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.
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