As an example, the sum \frac 34 + \frac 18 = \frac{3 \times 8+4 \times 1}{4 \times 8} = \frac{24 + 4}{32} = \frac{28}{32} = \frac{7}{8}.
A common construction of the set of real numbers is the Dedekind completion of the set of rational numbers.