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Concept

Lowest terms

A fraction written so that its numerator and denominator share no factor above one. Every fraction has exactly one such form, and nearly every argument about rational numbers begins by taking it, since it is where unique factorisation gets its grip.

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

The Stern–Brocot tree to depth 4. Every positive rational, each appearing exactly once, generated by taking mediants.

Every fraction, exactly once

Take two fractions, add the tops and add the bottoms. That is not how fractions are added, it is not an average, and repeating it produces every positive rational exactly once, already in lowest terms.

number · stern brocot 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.

Algebraic numberBijectionContinued fractionsCounting two waysDivisorFarey sequenceFord circlesInfinite descentIntegralityIrrationalityMediantPerfect square

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