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.

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