Question

minus root 6 is a rational number or irrational number​

Answer 1

Given:

• To state whether -√6 is a rational or a Irrational Number .

Solution:

-√6 has a non- terminating and a non- repeating decimal .So definitely it is a Irrational number.

Let's Prove it now :

♦Proof ♦

On the contrary let us assume -√6 to be a rational number . So , it can be expressed in the form of p/q where p and q are integers and q≠0. Also HCF of p and q is 1 .

So , as per our assumption :-

⇒ -√6 = p/q.

⇒ (-√6)² = (p/q)².

⇒ 6 = p²/q²

⇒ 6q² = p². .............................(i)

This implies that 6 is a factor of p². So by the " Fundamental Theorem of Arthemetic " we can say that 6 is also a factor of p.

⇒ p = 6k

Now put this value in equⁿ (i)

⇒ 6q² = (6k)².

⇒ 6q² = 36k²

⇒ q² = 36k²/6

⇒ q² = 6k²

This implies that 6 is a factor of q². So by the " Fundamental Theorem of Arthemetic " we can say that 6 is also a factor of q.

But this contradicts our assumption that p and q are co-primes . Hence our assumption was wrong -√6 is a Irrational number.

Hence Proved.