ABSTRACT
In the eyes of the ancient Greeks, there was a deep connection between mathematics and life, and numbers entered into every facet of philoso phy, science, and art, such as the “golden ratio” ( 1^ ν^ ), an important guide in architecture. The only real numbers of interest to them could be constructed, using a compass and straight edge, by means of certain well-defined rules, which in modern terms could be described as follows:
Construction by Straight Edge and Compass
Definition 1 . We work in the Euclidean plane, and construct real numbers in a sequence of steps. We start at Step 1 with two points which we call 0 and 1, and with the line L q defined by these two points (which we identify as the “real line” ), but with no other lines or circles. By convention we identify Lq with the X-axis in the plane, and we shall define any real number a in terms of its corresponding point a on L q. Thus the numbers 0 and 1 are defined by the points 0 and 1.
