I was trying to study Understanding Analysis by Stephen Abbott and I am stuck at the very second page. I feel stupid please help me. It says this:
Given two line segments $ab$ and $cd$ it would always be possible to find a third line segment whose length divides evenly into the first two.
How is this same as $cd=\frac{p}{q}ab$?
What the book says is that it was believed it was always possible to find a line segment whose length divides evenly into the first two. This would be true if all the numbers were rationals. However, since there are irrational numbers, this is not always the case. It's not possible, for example, to find a line segment whose lenght divide the sides and the diagonal of an unit square into pieces of same lenght, because there are no rational number whose square is $2$.