Constructible number

Builds on Straightedge and compass construction

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 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. The numbers $\sqrt{2}$ , $\sqrt{1 + \sqrt{3}}$ and $\cos(2\pi/5)$ are constructible. The numbers $\sqrt[3]{2}$ and $\pi$ are not.

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Straightedge and compass construction concept
Constructible number definition

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Tag	`0FMN2MN7TM`
URI	`m2://0FMN2MN7TM`
Kind	definition
Writing	W1 stated
MSC	12F05
Defines	Constructible number
Uses	Straightedge and compass construction

Constructible number

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