l Prove provesqrt(2)isirrational
Enter your statement to prove

Prove the following

provesqrt(2)isirrational

Use proof by contradiction.

Assume √2 is rational.

This means that √2 = p/q for
some integers p and q, with q <>0.
We assume p and q are in lowest terms.

Square both sides and we get:

2 = p2/q2

Cross multiply:

p2 = 2q2

This means p2 must be an even number
This means p is also even since the square of an odd is odd.

So we have p = 2k for some integer k:

2q2 = p2 = (2k)2 = 4k2

2q2 = 4k2

Divide each side by 2

2q2
2
=
  
4k2
2

q2 = 2k2

q2 is also even, therefore q must be even
Since the square of an odd number is odd.

So both p and q are even.
This contradicts are assumption that p and q were in lowest terms in p/q.

Final Answer


So √2 cannot be rational.