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Concept

Infinite descent

An argument that from any solution in whole numbers a strictly smaller one can be produced, which is impossible because the whole numbers do not descend forever. It proves an impossibility by exhibiting a procedure, and it survives into settings where no polynomial is available to argue about.

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

Two squares of side 12 inside one of side 17. Two overlapping squares laid into opposite corners of a larger one, with the overlap and the two uncovered corners marked.

The square that cannot shrink

The usual proof that the square root of two is irrational is about even and odd numbers. There is a proof about squares instead, in which a supposed solution is folded into a smaller one — and the folding is a drawing.

number · irrationality 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

Named alongside it

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

IrrationalityAlgebraic numberContinued fractionsCounting two waysDescentDivisorIncommensurabilityIntegralityLowest termsParityPell equationPerfect square

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