Constructible number

In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length |r| can be constructed with compass and straightedge in a finite number of steps. Not all real numbers are constructible and to describe those that are, algebraic techniques are usually employed. However, in order to employ those techniques, it is useful to first associate points with constructible numbers. a ⋅ b {displaystyle acdot b} based on the intercept theorem a b {displaystyle {frac {a}{b}}} based on the intercept theorem a {displaystyle {sqrt {a}}} based on the geometric mean theorem In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length |r| can be constructed with compass and straightedge in a finite number of steps. Not all real numbers are constructible and to describe those that are, algebraic techniques are usually employed. However, in order to employ those techniques, it is useful to first associate points with constructible numbers. A point in the Euclidean plane is a constructible point if it is either endpoint of the given unit segment, or the point of intersection of two lines determined by previously obtained constructible points, or the intersection of such a line and a circle having a previously obtained constructible point as a center passing through another constructible point, or the intersection of two such circles. Now, by introducing cartesian coordinates so that one endpoint of the given unit segment is the origin (0, 0) and the other at (1, 0), it can be shown that the coordinates of the constructible points are constructible numbers. In algebraic terms, a number is constructible if and only if it can be obtained using the four basic arithmetic operations and the extraction of square roots, but of no higher-order roots, from constructible numbers, which always include 0 and 1. The set of constructible numbers can be completely characterized in the language of field theory: the constructible numbers form the quadratic closure of the rational numbers: the smallest field extension that is closed under square roots. This has the effect of transforming geometric questions about compass and straightedge constructions into algebra. This transformation leads to the solutions of many famous mathematical problems, which defied centuries of attack. The traditional approach to the subject of constructible numbers has been geometric in nature, but this is not the only approach. However, the geometric approach does provide the motivation for the algebraic definitions and is historically the way the subject developed. In presenting the material in this manner, the basic ideas are introduced synthetically and then coordinates are introduced to transition to an algebraic setting. Let O and A be two given distinct points in the Euclidean plane. The set of points that can be constructed with compass and straightedge starting with O and A will be denoted by S and whose elements will be called constructible points. O and A are, by definition, elements of S. To more precisely describe the elements of S, we make the following two definitions: Then, the points of S, besides O and A are: As an example, the midpoint of constructed segment OA is a constructible point. To see this, note that the constructed circle C1 with center O and passing through A intersects the constructed circle C2 with center A and passing through O at the constructible points P and Q. The intersection of constructed segment PQ with constructed segment OA is the desired constructed midpoint. A Cartesian coordinate system can be introduced where the point O is associated to the origin having coordinates (0, 0) and the point A is associated with (1, 0). The points of S may now be used to link the geometry and algebra, namely, we define Due to point A, 0 and 1 are constructible numbers. Let P be a point in S, that is, a constructible point. If P lies on the x-axis, then OP is a constructed segment and the first coordinate of P is, in absolute value, the length of this constructed segment. If P does not lie on the x-axis then let the foot of the perpendicular from P to the x-axis be the point Q. The point Q is a constructed point, so PQ and OQ are constructed segments. The absolute values of the coordinates of the point P are therefore lengths of constructed segments. This process is reversible, so it is possible to use this device to provide an alternate characterization of constructible numbers, namely: