KnowledgeBoat Logo
|

Mathematics

Prove that 5−2\sqrt{5} - 2 is an irrational number.

Rational Irrational Nos

4 Likes

Answer

Let us assume 5−2\sqrt{5} - 2 is a rational number.

Let 5−2=x\sqrt{5} - 2 = x

Squaring on both sides, we get :

⇒(5−2)2=x2⇒(5)2+(2)2−2×5×2=x2⇒5+4−45=x2⇒9−45=x2⇒9−x2=45⇒5=9−x24.\Rightarrow (\sqrt{5} - 2)^2 = x^2 \\[1em] \Rightarrow (\sqrt{5})^2 + (2)^2 - 2 \times \sqrt{5} \times 2 = x^2 \\[1em] \Rightarrow 5 + 4 - 4\sqrt{5} = x^2 \\[1em] \Rightarrow 9 - 4\sqrt{5} = x^2 \\[1em] \Rightarrow 9 - x^2 = 4\sqrt{5} \\[1em] \Rightarrow \sqrt{5} = \dfrac{9 - x^2}{4}.

Here, x is rational,

∴ x2 is rational ………(1)

⇒ 9 - x2 is rational

So, 9−x24\dfrac{9 - x^2}{4} is rational.

⇒9−x24=5\Rightarrow \dfrac{9 - x^2}{4} = \sqrt{5} is rational

But 5\sqrt{5} is irrational

⇒9−x24\Rightarrow \dfrac{9 - x^2}{4} is irrational i.e., 9 - x2 is irrational and so x2 is irrational ……..(2)

From (1), x2 is rational, and

from (2), x2 is irrational

∴ We arrive at a contradiction.

So, our assumption that 5−2\sqrt{5} - 2 is a rational number is wrong.

∴ 5−2\sqrt{5} - 2 is irrational.

Hence, proved that 5−2\sqrt{5} - 2 is an irrational number.

Answered By

2 Likes

Related Questions