Prove That √p Is Irrational for Any Prime Positive Integer p

Video Explanation

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Solution

Statement: Prove that for any prime positive integer p, √p is an irrational number.

Proof:

Let us assume that √p is a rational number.

Then we can write:

√p = a / b

where a and b are positive integers having no common factor other than 1.

Squaring both sides, we get:

p = a² / b²

⇒ a² = p b²

This shows that a² is divisible by p, hence a is divisible by p.

Let a = p k for some integer k.

Substituting in the equation:

(p k)² = p b²

⇒ p k² = b²

This implies that b² is divisible by p, hence b is divisible by p.

Thus, both a and b are divisible by p.

This contradicts the assumption that a and b have no common factor other than 1.

∴ Our assumption is wrong.

Hence, √p is an irrational number.

Final Answer

∴ For any prime positive integer p, √p is irrational.

Conclusion

Thus, by using the method of contradiction, we have proved that the square root of any prime number is irrational.