1.5. Constructible points and constructible numbers 13 circle. This circle intersects the x-axis at I and also at (−1, 0). The point (−1, 0) is thus an example of a constructible point. We can henceforth use (−1, 0) as a point in future constructions. Now draw the (constructible) circle with center I and radius 1. This constructs the point (2, 0). Using this point as center, draw a circle of radius 1 this gives that (3, 0) has been constructed. Continue, and we can get that all the points (n, 0) with n Z are constructible. Now construct the line perpendicular to the x-axis and through O the y-axis. We can now continue this process as we did above to obtain that all the points (0,n), with n Z, are constructible. Then we can bisect the angle formed by the two axes and look at the points where this line intersects the various constructible circles. Clearly we can find a rich collection of constructible points. We have to give this process even more precision and so we use induction to define what we mean by constructible lines, circles, and points. 1.5.1. Definition. Say that a point in the plane R2 is a first order con- structible point if it is the point O or I. Say that a line is a first order constructible line if it passes through two first order constructible points. Say that a circle is a first order constructible circle if its center is a first or- der constructible point and its radius is the distance between two first order constructible points. Now assume that n 1 and we have the set Pn of n-th order constructible points, the set Ln of n-th order constructible lines, and the set Cn of n-th order constructible circles. Define Pn+1 is the union of Pn together with all points of intersection of two lines from Ln, two circles from Cn, or a line from Ln and a circle Cn Ln+1 is the union of Ln together with all lines that pass through two points in Pn+1 Cn+1 is the union of Cn together with all circles whose center is a point in Pn+1 and whose radius is the distance between two points in Pn+1. Let P = n=1 Pn, L = n=1 Ln, C = n=1 Cn be the sets of constructible points, lines, and circles, respectively. 1.5.2. Definition. A real number a is a constructible number if it is one of the coordinates of a constructible point. Denote the set of all constructible numbers by K. Of course both coordinates of a constructible point are constructible numbers. Also from the discussion above we have already proved the fol- lowing.
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