Prove That 2/√7 Is Irrational

Video Explanation

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

Solution

Question: Prove that the number 2/√7 is irrational.

Step 1: Assume the Contrary

Assume that 2/√7 is rational. Then it can be written as a fraction of two integers in lowest terms:

2/√7 = p/q, where p and q are integers with no common factors and q ≠ 0.

Step 2: Isolate √7

Rearrange the equation to isolate √7:

√7 = 2q/p

Now square both sides:

7 = (2q/p)²

7 = 4q²/p²

Multiply both sides by p²:

7p² = 4q²

Step 3: Analyze Divisibility

This equation shows that 7 divides the left side, so 7 must divide 4q². Since 7 is prime and doesn’t divide 4, it must divide q², and therefore q. Let q = 7k for some integer k.

Substitute back:

7p² = 4(7k)²

7p² = 4 × 49k²

7p² = 196k²

p² = 28k²

This shows p² is divisible by 7, so p is divisible by 7.

Step 4: Contradiction

We found both p and q are divisible by 7, which contradicts our assumption that p/q was in lowest terms.

Final Answer

∴ The number 2/√7 cannot be expressed as a ratio of two integers in lowest terms, so 2/√7 is irrational.

Conclusion

By assuming 2/√7 is rational and reaching a contradiction, we conclude that 2/√7 is irrational.