Question

Show that 2 + √2 is not a rational number.

Answer 1

Answer:

No 2+√2 is not a rational number

Answer 2

Assume that 2 + √2 is a rational number.

Then,

• It must be expressed in form of p/q where p & q are non-zero integers and p & q are co-primes also.

So,

$\implies \tt2 + \sqrt{2} = \frac{p}{q} \\ \tiny \text{transposing \: 2 \: to \: RHS} \\ \implies \tt \sqrt{2} = \frac{p}{q} - 2 \\ \\ \implies \tt \sqrt{2} = \frac{ p- 2q}{q}$

Here,

• RHS wil result as integer, but LHS is already an irrational number.

i.e.

$\huge{ \cal{LHS \: ≠ \: RHS}}$

• This has happened due to our wrong assumption that 2 + √2 is a rational number.

Now,

• It can be said that 2 + √2 is not a rational number.