Question
Show that the following numbers are irrational.
$3-\sqrt{5}$
$3-\sqrt{5}$
✓
Answer
Let assume that $(3-\sqrt{5})$ is rational than there exist co-prime a and b such that$(3-\sqrt{5})=\frac{\text{a}}{\text{b}}\ \dots(1)$
$\Rightarrow\ \sqrt{5}=3-\frac{\text{a}}{\text{b}}$
$\Rightarrow\ \sqrt{5}=\frac{3\text{b}-\text{a}}{\text{b}}\ \dots(2)$
Education (2) shows that $\sqrt{5}$ is rational. This is centradiction
Thus, $(3-\sqrt{5})$ is irrational.
$\Rightarrow\ \sqrt{5}=3-\frac{\text{a}}{\text{b}}$
$\Rightarrow\ \sqrt{5}=\frac{3\text{b}-\text{a}}{\text{b}}\ \dots(2)$
Education (2) shows that $\sqrt{5}$ is rational. This is centradiction
Thus, $(3-\sqrt{5})$ is irrational.
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