a center. Drag the center or a point on the Cartesian graph and the tangent stays perpendicular to it.
Ellipses in the plane
An ellipse keeps one constant sum of distances to two foci. Drag a point on the Cartesian graph and that sum stays the same.
Hyperbolas in the plane
A hyperbola keeps one constant difference of distances to two foci. Drag a point on the Cartesian graph and the branches stay beside their asymptotes.
Parabolas in the plane
A parabola is every point equally far from a focus and a directrix. Drag a point on the Cartesian graph and that equal distance stays.
Polar coordinates and conic sections
Polar coordinates (r, φ), the focus-directrix ratio ε, and the polar equation of a conic. Drag a point on the live board, then keep the figure or ask in chat.
Points, lines, and planes in space
Name a point by (x, y, z), read distance as a space diagonal, then a line as time t — or Ax+By+Cz=D for a plane. Drag a point on the live 3D board.
Powers, roots, and logarithms
Why exponents add for one positive base, why only the nonnegative root is kept, and why a logarithm is that curve read backwards. Drag the base on a live board.
Finite geometric and arithmetic series
Pair 1 through 40 by matching ends, then sum a finite row with a constant difference or a constant ratio. Drag the last term or the ratio on the live board.
The binomial theorem
The coefficient of each a^{n-k}b^k in (a+b)^n, read off a square and one Pascal row. Drag a side or a coefficient on the live board.
Sums of powers and Bernoulli numbers
Add the kth powers from 1 through n and get one polynomial in n. Bernoulli numbers fix the later coefficients. Drag n or the exponent on the live board.
Euler numbers
Odd reciprocal cubes alternate and still settle. Euler numbers from 1/cosh x on |x| < π/2 name that constant. Drag the partial sum on the board.
Important inequalities: the triangle inequality
Drag two complex numbers and keep |z−w| between ||z|−|w|| and |z|+|w|. Three steps make the broken path at least the straight modulus.
Important inequalities: the inequality for means
Drag two positive numbers and keep the harmonic, geometric, arithmetic, and quadratic means in that order between the smaller and the larger.
Important inequalities: Bernoulli's inequality
Drag n and x with x at least −1 and keep (1+x)^n on or above the line 1+nx. At n=3 and x=−4 the cubic drops below that line.
Important inequalities: Young's inequality
Young's inequality keeps |ab| at most |a|^p/p + |b|^q/q for conjugate exponents above 1. Exponent 2 recovers the two-square bound.
Important inequalities: Hölder's inequality
Drag two planar vectors. The absolute pairing stays under the product of the Euclidean lengths, and a conjugate exponent replaces those lengths.
Important inequalities: the Minkowski inequality
Drag two planar vectors and an exponent at least 1. The p-length of the sum stays under the sum of the p-lengths. The largest coordinate is the infinite case.
Important inequalities: Jensen's comparison of p-norms
Drag three real coordinates and two exponents. On a finite tuple the larger exponent gives the smaller length. On an interval of length 1 that readout grows.
Important inequalities: the convex inequality
Square an average of two real numbers, or average the squares. A convex function keeps the weighted average smaller. A nonnegative density does the same.
Important inequalities: the dual inequality
A dual function records the greatest surplus of an inner product over a given real function. For |x|^p/p that height is |y|^q/q, Young's inequality on the line.
Real functions and their graphs
The graph of a real function is the set of points (x, f(x)). Drag x along y=2x+1, watch a flat step break a strict rise, and flip y=x^2 for x≥0 into y=√x.
Transformation of functions
This transformation shifts the vertex of y=x^2 from (0, 0) to (1, 1), stretches it to y=2x^2, packs sin x into sin 2x, and reflects e^x across an axis.
Quadratic functions
A quadratic function moves the vertex of y=ax^2 to (-b/a, -D/a), where D=b^2-ac. For a>0 the sign of D decides two axis cuts, one touch, or a complex pair.
The power function
For integer n at least 2, y=ax^n matches the bowl of y=ax^2 when n is even and the crossing of y=ax^3 when n is odd. The sign of a turns either picture over.
The Euler e-function
The Euler e-function is the series for every complex e^x. On the reals it passes through height 1, outgrows every power, and its slope equals its height.
This chapter is the elementary layer: constants, plane and solid figures, series, and integral tables. The figures are live constructions, not stills.
Live topics so far: measuring angles, quadrilateral area and perimeter, triangle area and perimeter, regular n-gon area and perimeter, circle, sector, and ellipse area, prism, cylinder, box, and ball, pyramid, cone, and frustum volume, spherical cap, torus, and ellipsoid volume, regular polyhedra volume and surface, lines in the plane, circles in the plane, ellipses in the plane, hyperbolas in the plane, parabolas in the plane, polar coordinates and conic sections, points, lines, and planes in space, and powers, roots, and logarithms — each on a board you can drag.