Question
Prove that following numbers are irrationals:
$4+\sqrt{2}$
$4+\sqrt{2}$
✓
Answer
Let us assume that $4+\sqrt{2}$ is rational number.
$\therefore\ 4+\sqrt{2}=\frac{\text{a}}{\text{b}}$ where, a and b are positive co-prime numbers.
$\Rightarrow\ \sqrt{2}=\frac{\text{a}}{\text{b}}-4$
$\Rightarrow\ \sqrt{2}=\frac{\text{a}-4\text{b}}{\text{b}}$
We know that $\sqrt{2}$ in an irratonal.
This contradicts our assumption that $4+\sqrt{2}$ is a rational number.
Hence, $4+\sqrt{2}$ must be irrational.
$\therefore\ 4+\sqrt{2}=\frac{\text{a}}{\text{b}}$ where, a and b are positive co-prime numbers.
$\Rightarrow\ \sqrt{2}=\frac{\text{a}}{\text{b}}-4$
$\Rightarrow\ \sqrt{2}=\frac{\text{a}-4\text{b}}{\text{b}}$
We know that $\sqrt{2}$ in an irratonal.
This contradicts our assumption that $4+\sqrt{2}$ is a rational number.
Hence, $4+\sqrt{2}$ must be irrational.
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