Question

Use method of contradiction to show that $\sqrt{3}$ and $\sqrt{5}$ are irrational number.

Accepted Answer

Let us suppose that $\sqrt{3}$ and $\sqrt{5}$ are rational numbers

Therefore, $\sqrt{3} = \frac{a}{b}$ and $\sqrt{5} = \frac{x}{y}$ (Where $a, b \in 7$ and $b, y \neq 0 x, y$ )

Squaring both sides

$3 = \frac{a^{2}}{b^{2}}, 5 = \frac{x^{2}}{y^{2}}$

$3 b^{2} = a^{2}, 5 y^{2} = x^{2}$

$\Rightarrow a^{2}$ and $x^{2}$ are odd as $3 b^{2}$ and $5 y^{2}$ are odd .

$\Rightarrow a$ and $x$ are odd. $. . . (1)$

Let $a = 3 c, x = 5 z$

$a^{2} = 9 c^{2}, x^{2} = 25 z^{2}$

$3 b^{2} = 9 c^{2}, 5 y^{2} = 25 z^{2}$ (From equation $. . . . (1)$

$\Rightarrow b^{2} = 3 c^{2}, y^{2} = 5 z^{2}$

$\Rightarrow b^{2}$ and $y^{2}$ are odd as $3 c^{2}$ and $5 z^{2}$ are odd .

$\Rightarrow b$ and $y$ are odd $. . . (2)$

From equation $(1)$ and $(2)$ we get $a, b, x, y$ are odd integers.

i.e., $a, b$ , and $x, y$ have common factors $3$ and $5$ this contradicts our assumption that $\frac{a}{b} a n d \frac{x}{y}$ are rational

i.e, $a, b$ and $x, y$ do not have any common factors other than.

$\Rightarrow \frac{a}{b} a n d \frac{x}{y}$ is not rational

$\Rightarrow \sqrt{3}$ and $\sqrt{5}$ are irrational.

R K Bansal Solutions for Exercise 1: EXERCISE

R K Bansal Mathematics Solutions for Exercise - R K Bansal Solutions for Exercise 1: EXERCISE

Questions from R K Bansal Solutions for Exercise 1: EXERCISE with Hints & Solutions

Learn and Prepare for any exam you want !