constructible numbers

The smallest subfieldMathworldPlanetmath 𝔼 of ℝ over β„š such that 𝔼 is EuclideanMathworldPlanetmathPlanetmath is called the field of real constructible numbers. First, note that 𝔼 has the following properties:

  1. 1.


  2. 2.

    If a,bβˆˆπ”Ό, then also aΒ±b, a⁒b, and a/bβˆˆπ”Ό, the last of which is meaningful only when bβ‰ 0;

  3. 3.

    If rβˆˆπ”Ό and r>0, then rβˆˆπ”Ό.

The field 𝔼 can be extended in a natural manner to a subfield of β„‚ that is not a subfield of ℝ. Let 𝔽 be a subset of β„‚ that has the following properties:

  1. 1.


  2. 2.

    If a,bβˆˆπ”½, then also aΒ±b, a⁒b, and a/bβˆˆπ”½, the last of which is meaningful only when bβ‰ 0;

  3. 3.

    If zβˆˆπ”½βˆ–{0} and arg⁑(z)=ΞΈ where 0≀θ<2⁒π, then |z|⁒ei⁒θ2βˆˆπ”½.

Then 𝔽 is the field of constructible numbers.

Note that π”ΌβŠ‚π”½. Moreover, π”½βˆ©β„=𝔼.

An element of 𝔽 is called a constructible number. These numbers can be β€œconstructed” by a process that will be described shortly.

Conversely, let us start with a subset S of β„‚ such that S contains a non-zero complex numberMathworldPlanetmathPlanetmath. Call any of the binary operationsMathworldPlanetmath in condition 2 as well as the square root unary operation in condition 3 a ruler and compass operation. Call a complex number constructible from S if it can be obtained from elements of S by a finite sequencePlanetmathPlanetmath of ruler and compass operations. Note that 1∈S. If Sβ€² is the set of numbers constructible from S using only the binary ruler and compass operations (those in condition 2), then Sβ€² is a subfield of β„‚, and is the smallest field containing S. Next, denote S^ the set of all constructible numbers from S. It is not hard to see that S^ is also a subfield of β„‚, but an extensionPlanetmathPlanetmathPlanetmath of Sβ€². Furthermore, it is not hard to show that S^ is Euclidean. The general process (algorithm) of elementsMathworldMathworld in S^ from elements in S using finite sequences of ruler and compass operations is called a ruler and compass construction. These are so called because, given two points, one of which is 0, the other of which is a non-zero real number in S, one can use a ruler and compass to construct these elements of S^.

If S={1} (or any rational numberPlanetmathPlanetmathPlanetmath), we see that S^=𝔽 is the field of constructible numbers.

Note that the lengths of constructible line segmentsMathworldPlanetmath ( on the Euclidean planeMathworldPlanetmath are exactly the positive elements of 𝔼. Note also that the set 𝔽 is in one-to-one correspondence with the set of constructible points ( on the Euclidean plane. These facts provide a between abstract algebra and compass and straightedge constructions.

Title constructible numbers
Canonical name ConstructibleNumbers
Date of creation 2013-03-22 17:15:01
Last modified on 2013-03-22 17:15:01
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 17
Author CWoo (3771)
Entry type Definition
Classification msc 12D15
Related topic EuclideanField
Related topic CompassAndStraightedgeConstruction
Related topic TheoremOnConstructibleAngles
Related topic TheoremOnConstructibleNumbers
Defines ruler and compass operation
Defines compass and ruler operation
Defines compass and straightedge operation
Defines straightedge and compass operation
Defines constructible number
Defines constructible from
Defines constructible
Defines field of constructible numbers
Defines field of real constructible numbers