Constructible number
using only integers and the operations for addition, subtraction, multiplication, division, and square roots.
Alternatively and equivalently, taking the two endpoints of the given segment to be the points (0, 0) and (1, 0) of a Cartesian coordinate system, a point is constructible if and only if its Cartesian coordinates are both constructible numbers.
[4] The proof of the equivalence between the algebraic and geometric definitions of constructible numbers has the effect of transforming geometric questions about compass and straightedge constructions into algebra, including several famous problems from ancient Greek mathematics.
[7] The starting information for the geometric formulation can be used to define a Cartesian coordinate system in which the point
In the other direction, any formula for an algebraically constructible complex number can be transformed into formulas for its real and imaginary parts, by recursively expanding each operation in the formula into operations on the real and imaginary parts of its arguments, using the expansions[15] The algebraically constructible points may be defined as the points whose two real Cartesian coordinates are both algebraically constructible real numbers.
[16] In the other direction, a set of geometric objects may be specified by algebraically constructible real numbers: coordinates for points, slope and
It is possible (but tedious) to develop formulas in terms of these values, using only arithmetic and square roots, for each additional object that might be added in a single step of a compass-and-straightedge construction.
[17] The definition of algebraically constructible numbers includes the sum, difference, product, and multiplicative inverse of any of these numbers, the same operations that define a field in abstract algebra.
By the fundamental theorem of Galois theory, there is a corresponding tower of quadratic extensions
[24] Trigonometric numbers are the cosines or sines of angles that are rational multiples of
The ancient Greeks thought that certain problems of straightedge and compass construction they could not solve were simply obstinate, not unsolvable.
[26] However, the non-constructibility of certain numbers proves that these constructions are logically impossible to perform.
[27] (The problems themselves, however, are solvable using methods that go beyond the constraint of working only with straightedge and compass, and the Greeks knew how to solve them in this way.
One such example is Archimedes' Neusis construction solution of the problem of Angle trisection.
The restriction of using only compass and straightedge in geometric constructions is often credited to Plato due to a passage in Plutarch.
[36] However, this attribution is challenged,[37] due, in part, to the existence of another version of the story (attributed to Eratosthenes by Eutocius of Ascalon) that says that all three found solutions but they were too abstract to be of practical value.
[38] Proclus, citing Eudemus of Rhodes, credited Oenopides (c. 450 BCE) with two ruler and compass constructions, leading some authors to hypothesize that Oenopides originated the restriction.
[39] The restriction to compass and straightedge is essential to the impossibility of the classic construction problems.
The Quadratrix of Hippias of Elis, the conics of Menaechmus, or the marked straightedge (neusis) construction of Archimedes have all been used, as has a more modern approach via paper folding.
In 1796 Carl Friedrich Gauss, then an eighteen-year-old student, announced in a newspaper that he had constructed a regular 17-gon with straightedge and compass.
The argument was generalized in his 1801 book Disquisitiones Arithmeticae giving the sufficient condition for the construction of a regular
Gauss claimed, but did not prove, that the condition was also necessary and several authors, notably Felix Klein,[42] attributed this part of the proof to him as well.
[43] Alhazen's problem is also not one of the classic three problems, but despite being named after Ibn al-Haytham (Alhazen), a medieval Islamic mathematician, it already appears in Ptolemy's work on optics from the second century.
[21] Pierre Wantzel proved algebraically that the problems of doubling the cube and trisecting the angle are impossible to solve using only compass and straightedge.
In the same paper he also solved the problem of determining which regular polygons are constructible: a regular polygon is constructible if and only if the number of its sides is the product of a power of two and any number of distinct Fermat primes (i.e., the sufficient conditions given by Gauss are also necessary).
[44] An attempted proof of the impossibility of squaring the circle was given by James Gregory in Vera Circuli et Hyperbolae Quadratura (The True Squaring of the Circle and of the Hyperbola) in 1667.
Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of π.
[45][46] Alhazen's problem was not proved impossible to solve by compass and straightedge until the work of Jack Elkin.
[47] The study of constructible numbers, per se, was initiated by René Descartes in La Géométrie, an appendix to his book Discourse on the Method published in 1637.
Descartes associated numbers to geometrical line segments in order to display the power of his philosophical method by solving an ancient straightedge and compass construction problem put forth by Pappus.
