1.5 Incompleteness of RatIonal numbe Rs
As we have seen in the earlier section that, there exists a rational number between any two points on
the number line, and strictly speaking there are a lot of rational numbers. So, one can doubt whether
the entire real line consists of only rational numbers. And many people might wish this were the case
because each real number could be expressed not just as a point on the number line but also algebraically
as the ratio of integers, in such a number system. Before 500 BC, a group of mathematicians led by
Pythagoras believed in this. They believed that ...