Constructible number
In geometry and algebra, a real number formula_1 is constructible if and only if, given a line segment of unit length, a line segment of length formula_2 can be constructed with compass and straightedge in a finite number of steps. Equivalently, formula_1 is constructible if and only if there is a closed-form expression for formula_1 using only integers and the operations for addition, subtraction, multiplication, division, and square roots. The geometric definition of constructible numbers motivates a corresponding definition of constructible points, which can again be described either geometrically or algebraically. A point is constructible if it  can be produced as one of the points of a compass and straight edge construction (an endpoint of a line segment or crossing point of two lines or circles), starting from a given unit length segment. 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. Constructible numbers and points have also been called ruler and compass numbers and ruler and compass points, to distinguish them from numbers and points  that may be constructed using other processes. The set of constructible numbers forms a field: applying any of the four basic arithmetic operations to members of this set produces another constructible number. This field is a field extension of the rational numbers and in turn is contained in the field of algebraic numbers. It is the Euclidean closure of the rational numbers, the smallest field extension of the rationals that includes the square roots of all of its positive numbers. 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. The algebraic formulation of these questions led to proofs that their solutions are not constructible, after the geometric formulation of the same problems previously defied centuries of attack. Geometric definitions. Geometrically constructible points. Let formula_5 and formula_6 be two given distinct points in the Euclidean plane, and define formula_7 to be the set of points that can be constructed with compass and straightedge starting with formula_5 and formula_6. Then the points of formula_7 are called constructible points. formula_5 and formula_6 are, by definition, elements  of formula_7. To more precisely describe the remaining elements of formula_7, make the following two definitions: Then, the points of formula_7, besides formula_5 and formula_6 are: As an example, the midpoint of constructed segment formula_22 is a constructible point. One construction for it is to construct two circles with formula_22 as radius, and the line through the two crossing points of these two circles. Then the midpoint of segment formula_22 is the point where this segment is crossed by the constructed line. Geometrically constructible numbers. The starting information for the geometric formulation can be used to define a Cartesian coordinate  system in which the point formula_5 is associated to the origin having coordinates formula_26 and in which the point formula_6 is associated with the coordinates formula_28. The points of formula_7 may now be used to link the geometry and algebra by defining a constructible number to be a coordinate of a constructible point. Equivalent definitions are that a constructible number is the formula_30-coordinate of a constructible point formula_31 or the length of a constructible line segment. In one direction of this equivalence, if a constructible point has coordinates formula_32, then the point formula_31 can be constructed as its perpendicular  projection onto the formula_30-axis, and the segment from the origin to this point has length formula_30. In the reverse direction, if formula_30 is the length of a constructible line segment, then intersecting the formula_30-axis with a circle centered at formula_5 with radius formula_30 gives the point formula_31. It follows from this equivalence that every point whose Cartesian coordinates are geometrically constructible numbers is itself a geometrically constructible point. For, when formula_30 and formula_42 are geometrically constructible numbers, point formula_32 can be constructed as the intersection of lines through formula_31 and formula_45, perpendicular to the coordinate axes. Algebraic definitions . Algebraically constructible numbers. The algebraically constructible real numbers are the subset of the real numbers that can be described by formulas that combine integers using the operations of addition, subtraction, multiplication, multiplicative inverse, and square roots of positive numbers. Even more simply, at the expense of making these formulas longer, the integers in these formulas can be restricted to be only 0 and 1. For instance, the square root of 2 is constructible, because it can be described by the formulas formula_46 or formula_47. Analogously, the algebraically constructible complex numbers are the subset of complex numbers that have formulas of  the same type, using a more general version of the square root that is not restricted to positive numbers but can instead take arbitrary complex numbers as its argument, and produces the principal square root of its argument. Alternatively, the same system of complex numbers may be defined as the complex numbers whose real and imaginary parts are both constructible real numbers. For instance, the complex number formula_48 has the formulas formula_49 or formula_50, and its real and imaginary parts are the constructible numbers 0 and 1 respectively. These two definitions of the constructible complex numbers are equivalent. In one  direction, if formula_51 is a complex number whose real part formula_30 and imaginary part formula_42 are both constructible real numbers, then replacing formula_30 and formula_42 by their formulas within the larger formula formula_56 produces a formula for formula_57 as a complex number. 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 Algebraically constructible points. The algebraically constructible points may be defined as the points  whose two real Cartesian coordinates are both algebraically constructible real numbers. Alternatively, they may be defined as the points in the complex plane given by algebraically constructible complex numbers. By the equivalence between the two definitions for algebraically constructible complex numbers, these two definitions of algebraically constructible points are also equivalent. Equivalence of algebraic and geometric definitions. If formula_64 and formula_65 are the non-zero lengths of geometrically constructed segments then elementary compass and straightedge constructions can be used to obtain constructed segments of lengths formula_66, formula_67, formula_68, and formula_69. The latter two can be done with a construction based  on the intercept theorem. A slightly less elementary construction using these tools is based on the geometric mean theorem and will construct a segment of length formula_70 from a constructed segment of length formula_64. It follows that every algebraically constructible number is geometrically constructible, by using these techniques to translate a formula for the number into a construction for the number. In the other direction, a set of geometric objects may be specified by algebraically constructible real numbers: coordinates for points, slope and formula_42-intercept for lines, and center and radius for circles. 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. It follows from these formulas that every geometrically constructible number is algebraically constructible. Algebraic properties. 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. Thus, the constructible numbers (defined in any of the above ways) form a field. More specifically, the constructible real numbers form a Euclidean field, an ordered  field containing a square root of each of its positive elements. Examining the properties of this field and its subfields leads to necessary conditions on a number to be constructible, that can be used to show that specific numbers arising in classical geometric construction problems are not constructible. It is convenient to consider, in place of the whole field of constructible numbers, the subfield formula_73 generated by any given constructible number formula_74, and to use the algebraic construction of formula_74 to decompose this field. If formula_74 is a constructible real number, then the values occurring within a formula constructing it  can be used to produce a finite sequence of real numbers formula_77 such that, for each formula_48, formula_79 is an extension of formula_80 of degree 2. Using slightly different terminology, a real number is constructible if and only if it lies in a field at the top of a finite tower of real quadratic extensions, formula_81 starting with the rational field formula_82 where formula_74 is in formula_84 and for all formula_85, formula_86. It follows from this decomposition that the degree of the field extension formula_87 is formula_88, where formula_1 counts the number of quadratic extension steps. Analogously to the real  case, a complex number is constructible if and only if it lies in a field at the top of a finite tower of complex quadratic extensions. More precisely, formula_74 is constructible if and only if there exists a tower of fields formula_91 where formula_74 is in formula_93, and for all formula_94, formula_95. The difference between this characterization and that of the real constructible numbers is only that the fields in this tower are not restricted to being real. Consequently, if a complex number formula_74 is constructible, then formula_87 is a power of two. However, this necessary condition is not sufficient : there exist field extensions whose degree is a power of two that cannot be factored into a sequence of quadratic extensions. The fields that can be generated in this way from towers of quadratic extensions of formula_82 are called "iterated quadratic extensions" of formula_82. The fields of real and complex constructible numbers are the unions of all real or complex iterated quadratic extensions of formula_82. Trigonometric numbers. Trigonometric numbers are the cosines or sines of angles that are rational multiples of formula_101. These numbers are always algebraic, but they may not be constructible. The cosine or sine of the  angle formula_102 is constructible only for certain special numbers formula_103: Thus, for example, formula_104 is constructible because 15 is the product of two Fermat primes, 3 and 5. Impossible constructions. The ancient Greeks thought that certain problems of straightedge and compass construction they could not solve were simply obstinate, not unsolvable. However, the non-constructibility of certain numbers proves that these constructions are logically impossible to perform. (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.) In particular, the algebraic formulation of constructible numbers leads to a proof of the impossibility of the following construction problems: History. The birth of the concept of constructible numbers is inextricably linked with the history of the three impossible compass and straightedge constructions: duplicating the cube, trisecting an angle, and squaring the circle. The restriction of using only compass and straightedge in geometric constructions is often credited to Plato due to a passage in Plutarch. According to Plutarch, Plato gave the duplication of the cube (Delian) problem to  Eudoxus and Archytas and Menaechmus, who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using pure geometry. However, this attribution is challenged, 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. Proclus, citing Eudemus of Rhodes, credited Oenopides (circa 450 BCE) with two ruler and compass constructions, leading some authors to hypothesize that Oenopides originated the restriction. The restriction to compass and straightedge is essential  to the impossibility of the classic construction problems. Angle trisection, for instance, can be done in many ways, several known to the ancient Greeks. 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. Although not one of the classic three construction problems, the problem of constructing regular polygons with straightedge and compass is often treated alongside them. The Greeks knew how to construct regular -gons with (for any integer ) or the product of any two or three of  these numbers, but other regular -gons eluded them. 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. Gauss's treatment was algebraic rather than geometric; in fact, he did not actually construct the polygon, but rather showed that the cosine of a central angle was a constructible number. The argument was generalized in his 1801 book "Disquisitiones Arithmeticae" giving the "sufficient" condition for the construction of a regular -gon. Gauss claimed, but did not prove, that the condition was also necessary and  several authors, notably Felix Klein, attributed this part of the proof to him as well. 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 appear's in Ptolemy's work on optics from the second century. are impossible to solve if one uses 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). 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 . It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility, by extending the work of Charles Hermite and proving that is a transcendental number . Alhazen's problem was not proved impossible to solve by compass and straightedge until the work of . 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.