Question

Proof that 6+√2 is irrational

Answer 1

Step-by-step explanation:

Let us assume that 6+√2 is rational.

That is , we can find coprimes a and b (b≠0) such that

6+\sqrt{2}=\frac{a}{b}6+

2

=

b

a

\implies \sqrt{2}=\frac{a}{b}-6⟹

2

=

b

a

−6

\implies \sqrt{2}=\frac{a-6b}{b}⟹

2

=

b

a−6b

Since , a and b are integers , \frac{a-6b}{b}

b

a−6b

is rational ,and so √2 is rational.

But this contradicts the fact that √2 is irrational.

So, we conclude that 6+√2 is irrational