---
tag: 0FMN2MN7TM
kind: definition
name: Constructible number
msc: [12F05]
defines: 0QNFRBJF4Q
uses: [0DAECMZVRT]
sources:
  - url: https://en.wikipedia.org/wiki/Constructible_number
    locator: Geometrically constructible points
  - url: https://en.wikipedia.org/wiki/Constructible_number
    locator: Geometrically constructible numbers
---

## Definition

Identify the plane with $\mathbb{C}$.
A complex number $z$ is constructible if the point $z$ can be obtained from $0$ and $1$ by finitely many [[straightedge and compass|0DAECMZVRT]] steps.
In each step, one adds an intersection point of two distinct lines or circles drawn through, or centred at, points already obtained.

## Remarks

A real number $x$ is constructible if and only if a segment of length $|x|$ can be constructed from a segment of length $1$.
A complex number $a + b i$ is constructible if and only if $a$ and $b$ are constructible.
The constructible numbers [[form a field closed under square roots|0PN9WMRWT7]].
The numbers $\sqrt{2}$, $\sqrt{1 + \sqrt{3}}$ and $\cos(2\pi/5)$ are constructible. The numbers $\sqrt[3]{2}$ and $\pi$ are not.

## Explanations

### L1 human

Start with two points, $0$ and $1$.
Draw lines through points you have, and circles centred at points you have, and mark the new intersections.
The constructible numbers are all the points you can reach in this way.