Real Number System

Before working with real numbers formally, it helps to see how the number system was actually built up, one enlargement at a time, starting from counting.

Each inclusion above solves a problem the smaller system could not: integers let you subtract past zero (debts), rationals let you divide evenly (splitting a cake into 5 equal pieces needs $x=\frac15$, which has no solution in $Z$), and finally the real numbers fill in every remaining point of the …