Matpic
mathematics, in pictures
Concept

Unit — where it appears

An element of a number system with a multiplicative inverse inside that same system, as 1 and −1 are among the integers. Among numbers a + b√D the units are the solutions of x² − Dy² = ±1, generated by −1 and one fundamental unit.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Whole-number points on x² − 2y² = 1. The branch of the hyperbola x² − 2y² = 1 in the first quadrant, with the whole-number points on it marked and labelled, and the lattice drawn faintly behind.

One solution that makes all the others

The equation x² − 2y² = 1 has infinitely many whole-number solutions, and every one of them is a power of the smallest. The multiplication that produces them is what multiplying two numbers of the form a + b√2 comes to when the √2 terms are collected — so an equation about a hyperbola turns out to carry a group.

number · Pell Bhāskara's cyclic method on x² − 61y² = 1, step by step. A table of the cyclic method's rows: the helper m chosen at each step and the near miss a² − Db² = k it produces, ending at k = 1 with the fundamental solution.

A method that is allowed to miss

Bhāskara's cyclic method solves x² − Dy² = 1 by aiming at the wrong target. It keeps a pair a, b with a² − Db² = k for some small k, combines it with a helper chosen so that k can be divided out, and repeats until k is 1. For D = 61 it reaches the ten-digit fundamental solution in 13 steps, where walking the convergents of √61 takes 22 — and for every D up to 100 it is faster.

number · Pell The integers of ℚ(√5), with ℤ[√5] inside them. Points a + bφ plotted against their conjugates for small whole a and b, with the index-two sublattice ℤ[√5] filled and the basic cells of both lattices shaded, of areas √5 and 2√5.

The integers a field contains

Inside the field of numbers a + b√5, the obvious integers are those with whole a and b. They are not all of them: the golden ratio has a one-half in it and satisfies x² = x + 1, a monic equation with whole coefficients, exactly as an integer should. The right integers form a lattice twice as dense as the obvious one — and a whole-number matrix proves they are closed under addition.

algebra · Field extensions The units of a cubic field, as a lattice of logarithms. Points for 66 units of the field of a root of x³ − 3x + 1, plotted by the logarithms of two of their conjugates, lying on the lattice spanned by the logarithms of θ and θ − 1.

Units that form a lattice

In the whole numbers only 1 and −1 have whole-number reciprocals. In the integers of a bigger field there can be infinitely many such units, and they are not scattered: take logarithms of their sizes under each way of placing the field in the real or complex numbers, and the units land exactly on a lattice. How many dimensions that lattice has is a count of those placements, and the area of its cell is a number no formula gives.

algebra · Field extensions

Named alongside it

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

Algebraic integerContinued fractionsConvergentDeterminantField extensionFundamental solutionLatticePell equationAlgorithmCharacteristic polynomialGolden ratioHyperbola

All concepts