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Concept

Algebraic number

A number that is a root of some polynomial with whole-number coefficients. They include every fraction and every root of a whole number, they arrive in finite batches when sorted by the size of the polynomial that catches them, and so they can be listed — which is why almost every real number is not one.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

Looking for a polynomial with π as a root. A table of the closest an integer polynomial of each degree comes to vanishing at the number, over a bounded search.

The circle that will not square

The other three impossibilities are a number having the wrong degree. This one is a number having no degree at all — and that is a claim no finite search can establish, which makes it the one place in this field where the picture has to admit what it is not doing.

computation · Constructible numbers Every candidate for a rational square root, tried. A column for each of 2, 3, 4, 5, 6, 7, 8, 9, listing the whole numbers that divide it with their squares, and the verdict the search returns.

Which roots refuse to be fractions

The square root of two is not a fraction, and neither is the square root of three, five, six or seven. The rule behind the list turns an infinite question into a search over the divisors of a single number — and the search finishes.

number · Irrationality How closely a fraction can come, and the barrier that says no closer. Two panels at very different scales: the approximations to √2, which stay above the barrier a degree-two number obeys, and the truncations of a constructed number, which fall below every barrier drawn.

Approached too fast to be algebraic

An algebraic number of degree d cannot be approached by fractions faster than the denominator's dth power. So a number that is approached faster than that is the root of no polynomial at all — and one can be built by choosing where its decimal digits go.

number · Irrationality The algebraic numbers, arriving in finite batches. A stretch of the number line with the roots of integer polynomials marked, each at the height of the smallest polynomial that catches it, and the count of polynomials at each height.

Countable, and everywhere

The numbers a polynomial can catch arrive in finite batches, so they can be listed. They are also in every interval, however short. Being listable turns out to say nothing whatever about being sparse.

logic · Cardinality A countable structure grown by adding witnesses. Stages of a structure built from 0 and 1 by adding sums, products, negatives and roots of quadratics: 2, 4, 12, 158 elements between −3 and 3.

A countable field that passes for the line

The real numbers are uncountable, and every first-order sentence about their addition, multiplication and order is also true of a countable field inside them — the real algebraic numbers. Löwenheim and Skolem showed this is no quirk of the reals: every theory with an infinite model has a countable one, including set theory, which then contains sets it calls uncountable.

logic · Models Seven powers of √2 + ∛3 in a space of six. A table of the powers 1 to (√2 + ∛3)⁶ as coordinate vectors over a six-element basis, with the coefficients of the dependency among them: the minimal polynomial x⁶ − 6x⁴ − 6x³ + 12x² − 36x + 1.

Seven powers in a space of six

Is √2 + ∛3 a root of some polynomial with whole-number coefficients? It lives in a field of dimension six, so its first seven powers are seven vectors in a six-dimensional space and must be dependent — and the dependency, solved exactly, is the polynomial. The same count shows every sum, product and quotient of algebraic numbers is algebraic, without ever needing a formula.

algebra · Field extensions The first 600 terms of the continued fraction of ∛2. Partial quotients of ∛2 computed exactly: 1, 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3 …; the largest in the first 600 is 7451.

A cube root that looks like chance

The continued fraction of √61 repeats after eleven terms, and every square root's must. The cube root of 2 obeys no such law. Computed exactly, by a method of Lagrange's that never leaves the whole numbers, its terms run 1, 3, 1, 5, 1, 1, 4 … and throw up a 534 at the thirty-fifth place and a 7,451 at the 571st — and in every statistic anyone has measured, they look like the terms of a number picked at random.

number · Pell A diagonal walk through countably many listed sets. A grid of six rows and seven columns, each cell numbered by Cantor's diagonal order, with the path of the first twenty-one cells drawn.

The choice inside a countable union

A countable union of countable sets is countable: list each set, then walk the grid of all their members along its diagonals. The proof is two lines and every student meets it early. It also makes infinitely many arbitrary choices at once — one listing for each set — and without the axiom of choice the theorem can fail: there are consistent worlds in which the real numbers are a countable union of countable sets, and worlds in which countably many pairs of socks cannot be counted.

logic · Axiom of choice

Named alongside it

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

CountabilityCardinalityContinued fractionsDegree of an extensionMinimal polynomialTranscendenceAlgorithmAxiom of choiceAxiomatic set theoryBaire categoryBasisChoice function

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