Chapter 1

The Real Line

Stub chapter proving the build pipeline; real content gets rebuilt from the planning material (see plan/MAP.md).

1.1Incompleteness of the Rationals

The diagonal of a unit square has length $\sqrt 2$ , and no fraction represents it exactly: the approximations

𝑥 𝑛 + 1 = 12 (𝑥 𝑛 + 2 𝑥 𝑛)

(1.1)

are each rational, each closer to $\sqrt 2$ than the last, and their target is not in $ℚ$ .

Theorem 1.1 (Irrationality of the Square Root of Two). There is no rational number $𝑝 / 𝑞$ with $(𝑝 / 𝑞) 2 = 2$ .

Proof. Suppose $𝑝 / 𝑞$ is in lowest terms and $𝑝 2 = 2 𝑞 2$ . Then $𝑝$ is even, so $𝑝 = 2 𝑘$ and $𝑞 2 = 2 𝑘 2$ , making $𝑞$ even too — contradicting lowest terms. ∎

1.2Completeness

By theorem 1.1, the iteration (1.1) converges to no rational number: the real line is the completion that gives such processes a home.