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单词 ConstructibleNumbers
释义

constructible numbers


The smallest subfieldMathworldPlanetmath 𝔼 of over such that 𝔼 isEuclideanMathworldPlanetmathPlanetmath is called the field of real constructible numbers. First, note that 𝔼 has the following properties:

  1. 1.

    0,1𝔼;

  2. 2.

    If a,b𝔼, then also a±b, ab, and a/b𝔼, the last of which is meaningful only when b0;

  3. 3.

    If r𝔼 and r>0, then r𝔼.

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

  1. 1.

    0,1𝔽;

  2. 2.

    If a,b𝔽, then also a±b, ab, and a/b𝔽, the last of which is meaningful only when b0;

  3. 3.

    If z𝔽{0} and arg(z)=θ where 0θ<2π, then |z|eiθ2𝔽.

Then 𝔽 is the field of constructible numbers.

Note that 𝔼𝔽. Moreover, 𝔽=𝔼.

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

Conversely, let us start with a subset S of 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 1S. If S is the set of numbers constructible from S using only the binary ruler and compass operations (those in condition 2), then S is a subfield of , 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 , but an extensionPlanetmathPlanetmathPlanetmath of S. 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 (http://planetmath.org/Constructible2) 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 (http://planetmath.org/Constructible2) on the Euclidean plane. These facts provide a between abstract algebra and compass and straightedge constructions.

Titleconstructible numbers
Canonical nameConstructibleNumbers
Date of creation2013-03-22 17:15:01
Last modified on2013-03-22 17:15:01
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id17
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 12D15
Related topicEuclideanField
Related topicCompassAndStraightedgeConstruction
Related topicTheoremOnConstructibleAngles
Related topicTheoremOnConstructibleNumbers
Definesruler and compass operation
Definescompass and ruler operation
Definescompass and straightedge operation
Definesstraightedge and compass operation
Definesconstructible number
Definesconstructible from
Definesconstructible
Definesfield of constructible numbers
Definesfield of real constructible numbers
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更新时间:2025/5/4 12:49:59