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.

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)


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


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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