Let alpha a bi be a complex number Then alpha is construct

Let alpha = a + bi be a complex number. Then alpha is constructible if and only if both a and b are constructible real numbers.

Solution

Lemma 1.2 If U, V, P are distinct constructible points with P L (U, V ), then there is a constructible point Q with L(P, Q) parallel to L(U, V ).

If is constructible, then Lemma 1.2 draws the vertical line L through (a,b) which is parallel to the y-axis. It follows that x is a constructible number, for the point (a, 0) is constructible, being the intersection of L and the x-axis. Similarly, the point (0, b) is the intersection of the y-axis and a line through (a,b) which is parallel to the x-axis. It follows that P = (b, 0) is constructible, for it is an intersection point of the x-axis and C (O, P). Hence, b is a constructible number.

Conversely, assume that a and b are constructible numbers; that is, Q = (a, 0) and P = (b, 0) are constructible points. The point (0, b) is constructible, being the intersection of the y-axis and C(O, P). By Lemma 1.2, the vertical line through (a, 0) as well as the horizontal line through (0, b) can be drawn, and (a,b) is the intersection of these lines. Therefore, (a,b) is a constructible point, and so = a + i b is a constructible number.