Prove That √p + √q Is an Irrational Number
Video Explanation
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Solution
Statement: If p and q are prime positive integers, prove that √p + √q is an irrational number.
Proof:
Let us assume that √p + √q is a rational number.
Then, squaring both sides, we get:
(√p + √q)2 is rational
⇒ p + q + 2√pq is rational
Since p and q are integers, p + q is a rational number.
Subtracting p + q from both sides, we get:
2√pq is rational
Dividing both sides by 2 (a non-zero rational number), we get:
√pq is rational
But since p and q are prime numbers, their product pq is not a perfect square.
Therefore, √pq is irrational.
This is a contradiction.
∴ Our assumption is wrong.
Hence, √p + √q is an irrational number.
Final Answer
∴ √p + √q is an irrational number.
Conclusion
Thus, using the method of contradiction, we have proved that the sum of square roots of two prime positive integers is always irrational.