Proof that sqrt(p) is irrational if p is a prime number:

Assumption: Let's assume that sqrt(p) is rational, where p is a prime number.

Definition of rational number: A rational number is any number that can be expressed as a fraction of two integers.

Definition of irrational number: An irrational number is any number that cannot be expressed as a fraction of two integers.

Proof:

Let's assume that sqrt(p) is rational. Therefore, it can be expressed as a fraction of two integers a and b, where a and b have no common factors.

Therefore, we can write:

sqrt(p) = a/b

Squaring both sides, we get:

p = a^2/b^2

Multiplying both sides by b^2, we get:

p * b^2 = a^2

Conclusion: Since p is a prime number, it can only be factored into p and 1. Therefore, a^2 can only have p as a prime factor or 1 as a factor.

Case 1: If a^2 has p as a prime factor, then b^2 must also have p as a prime factor.

Case 2: If a^2 has 1 as a factor, then b^2 must also have 1 as a factor.

Contradiction: In both cases, a and b have a common factor of p, which contradicts our initial assumption that a and b have no common factors.

Therefore, our assumption that sqrt(p) is rational is incorrect. Hence, sqrt(p) is irrational if p is a prime number.