Show That 1/√2 Is Irrational

Video Explanation

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

Solution

Question: Show that the number 1/√2 is irrational.

Step 1: Assume the Contrary

Assume that 1/√2 is rational. Then it can be written as a fraction of two integers in lowest terms:

1/√2 = p/q, where p and q are integers with no common factors and q ≠ 0.

Step 2: Manipulate the Equation

From 1/√2 = p/q, cross-multiply:

q = p√2

Now square both sides:

q² = p² × 2

So, q² = 2p²

Step 3: Analyze Parity

This equation implies that q² is even (because it equals 2×p²). Therefore q must be even.

Let q = 2k for some integer k. Substitute back:

(2k)² = 2p²

4k² = 2p²

2k² = p²

This shows that p² is even, which means p is also even.

Step 4: Contradiction

We found that both p and q are even, meaning they have a common factor 2. This contradicts our assumption that p/q was in lowest terms.

Final Answer

∴ The number 1/√2 cannot be expressed as a ratio of two integers in lowest terms, so 1/√2 is irrational.

Conclusion

By assuming 1/√2 is rational and reaching a contradiction, we conclude that 1/√2 is irrational.