Question

root 12 is irrational Prove it​

Answer 1

Answer:

Let’s try to prove this via proof by contradiction.

Let  ∈ℕ , and let    not be the square of an integer.

Assume  √  is rational:

√=

with    and    non-negative integers,  >0  and    and    being co-prime, i.e.  gcd(,)=1 .

Then

=22

As  2  and  2  still cannot share a common factor,  2  and therefore    has to be  1 , otherwise the fraction cannot be an integer.

Therefore

=2

But we assumed    to not be the square of an integer. This is a contradiction, therefore the square root of such    cannot be rational, i.e. it is irrational.

This proof directly implies that the square root of a rational number,  ‾‾√ , is rational if and only if    and   are co-prime and both   and    are a perfect square. This is due to the fact that  ‾‾√=√√  and we have already shown when  √  and  √  are rational.

Step-by-step explanation:

Answer 2

Answer:

if root 2 could be writen as a irrational no. the consequences would be ansurd. so it is true to say that root 2 cannot be written in the form p/q . Hence tor root in an irrational