Editing Constructible number (section)

{{short description|Number constructible via compass and straightedge}}
{{redirect-distinguish|Euclidean number|Euclid number}}
{{distinguish|computable number}}
{{for|numbers "constructible" in the sense of set theory|Constructible universe}}
{{good article}}

[[File:Square root of 2 triangle.svg|thumb|250px|The [[square root of 2]] is equal to the length of the [[hypotenuse]] of a [[right triangle]] with legs of length 1 and is therefore a '''constructible number''']]

In [[geometry]] and [[algebra]], a [[real number]] <math>r</math> is '''constructible''' if and only if, given a [[line segment]] of unit length, a line segment of length <math>|r|</math> can be constructed with [[compass and straightedge constructions|compass and straightedge]] in a finite number of steps. Equivalently, <math>r</math> is constructible if and only if there is a [[closed-form expression]] for <math>r</math> 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 straightedge 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.{{sfnmp|Kazarinoff|2003|1pp=10, 15|Martin|1998|2p=41|2loc=Corollary 2.16}} 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.{{sfnp|Martin|1998|pp=31–32}}

The set of constructible numbers forms a [[field (algebra)|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 number]]s and in turn is contained in the field of [[algebraic number]]s.<ref>{{harvp|Courant|Robbins|1996|pp=133–134|loc=Section III.2.2: All constructible numbers are algebraic}}</ref> It is the [[Euclidean closure]] of the [[rational number]]s, the smallest field extension of the rationals that includes the [[square root]]s of all of its positive numbers.{{sfnp|Kazarinoff|2003|p=46}}

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 [[abstract algebra|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.