Concept

P adic numbers

Numbers built by measuring size through divisibility by a prime p, so that high powers of p count as small. In them 1 + 2 + 4 + … converges to −1, and equations can be solved digit by digit by lifting a solution modulo p.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

Square roots of 2, lifted from one power of 7 to the next. A tree whose rows are the square roots of 2 modulo 7 to the powers 1 to 4, each joined to the root it reduces to; counts 2, 2, 2, 2.

A root lifted one digit at a time

On a dial of seven, 3 × 3 is 2. On a dial of forty-nine the square root of 2 must reduce to 3, so there are only seven candidates, and exactly one of them works: 10. On a dial of 343 exactly one lift of 10 works: 108. Each step adds one digit on the left, found by solving a linear equation, and the digits go on for ever — a number …21216213 whose square is 2, in a world where closeness means divisibility by seven.

discrete · Modular arithmetic

1 + 2 + 4 + … in two notions of size. Two sets of points against the number of terms up to 20 on an axis whose gridlines are powers of 2: the partial sums' ordinary size rising, their 2-adic distance from −1 falling.

A series that converges to minus one

1 + 2 + 4 + 8 + … runs off to infinity, and yet the formula for a geometric series says it should equal 1/(1 − 2) = −1. Measure size by how many factors of 2 a number has, instead of how large it is, and powers of 2 become small: the series converges, and to exactly −1. The same change of ruler explains why every repeating decimal is a fraction, and why repeating binary digits running off to the left are fractions too.

analysis · Geometric series

A triangulation of the square with 7 three-coloured triangles. A triangulation of the unit square into 34 triangles, vertices coloured by Monsky's rule, with the 7 triangles carrying all three colours shaded.

No odd number of equal triangles

A square can be cut into two triangles of equal area, or four, or any even number. It cannot be cut into three, or five, or any odd number — whatever shapes the triangles take. Paul Monsky's proof of 1970 has no geometry in its engine at all: it colours the points of the plane by how divisible their coordinates are by two.

computation · Scissors congruence

The Collatz map on the 2-adic integers, before and after changing to parity coordinates. Two scatter plots of 2048 points: the Collatz map on 2-adic integers in binary-digit coordinates, a scattered cloud, and the same map in parity-vector coordinates, where every point lies on the two lines of the doubling map.

Where the Collatz map is a coin

Extend the Collatz map from the whole numbers to the 2-adic integers — binary strings that run on for ever to the left — and it stops being mysterious. It becomes, after a change of coordinates, the simplest chaotic system there is: shifting a string of coin tosses one place. Everything about it is then known, and none of it says anything about the whole numbers, which is the most instructive failure in the whole story.

dynamics · Collatz

The sum of all factorials in base 5: digits that stop changing. Partial sums of n! for N = 0 to 21, last 14 base-5 digits; the 5-adic limit ends …13010230042224.

A sum of factorials that converges at every prime

1 + 1 + 2 + 6 + 24 + 120 + … grows faster than any geometric series and has no sum in the real numbers. Measure size by divisibility instead and the same series converges at every prime at once, to a different number each time. Whether any of those numbers ends in zero is a question Đuro Kurepa asked in 1971, and a search through every prime below 100,000 says no while a coin-toss model says it should have said yes about twice.

analysis · Geometric series

Named alongside it

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

Modular arithmetic Convergence Parity Area Binary Collatz Conjugacy Derivative Dissection Factorial Geometric series Hensel lemma

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