Show That 2 − √3 Is an Irrational Number

Video Explanation

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

Solution

Question: Show that the number 2 − √3 is irrational.

Step 1: Assume the Contrary

Assume that 2 − √3 is rational. Then it can be written in the form:

2 − √3 = p/q,

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

Step 2: Isolate √3

Rearranging the equation:

√3 = 2 − (p/q)

Step 3: Square Both Sides

3 = [2 − (p/q)]²

3 = 4 − (4p/q) + (p²/q²)

Multiply both sides by q²:

3q² = 4q² − 4pq + p²

Rearranging:

0 = p² − 4pq + q²

Step 4: Analyze the Equation

The equation

p² − 4pq + q² = 0

is a quadratic equation in p. Its discriminant is:

D = (−4q)² − 4(1)(q²)

D = 16q² − 4q²

D = 12q²

Since 12q² 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 2 − √3 cannot be expressed as a ratio of two integers. Hence, 2 − √3 is an irrational number.

Conclusion

By assuming 2 − √3 to be rational and reaching a contradiction, we conclude that 2 − √3 is irrational.