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  • Let x and y be rational and irrational number respectively. Is x + y necessarily an irrational number? Give an example in support of your answer.

Let x and y be rational and irrational number respectively. Is x + y necessarily an irrational number?
Give an example in support of your answer.

Answers (1)

Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. So it is basically a fraction with non-zero denominator.
Irrational numbers are real numbers which cannot be represented as simple fractions.
Given that x and y be rational and irrational number respectively.
So, (x + y)  is necessarily an irrational number.
For example, let x=2,y=\sqrt{3}
Then, x + y = 2+\sqrt{3}
If possible, let us assume x + y = 2+\sqrt{3} be a rational number.
Consider a = 2 + \sqrt{3}
On squaring both sides, we get
a^{2}=(2+\sqrt{3})^{2}  (using identity (a + b)^{2} = a^{2} + b^{2} + 2ab)
\Rightarrow a^{2}=2^{2}+(\sqrt{3})^{2}+2(2)(\sqrt{3})
\Rightarrow a^{2}=4+3+4\sqrt{3}
\Rightarrow \frac{a^{2}-7}{4}=\sqrt{3}
But we have assumed a is rational
\Rightarrow \frac{a^{2}-7}{4} is rational
\Rightarrow \sqrt{3} is rational which is not true.
Hence our assumption was incorrect, so 2+ \sqrt{3} is irrational.
Hence proved

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