# Prove the following statement: if x is irrational and y is irrational then x+y is irrational

Prove the following statement: if x is irrational and y is irrational then x+y is irrational

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Aniqa O'Neill

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.