Editing Constructible number (section)

===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}}