If x and yy are both irrational numbers then x+y need not to be irrational number. As a
example, take \(\displaystyle{x}=−\sqrt{{{2}}}{\quad\text{and}\quad}{y}={1}+\sqrt{{{2}}},{t}{h}{e}{n}{x}+{y}=−\sqrt{{{2}}}+{\left({1}+\sqrt{{{2}}}\right)}={1},\)

which is rational. But if xx and yy bothrational, say

\(\displaystyle{x}={\frac{{{a}}}{{{b}}}},{y}={\frac{{{c}}}{{{d}}}},\)

where a,b,ca,b,c and dd are integers with b,db,d not equal to 0.0. Now

\(\displaystyle{x}+{y}={\frac{{{a}}}{{{b}}}}+{\frac{{{c}}}{{{d}}}}={\frac{{{a}{d}+{b}{c}}}{{{b}{d}}}}.\)

Therefore, x+y is rational since ad+bc,bd are integers and bd not equal to 0.0.

which is rational. But if xx and yy bothrational, say

\(\displaystyle{x}={\frac{{{a}}}{{{b}}}},{y}={\frac{{{c}}}{{{d}}}},\)

where a,b,ca,b,c and dd are integers with b,db,d not equal to 0.0. Now

\(\displaystyle{x}+{y}={\frac{{{a}}}{{{b}}}}+{\frac{{{c}}}{{{d}}}}={\frac{{{a}{d}+{b}{c}}}{{{b}{d}}}}.\)

Therefore, x+y is rational since ad+bc,bd are integers and bd not equal to 0.0.