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.
The logarithm
A positive height of the Euler e-function comes from one real input. For positive numbers the logarithm of a product is a sum, and near zero it falls slowly.
The general exponential function
For a positive base, the general exponential function is the Euler e-function at a scaled input. Off base 1, its logarithm is a quotient of natural logarithms.
Sine and cosine
Sine and cosine are the imaginary and real parts of the exponential at an imaginary input. A real angle reads them as coordinates on the unit circle.
Tangent and cotangent
Tangent and cotangent are the quotients of sine and cosine. With the adjacent side frozen at 1, tangent is the opposite side. On the unit circle they are cuts.
Tangent zeros, poles, and period
Simple zeros and poles of tangent repeat after a shift by pi. Cotangent swaps the lattices. Both are odd, and the slopes are 1/cos^2 and -1/sin^2.
Tangent addition formulas
The tangent of a sum is not a sum of tangents. Drag two angles to match the formula, then freeze them for the double angle and the half-angle.
Hyperbolic sine and cosine
Hyperbolic sine and cosine split the exponential into two heights. Drag the real input and see both graphs, then the identity on x^2 - y^2 = 1.
Hyperbolic addition formulas
Hyperbolic addition reads sinh and cosh from a product of exponential heights. Drag two inputs, then freeze them, and the half-parameter changes sign.
Hyperbolic tangent and cotangent
Tanh and coth are the quotients of sinh and cosh. On the real line tanh stays strictly between −1 and 1; coth has a pole at 0. Slopes are 1/cosh² and −1/sinh².
Hyperbolic tangent addition
Tanh of a sum uses the denominator 1 + tanh x tanh y, where tangent uses a minus. Drag two real inputs on the right branch, then freeze them.
Inverse trigonometric functions
Each inverse trig function is the unique angle on a principal branch. Reflect the restricted graph across y = x, then solve equations by adding the period.
Inverse trigonometric identities
On the principal branches, arcsin and arccos of one number add to a right angle. Arctan and arccot do the same. Slopes are 1/√(1−x²) and 1/(1+x²).
Inverse hyperbolic functions
Arsinh needs no freeze on the real line. Arcosh freezes cosh to [0, ∞). Artanh and arcoth reflect tanh and coth, then each inverse also has an explicit ln formula.
Inverse hyperbolic identities
On the principal branches, one horizontal step on arsinh rises by 1/√(1+x²). Artanh and arcoth share 1/(1−x²) on complementary domains. Each branch also rewrites into the others.
Polynomials
A polynomial is a finite sum a_n x^n + … + a_0 with a_n ≠ 0. Far away it matches a_n x^n; odd degree meets the axis at least once; even degree has a global min or max; local extrema and inflection points stay bounded by degree.
Rational functions
A rational function is a quotient of polynomials. From y=b/x to even or odd powers, a smooth or two-pole denominator, and far-away height c x^{n-m}.
Empirical data for sequences of measurements
A measurement sequence has an empirical mean and an empirical standard deviation with n−1. Same mean can hide different spread; a histogram reads relative frequencies by class.
The theoretical distribution function
Φ(x) = P(X < x) is the left-tail law of a random variable. Under a normal density, area is probability, and μ ± σ z_α is a window of probability 1 − α.
Checking for a normal distribution
On probability paper, Table 0.29 makes Φ a straight line in z. Standardize with x̄ and Δx, and call X approximately normal when empirical points hug that line.
The statistical evaluation of a sequence of measurements
When a normal series has unknown μ and σ, the mean lies in x̄ ± t·Δx/√n and σ² lies in a χ² interval. Eight heights give 169 ≤ μ ≤ 181 and 2.8 ≤ σ ≤ 11.4.
Empirical correlation and regression
Equal-length pairs give a correlation ρ between −1 and 1, and the line y = ȳ + ρ(Δy/Δx)(x − x̄). Eight height pairs give ρ = 0.96.
Comparing two means with t and F tests
An F-ratio checks whether two normal spreads match; the two-sample t then compares the means. Medicine A and B give F = 1.6 < 3.7 and t = 3.6 > 2.8.
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.