Matpic
mathematics, in pictures
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 5 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 The tree as words in two matrices. 6 nodes of the Stern–Brocot tree, each as the word of turns reaching it, the matrix that word multiplies out to, its two columns as fractions, and the mediant of those columns.

Two matrices that generate the tree

A node of the Stern–Brocot tree is not really a fraction — it is the pair of fractions it lies between. Written as the columns of a matrix, the two turns of the tree become two multiplications, and the determinant that kept everything in lowest terms becomes a property of a product.

number · Stern brocot Every positive rational, in one sequence. The first 32 terms of Stern's diatomic sequence as bars, with the ratios of consecutive terms beneath. Every ratio is in lowest terms, no two agree, and each term counts the hyperbinary representations of its index.

Every rational in one sequence

The tree lists every positive fraction once and needs a tree to do it. One recursion on the whole numbers lists them in a single row — and each term of it counts something nobody was asking about, which is why the enumeration works.

number · Stern brocot 13 record approximations in 26 turns. The distance from π to each fraction the descent passes, against its denominator, on logarithmic axes. 13 of them beat every fraction with a smaller denominator.

The fractions that beat every smaller one

Walking down the tree towards a number produces a sequence of fractions closing in on it. Most of them are steps along the way; a few are the best approximations there are — closer than every fraction with a smaller denominator — and which few is decided by where the turns change direction.

number · Stern brocot

Named alongside it

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

MediantStern brocot treeBijectionContinued fractionsCounting two waysFarey sequenceIrrationalityUnimodularAlgebraic numberBinaryCountabilityDeterminant

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