Prove That 4 + √2 Is Irrational

Video Explanation

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

Solution

Question: Prove that the number 4 + √2 is irrational.

Step 1: Assume the Contrary

Assume that 4 + √2 is rational. Then it can be written in the form:

4 + √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/q) − 4

Step 3: Square Both Sides

2 = [(p/q) − 4]²

2 = (p²/q²) − (8p/q) + 16

Multiply both sides by q²:

2q² = p² − 8pq + 16q²

Rearranging:

0 = p² − 8pq + 14q²

Step 4: Analyze the Equation

The equation

p² − 8pq + 14q² = 0

is a quadratic equation in p. Its discriminant is:

D = (−8q)² − 4(1)(14q²)

D = 64q² − 56q²

D = 8q²

Since 8q² is not a perfect square for any non-zero integer q, this equation has no integer solution. This contradicts our assumption.

Final Answer

∴ The number 4 + √2 cannot be expressed as a ratio of two integers. Hence, 4 + √2 is irrational.

Conclusion

By assuming 4 + √2 to be rational and reaching a contradiction, we conclude that 4 + √2 is irrational.