Constructible number

A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), the point can be constructed with unruled straightedge and compass. A complex number is a constructible number if its corresponding point in the Euclidean plane is constructible from the usual x- and y-coordinate axes.

It can then be shown that a real number r is constructible if and only if, given a line segment of unit length, a line segment of length |r | can be constructed with compass and straightedge.[1] It can also be shown that a complex number is constructible if and only if its real and imaginary parts are constructible.

The set of constructible numbers can be completely characterized in the language of field theory: the constructible numbers form the smallest field extension of the rational numbers which is closed under square root and complex conjugation. This has the effect of transforming geometric questions about compass and straightedge constructions into algebra. This transformation leads to the solutions of many famous mathematical problems, which defied centuries of attack.


Geometric definitions

The geometric definition of a constructible point is as follows. First, for any two distinct points P and Q in the plane, let L(P, Q ) denote the unique line through P and Q, and let C (P, Q ) denote the unique circle with center P, passing through Q. (Note that the order of P and Q matters for the circle.) By convention, L(P, P ) = C (P, P ) = {P }. Then a point Z is constructible from E, F, G and H if either

  1. Z is in the intersection of L(E, F ) and L(G, H ), where L(E, F ) ≠ L(G, H );
  2. Z is in the intersection of C (E, F ) and C (G, H ), where C (E, F ) ≠ C (G, H );
  3. Z is in the intersection of L(E, F ) and C (G, H ).

Since the order of E, F, G, and H in the above definition is irrelevant, the four letters may be permuted in any way. Put simply, Z is constructible from E, F, G and H if it lies in the intersection of any two distinct lines, or of any two distinct circles, or of a line and a circle, where these lines and/or circles can be determined by E, F, G, and H, in the above sense.

Now, let A and A′ be any two distinct fixed points in the plane. A point Z is constructible if either

  1. Z = A;
  2. Z = A′;
  3. there exist points P1, ..., Pn, with Z = Pn, such that for all j ≥ 1, Pj + 1 is constructible from points in the set {A, A′, P1, ..., Pj }.

Put simply, Z is constructible if it is either A or A′, or if it is obtainable from a finite sequence of points starting with A and A′, where each new point is constructible from previous points in the sequence.

For example, the center point of A and A′ is defined as follows. The circles C (A, A′) and C (A′, A) intersect in two distinct points; these points determine a unique line, and the center is defined to be the intersection of this line with L(A, A′).

Transformation into algebra

All rational numbers are constructible, and all constructible numbers are algebraic numbers. Also, if a and b are constructible numbers with b ≠ 0, then ab and a/b are constructible. Thus, the set K of all constructible complex numbers forms a field, a subfield of the field of algebraic numbers.

Furthermore, K is closed under square roots and complex conjugation. These facts can be used to characterize the field of constructible numbers, because, in essence, the equations defining lines and circles are no worse than quadratic. The characterization is the following: a complex number is constructible if and only if it lies in a field at the top of a finite tower of quadratic extensions, starting with the rational field Q. More precisely, z is constructible if and only if there exists a tower of fields

\mathbb{Q} = K_0 \subseteq K_1 \subseteq \dots \subseteq K_n

where z is in Kn and for all 0 ≤ j < n, the dimension [Kj + 1 : Kj ] = 2.

Impossible constructions

The algebraic characterization of constructible numbers provides an important necessary condition for constructibility: if z is constructible, then it is algebraic, and its minimal irreducible polynomial has degree a power of 2, or equivalently, the field extension Q(z)/Q has dimension a power of 2. One should note that it is true, (but not obvious to show) that the converse is false — this is not a sufficient condition for constructibility. However, this defect can be remedied by considering the normal closure of Q(z)/Q.

The non-constructibility of certain numbers proves the impossibility of certain problems attempted by the philosophers of ancient Greece. In the following chart, each row represents a specific ancient construction problem. The left column gives the name of the problem. The second column gives an equivalent algebraic formulation of the problem. In other words, the solution to the problem is affirmative if and only if each number in the given set of numbers is constructible. Finally, the last column provides the simplest known counterexample. In other words, the number in the last column is an element of the set in the same row, but is not constructible.

Construction problem Associated set of numbers Counterexample
Doubling the cube \left \{ \sqrt[3]{x} : x \mbox{ is constructible} \right \} \sqrt[3]{2} is not constructible, because its minimal polynomial has degree 3 over Q
Trisecting the angle \left \{ \cos \left( \frac{\arccos x}{3} \right) : x \mbox{ is constructible} \right \} \cos \left( \frac{\arccos (1/2)}{3} \right) = \frac{1}{2} \left( 2\cos \left( \frac{\pi}{9} \right) \right) is not constructible, because 2\cos \left( \frac{\pi}{9} \right) has minimal polynomial of degree 3 over Q
Squaring the circle \left \{ \sqrt{\pi} \right \} \sqrt{\pi} is not constructible, because it is not algebraic over Q
Constructing all regular polygons \left \{ e^{2\pi i/n} : n \in \mathbb{N}, n \geq 3 \right \} ei / 7 is not constructible, because 7 is not a Fermat prime, nor is 7 the product of 2^k and one or more Fermat primes

See also


  1. ^ John A. Beachy, William D. Blair; Abstract Algebra; Definition 6.3.1


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Constructible polygon — Construction of a regular pentagon In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular… …   Wikipedia

  • Constructible universe — Gödel universe redirects here. For Kurt Gödel s cosmological solution to the Einstein field equations, see Gödel metric. In mathematics, the constructible universe (or Gödel s constructible universe), denoted L, is a particular class of sets… …   Wikipedia

  • Constructible sheaf — In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space X, such that X is the union of a finite number of locally closed subsets on each of which the sheaf is a twisted constant sheaf. It is a generalization …   Wikipedia

  • Fermat number — In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form:F {n} = 2^{2^{ overset{n} {} + 1where n is a nonnegative integer. The first nine Fermat numbers are OEIS|id=A000215:As of|2008 …   Wikipedia

  • Definable real number — A real number a is first order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds in the standard… …   Wikipedia

  • Algebraic number — In mathematics, an algebraic number is a complex number that is a root of a non zero polynomial in one variable with rational (or equivalently, integer) coefficients. Complex numbers such as pi that are not algebraic are said to be transcendental …   Wikipedia

  • 65535 (number) — Number number= 65535 range = 10000 100000 cardinal = sixty five thousand five hundred thirty five ordinal = th ordinal text = sixty five thousand five hundred thirty fifth numeral = factorization = 3 x 5 x 17 x 257 prime = divisor = 16 roman =… …   Wikipedia

  • 65537 (number) — Number number= 65537 range = 10000 100000 cardinal = sixty five thousand five hundred thirty seven ordinal = th numeral = factorization = prime prime = divisor = 2 roman = unicode = greek prefix = latin prefix = bin = 10000000000000001 oct = duo …   Wikipedia

  • Pirates Constructible Strategy Game — Infobox Game subject name=Pirates Constructible Strategy Game image link= image caption=Pirates of the Cursed Seas is a tabletop strategy game depicting naval battles and hunt for treasure in the Caribbean in the 17th century. players= 2 ndash;?… …   Wikipedia

  • Prime number — Prime redirects here. For other uses, see Prime (disambiguation). A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.