Prove That 5√2 Is Irrational

Video Explanation

Watch the video below to understand the complete solution step by step:

Solution

Question: Prove that the number 5√2 is irrational.

Step 1: Assume the Contrary

Assume that 5√2 is rational. Then it can be written in the form:

5√2 = p/q,

where p and q are integers having no common factors and q ≠ 0.

Step 2: Isolate √2

Rearranging the equation:

√2 = p / (5q)

Step 3: Square Both Sides

2 = p² / (25q²)

Multiply both sides by 25q²:

50q² = p²

Step 4: Analyze Divisibility

From the equation 50q² = p², we see that p² is divisible by 2 and 25. Hence, p² is even, which implies that p is even.

Let p = 2k, where k is an integer.

Substitute p = 2k:

50q² = (2k)²

50q² = 4k²

25q² = 2k²

This shows that q² is even, so q is also even.

Step 5: Contradiction

Thus, both p and q are even, which means they have a common factor 2. This contradicts the assumption that p/q is in the lowest terms.

Final Answer

∴ The number 5√2 cannot be expressed as a ratio of two integers. Hence, 5√2 is irrational.

Conclusion

By assuming 5√2 to be rational and reaching a contradiction, we conclude that 5√2 is irrational.