To prove that the product of √2 and √5 is irrational, we can use a proof by contradiction. This method helps us show that assuming the opposite leads to an impossible situation.
1. **Assumption**: First, let’s suppose that √2 * √5 is rational. This means we can express it as a fraction of two integers p/q, where p and q are integers, and q ≠ 0.
2. **Simplifying the Expression**: The product can be simplified as follows:
√2 * √5 = √(2 * 5) = √10
3. **Analyzing the New Expression**: Now we have the expression as √10. If our initial assumption is correct, then we can also express it as a rational number:
√10 = p/q
4. **Squaring Both Sides**: Next, square both sides to eliminate the square root:
10 = p²/q²
Thus, we have 10 * q² = p².
5. **Divisibility by 10**: From this equation, we see that p² is divisible by 10. Since 10 = 2 * 5, p² must also be divisible by both 2 and 5.
6. **Conclusion on p**: If p² is divisible by 2, then p itself must also be divisible by 2 (since the square of an odd number is odd). Thus, we can express p as:
p = 2k, where k is some integer.
7. **Substituting Back**: Now substitute p = 2k back into the earlier equation:
10 * q² = (2k)² = 4k²
8. **Rearranging**: This gives us:
10 * q² = 4k²
Hence, q² = (4/10) * k² = (2/5) * k².
9. **Divisibility of q**: This implies that q² is also divisible by 2. Therefore, q must also be divisible by 2.
10. **Contradiction**: We have reached a contradiction. Both p and q cannot be even, as it violates the definition of a fraction being in its simplest form (i.e., having no common factors other than 1). Hence, our initial assumption that √2 * √5 is rational must be incorrect.
11. **Conclusion**: Therefore, we conclude that √2 * √5, or equivalently √10, is indeed irrational.