# Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.3

3. Prove that both the sum and the product of two rational numbers are rational.
Solution:

We assume that x and y are rational numbers. Then these rational numbers will be of form $x=\frac{a}{b}$ and $y=\frac{c}{d}$. Where a,b,c,d are integers with $b\neq0$ and $d\neq0$.

Now,  $xy=\left(\begin{array}{c}\frac{a}{b}\end{array}\right)\cdot\left(\begin{array}{c}\frac{c}{d}\end{array}\right)$

$xy=\frac{ac}{bd}$

And $x+y=\frac{a}{b}+\frac{c}{d}$

$x+y=\frac{(ad+bc)}{bd}$

Where $bd\neq0$ Also both x+y and xy are ratios of integrs.Therefore they are both rational numbers. Hence our proof is complete.