Question

Prove that 1 + √2 is irrational.​

Answer 1

Step-by-step explanation:

Sol:

→ If possible let 1 + √2 is rational number.

→ 1 + √2 - 1 is also a rational number.

→ but √2 is irrational

★ Our contradiction arises by assuming 1 + √2 rational, that is wrong...

so, 1 + √2 is irrational number.

#hoPe #iT #heLps #yOu..

Answer 2

$\mathfrak{\underline{\underline{\green{Solution:-}}}}$

To Prove:

$\mathsf{ 1+\sqrt{2} \: \: is\: irrational }$

Proof:

Let us assume, $\</strong><strong>mathsf{</strong><strong>1</strong><strong>+</strong><strong> </strong><strong>\</strong><strong>sqrt{</strong><strong>2</strong><strong>}</strong><strong> </strong><strong>\</strong><strong>:</strong><strong>\</strong><strong>:</strong><strong> </strong><strong>is </strong><strong>\</strong><strong>:</strong><strong> </strong><strong>rational</strong><strong>}</strong><strong>$

Therefore

$\mathsf{ 1+\sqrt{2} = \dfrac{a}{b} \:\:\:\: (a,b \: are \: integers)}$

$\mathsf{\sqrt{2}= \dfrac{a}{b}-1}$

$\mathsf{\sqrt{2}= \dfrac{a-b}{b}}$

Here,

RHS is rational

But, LHS is irrational

A rational and irrational are never equal!

So, our assumption is False

Therefore, $\mathsf{ 1+\sqrt{2}}$ is irrational

HENCE PROVED!