Editing Constructible number (section)

== Geometric definitions ==
===Geometrically constructible points===
Let <math>O</math> and <math>A</math> be two given distinct points in the [[plane (geometry)|Euclidean plane]], and define <math>S</math> to be the set of points that can be constructed with compass and straightedge starting with <math>O</math> and <math>A</math>. Then the points of <math>S</math> are called '''constructible points'''. <math>O</math> and <math>A</math> are, by definition, elements of <math>S</math>. To more precisely describe the remaining elements of <math>S</math>, make the following two definitions:{{sfnp|Kazarinoff|2003|p=10}}
* a line segment whose endpoints are in <math>S</math> is called a '''constructed segment''', and
* a circle whose center is in <math>S</math> and which passes through a point of <math>S</math> (alternatively, whose radius is the distance between some pair of distinct points of <math>S</math>) is called a '''constructed circle'''.
Then, the points of <math>S</math>, besides <math>O</math> and <math>A</math> are:{{sfnmp|Kazarinoff|2003|1p=10|Martin|1998|2pp=30–31|2loc=Definition 2.1}}
* the [[Line-line intersection|intersection]] of two non-parallel constructed segments, or lines through constructed segments,
* the intersection points of a constructed circle and a constructed segment, or line through a constructed segment, or
* the intersection points of two distinct constructed circles.

As an example, the midpoint of constructed segment <math>OA</math> is a constructible point. One construction for it is to construct two circles with <math>OA</math> as radius, and the line through the two crossing points of these two circles. Then the midpoint of segment <math>OA</math> is the point where this segment is crossed by the constructed line.<ref>This construction for the midpoint is given in Book I, Proposition 10 of [[Euclid's Elements|Euclid's ''Elements'']].</ref>

===Geometrically constructible numbers===
The starting information for the geometric formulation can be used to define a [[Cartesian coordinate system]] in which the point <math>O</math> is associated to the origin having coordinates <math>(0,0)</math> and in which the point <math>A</math> is associated with the coordinates <math>(1, 0)</math>. The points of <math>S</math> may now be used to link the geometry and algebra by defining a '''constructible number''' to be a coordinate of a constructible point.{{sfnp|Kazarinoff|2003|p=18}}

Equivalent definitions are that a constructible number is the <math>x</math>-coordinate of a constructible point <math>(x,0)</math>{{sfnp|Martin|1998|pp=30–31|loc=Definition 2.1}} or the length of a constructible line segment.<ref>{{harvp|Herstein|1986|p=237}}. To use the length-based definition, it is necessary to include the number zero as a constructible number, as a special case.</ref> In one direction of this equivalence, if a constructible point has coordinates <math>(x,y)</math>, then the point <math>(x,0)</math> can be constructed as its perpendicular projection onto the <math>x</math>-axis, and the segment from the origin to this point has length <math>x</math>. In the reverse direction, if <math>x</math> is the length of a constructible line segment, then intersecting the <math>x</math>-axis with a circle centered at <math>O</math> with radius <math>x</math> gives the point <math>(x,0)</math>. It follows from this equivalence that every point whose Cartesian coordinates are geometrically constructible numbers is itself a geometrically constructible point. For, when <math>x</math> and <math>y</math> are geometrically constructible numbers, point <math>(x,y)</math> can be constructed as the intersection of lines through <math>(x,0)</math> and <math>(0,y)</math>, perpendicular to the coordinate axes.{{sfnmp|Moise|1974|1p=227|Martin|1998|2p=33|2loc=Theorem 2.4}}