Concept

Norm — where it appears

A measure of the size of a vector or a number, and the notion of distance that follows from it. Which one is chosen decides what nearest means, and only the ordinary one is unchanged by rotation.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Two factor trees of 360. The same number split two different ways, both ending in the same primes.

One way to factor, and no other

Every number breaks into primes in exactly one way. That is so familiar it is hard to see as a claim at all — until it is put beside an arithmetic where it is false, and where six has two different factorisations that cannot be reconciled.

number · Unique factorisation

The circle of radius √25 on the integer lattice. A circle drawn on the whole-number grid, with the lattice points it passes through marked.

Two squares, and a lattice

Whether a prime is the sum of two squares is decided entirely by its remainder on division by four. A fact about circles is settled by a fact about remainders, and neither statement contains any hint of the other.

number · Sums of two squares

The unit ball at p = 2.00. The set of points one unit from the origin, when distance is measured by the p-th power sum. At p = 1 it is a diamond, at p = 2 a circle, and as p grows it fills out a square.

Circles that are diamonds and squares

The theorem hands over a formula for distance. Take the formula as a definition, change the exponent in it, and the set of points one unit from the origin stops being round — while remaining, in every sense that matters, a circle.

geometry · Pythagoras

The four-square identity, and how few squares a number needs. A product of two whole quaternions with both sides of the four-square identity evaluated, above a strip colouring every number by the fewest squares that add to it.

The identity that multiplies sums of squares

A sum of two squares times a sum of two squares is a sum of two squares, and the same is true of four and of eight. It is true of three of nothing: 3 and 5 are each a sum of three squares, 15 is not, and one small pair of numbers rules the case out forever.

algebra · Quaternions

The twenty-four unit quaternions, at the corners of a four-dimensional solid. The twenty-four units of the Hurwitz quaternions drawn as the vertices of a 24-cell projected into three-space, with the ninety-six edges joining units at distance one.

The integers among the quaternions

The obvious integer quaternions are the ones with whole coordinates, and they are the wrong ones. Sixteen more, with every coordinate a half, have to be let in — and with them comes a division algorithm, twenty-four units instead of eight, and a regular solid that exists only in four dimensions.

algebra · Quaternions

What is lost at eight

Double the quaternions and division still works, but ab times c and a times bc are no longer the same number. What survives is a weaker law that turns out to be enough for a great deal — and the whole multiplication table fits into a picture of seven lines.

algebra · Quaternions

The parabola that proves |a·b| ≤ |a||b|. The squared length of a − t b plotted against t. It is a parabola opening upward whose least value is 7.118; that this is never negative is exactly the Cauchy–Schwarz inequality.

The square that cannot be negative

Cauchy–Schwarz is the load-bearing inequality of the whole subject and is nearly always asserted. It is one line away from a fact nobody would argue with, and the line is a parabola with no room to cross the axis.

algebra · Inner product

Named alongside it

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

Primes Quaternion Unique factorisation Counting two ways Division algebra Inner product Lattice Sums of two squares Symmetry group Associativity Cauchy schwarz Complex numbers

All concepts