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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 4 essays across 3 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

Named alongside it

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

TranscendenceBaire categoryConstructible numberConstructionContinued fractionsCountabilityDegreeDegree of an extensionDensityDiophantine approximationDivisorEnumeration

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