Question

Prove that (2 + √2 ) is irrational.

Answer 1

Let 2 + √2 be rational no.

2 + √2 = p/q

where p and q are co prime and and q≠0

√2 = p/q - 2

In LHS there is Irrational no. and in RHS there is rational no. This is not possible. This contradiction arise due to our wrong assumption. Thus our assumption is wrong and 2 + √2 is Irrational no.

Hence proved

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Answer 2

${\huge{\overbrace{\underbrace{\red{Answer:}}}}}$

Let's assume 2 + √2 as ${\frac{p}{q}$ where p and q are comprise and q ≠ 0.

2 + √2 = ${\frac{p}{q}}$

√2 = ${\frac{p}{q}}$ - 2

√2 = ${\frac{p-2q}{q}}$

→ As we can see, the value of √2 is coming rational but we know that it's irrational.

→ This contradicts our assumption.

→ Therefore, this needs to be concluded that √2 is irrational.

${\bold{\huge{Hence Proved ✓}}}$