Matpic
mathematics, in pictures
Concept

Inverse

The element that undoes another under an operation, returning the identity. Modulo a whole number it exists exactly for the residues coprime to the modulus, and the extended Euclidean algorithm finds it in a few steps.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

eˣ and its inverse, reflected in the diagonal. A curve, the line y = x, and the curve reflected in it — which is the graph of the inverse function. Tangents are drawn at matched pairs of points, and the two slopes at each pair multiply to one.

The slope of the mirror image

Undoing a function is reflecting its graph in the diagonal, and a reflection turns a slope into its reciprocal. That single observation supplies the derivative of every inverse — the logarithm, the roots, the inverse trigonometric functions — without differentiating any of them.

analysis · The derivative Four outputs are enough to find the rule. A row of 12 outputs of a linear generator, with the first 4 marked as given and the rest as predicted. The multiplier and increment recovered from the given ones reproduce every later output exactly.

Four numbers and the rule is yours

A linear generator can be solved. Given a few of its outputs, the multiplier and the increment fall out of two congruences, and every future output is then known exactly — which is a failure of a completely different kind from the lattice defect, and is not detected by any test of how evenly the points are spread.

computation · Pseudorandomness Powers of a matrix that shrink in the end. Three curves of the norm of the n-th power of a two-by-two matrix against n on a logarithmic axis: one decays steadily, two rise to peaks of about 18 and 7 before decaying.

A geometric series whose ratio is a matrix

1 + r + r² + … adds to 1/(1 − r) when r is smaller than one. Put a matrix in place of r and the same formula holds, with the inverse matrix in place of the fraction — but what must be smaller than one is not the matrix's size. It is its largest eigenvalue. A matrix whose eigenvalues are 0.9 and 0.8 can stretch vectors ten times over before its powers begin to shrink, and the series still converges, after a detour the eigenvalues say nothing about.

analysis · Geometric series Secants closing on the slope of the folium x³ + y³ = 3xy. The curve x³ + y³ = 3xy with secants from (1.001, 0.348) of slopes 1.274, 0.953, 0.830, 0.777, approaching the tangent slope 0.744 given by the equation's partial derivatives.

A slope for a curve that is no function

The folium x³ + y³ = 3xy loops back over itself, so no formula y = f(x) describes it, and yet at almost every point it has a perfectly good tangent. Differentiating the equation as it stands gives the slope, −(∂F/∂x) ÷ (∂F/∂y), and the only points where that fails are the ones where the curve turns vertical or crosses itself — which are exactly the points where it stops being a graph.

analysis · The derivative Walking a figure eight, and the sine it makes. Bernoulli's lemniscate with 24 equally spaced stations, and the graph of the distance from its centre against arc length walked: the lemniscatic sine, period 2ϖ ≈ 5.2441, beside an ordinary sine of the same period.

The sine of a figure eight

Walk round a circle and read the height against the distance walked, and the reading is the sine. Do the same on Bernoulli's figure eight and the reading is a new function, with an arc integral that has t⁴ where the circle's has t², a length Gauss found inside an average, two periods instead of one — and exactly the same list of equal divisions that compass and straightedge can draw.

analysis · Circular functions

Named alongside it

The objects these essays reach for when they reach for this one.

ApproximationDerivativeLimitTangencyArc lengthComplex numbersContinuityConvergenceCryptographyEigenvalueElliptic functionGeometric series

All concepts