The number line, zoomed
Six kinds of number on one line, each set inside the next. Zoom in anywhere and the fractions never run out. Yet almost none of the line belongs to them, or to any number you could write down.
Scroll or pinch to zoom, drag to pan. Hover or tap a mark to name it.
Where the dart landed
Those are its first 50 decimals. A real dart has infinitely many, each one random. With probability 1 this number is not rational, not algebraic and not computable: no program prints its digits.
That isn't because the digits look messy. It's because each of those three sets is countable: you can list its members as a first, second, third, and so on. Any list, however dense on the line, has total length zero. Watch it happen to the rationals.
The rationals in view are listed by denominator: r₁, r₂, r₃, … Interval k has length ε/2ᵏ and sits on rₖ. Every rational gets an interval, since every rational is somewhere on the list. The lengths add up to at most ε/2 + ε/4 + ε/8 + … = ε, and overlaps only make it less.
ε can be as small as you like, so the rationals have length zero. Between any two points there are infinitely many of them, and still they take up no room. The same covering works for any list: the algebraic numbers, and the computable ones too.
The layers
Each band shows what is new at that level. Everything in a band also belongs to every band above it.
Naturals 0, 1, 2, 3, …
Counting. Some books start at 1; here 0 is in.
Countable: this is what countable means.
Integers adds −1, −2, −3, …
Negatives, so subtraction always works. Zero and negatives on the map.
Countable: list them 0, 1, −1, 2, −2, …
Rationals adds p/q
Fractions, so division works. They are dense: between any two reals there is one. The band shows every p/q in view up to a denominator that grows as you zoom, taller for smaller q. Fractions on the map.
Countable: list by denominator, or walk the Stern–Brocot tree.
Algebraic adds √2, φ, ∛2, …
Roots of polynomials with integer coefficients. √2 was the first number shown not to be a fraction (√2 is irrational). The band shows real roots of irreducible polynomials of degree 2 (dots) and 3 (rings) with coefficients up to 10 in size, bigger for smaller coefficients.
Countable: finitely many polynomials of each degree and coefficient size, each with finitely many roots.
Computable adds π, e, ln 2, …
Numbers some program can print to any precision you ask for. π and e are transcendental (π is transcendental) but perfectly computable. Every algebraic number is computable too. Hollow marks are constants that might belong lower: nobody knows whether Euler's γ is even irrational.
Countable: a program is a finite string, so there are countably many programs, and each prints at most one number. See computability.
Reals adds everything else
Every point of the line: Dedekind's cuts in the rationals, or limits of Cauchy sequences (construction of the reals). No gaps. The pink haze is Chaitin's Ω, the probability that a randomly generated program halts on a fixed universal machine. It is a definite number between 0 and 1 with a precise definition, but not computable: its first n bits would settle the halting problem for every program up to n bits long.
Uncountable (Cantor, 1874).
Why a list can't fill a line
A countable set can be covered by intervals of total length as small as you like, as the dart shows. The interval [0, 1] has length 1. So it can't be countable: there are more reals than any list can hold. Cantor proved this first with nested intervals and later with the diagonal argument, which grew into Cantor's theorem. Length made precise is measure theory.
Put the layers together. Each of ℕ, ℤ, ℚ, 𝔸 and ℝc is a list, so their union has length zero. Everything else, the whole of the line in any sense that measures size, is numbers no program can print. Finite descriptions of any kind are strings too, so only countably many reals can ever be named at all. Almost every real number can never be written down.
Fractions still come closest to the ones we do write down. The continued fraction of a number gives its best approximations, which is what lights up when you fly to √2, φ, e or π.