Question

Given that √2 is irrational. Show that 5+√2 is irrational

Answer 1

Answer:

5+root2 is also irrational number because

irrational number is a number which cannot be written in the form of p/q and is non-terminating and non-repeating.

so root 2 is approximately 1.414....... and afer adding we get 6.414........ which is in decimal form and not in p/q form

Hence, 5+root2 is also an irrational number.....

or

by the irrational number property

rational+irrational= irrational

Hope this helps you

Answer 2

Given:

• √2 is an irrational number.

To prove:

• 5 + √2 is irrational.

Solution:

Let us assume 5 + √2 to be a rational number. This implies that 5 + √2 can be expressed in the form p/q where q ≠ 0 and p and q are co-primes.

[Co-primes are numbers that are divisible by 1 the number itself and no other integer]

$\sf \Longrightarrow 5 + \sqrt{2} = \dfrac{p}{q}$

$\sf \Longrightarrow \sqrt{2} = \dfrac{p}{q} - 5$

$\sf \Longrightarrow \sqrt{2} = \dfrac{p - 5q}{q}$

Here, we know that √2 is an irrational number (given in the question), and (p - 5q)/q is a rational number.

But;

Irrational number ≠ Rational number.

Therefore, √2 cannot be equal to (p - 5q)/q. This contradiction is due to my incorrect assumption that 5 + √2 is a rational number.

∴ 5 + √2 is an irrational number.