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Concept

Integrality

The fact that a rational root of a polynomial whose leading coefficient is one must be a whole number. It turns an infinite search over fractions into a finite search over the divisors of the constant term, and it is what decides which roots of whole numbers are irrational.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

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 The tail that would have to be a whole number. For each denominator, the value of q! times the tail of the series for e, plotted against the band between zero and one where no whole number lies, with the bound 1/q above it.

A tail too small to be a whole number

If e were a fraction with denominator q, then q! times e would be a whole number. It splits into a whole part and a tail, the tail is squeezed strictly between nothing and one, and there is no whole number there.

number · Irrationality The polynomial that squeezes π. On the left, xⁿ(π − x)ⁿ/n! drawn at several degrees, its largest value falling toward nothing; on the right, the derivatives of the same polynomial at zero, every one a whole number.

An integral that cannot be a whole number

Niven's proof that π is not a fraction is the same squeeze as the one for e, with a much harder multiplier. A polynomial supplies the whole number; its own smallness supplies the contradiction; and both halves are computable.

number · Irrationality The 6 corners, and nothing in between. The 6 permutation matrices of size 3, drawn as grids. A search over every table of shares on a fine grid finds these and only these as corners of the set.

The corners are whole assignments

A table of shares can be written as a lottery over whole assignments, which one worked example shows. The general statement is that the corners of the set of such tables are exactly the whole assignments, and that single fact is why the whole subject is easy.

applied · Assignment The corner that is a half on every edge. A 3-vertex graph beside a table of the 5 corners of its matching relaxation. 4 are whole and one assigns a half to every edge.

Where the corners stop being whole

The easy theory of assignment rests on one property — the relaxation of the assignment problem has whole-numbered corners. Add a single edge that closes an odd cycle and the property fails, a corner appears with a half in every coordinate, and the problem changes character completely.

applied · Assignment Three pairs of places sharing one hub. A hub with three spokes of capacity one, the three pairs of outer places each routing half a unit through it, beside the total that can be sent fractionally, in whole units, and the cost of the cheapest set of roads separating every pair.

When several pairs share the roads

For one pair of places, the most that can travel between them equals the cheapest cut that separates them. Give three pairs a hub of three roads to share and the two numbers come apart: the pairs can send 3/2 between them, while separating every pair costs 2. Two pairs still meet their cut, but only by splitting units in half.

discrete · Network flow

Named alongside it

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

IrrationalityMatchingAssignmentFactorialLinear programmingPolytopeProof by contradictionRelaxationSqueezeAlgebraic numberBipartiteCertificate

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