Question

Prove that the sum of 1/2 and square root of 2 is irrational show your work

Answer 1

To prove that \( \frac{1}{2} + \sqrt{2} \) is irrational, we will proceed by contradiction. We will assume that \( \frac{1}{2} + \sqrt{2} \) is rational and show that this leads to a contradiction.

1. **Assume that \( \frac{1}{2} + \sqrt{2} \) is rational**:
Let \( \frac{1}{2} + \sqrt{2} = r \), where \( r \) is a rational number.

2. **Rearranging the equation**:
We can rearrange the equation to isolate \( \sqrt{2} \):
\[
\sqrt{2} = r - \frac{1}{2}
\]

3. **Expressing \( \sqrt{2} \) in terms of rational numbers**:
Since \( r \) is rational, \( r - \frac{1}{2} \) is also rational because the difference of two rational numbers (here \( r \) and \( \frac{1}{2} \)) is rational.

4. **Conclusion about \( \sqrt{2} \)**:
We have shown that \( \sqrt{2} \) equals a rational number, \( r - \frac{1}{2} \). However, we know that \( \sqrt{2} \) is an irrational number.

5. **Contradiction**:
This leads to a contradiction because we assumed \( \frac{1}{2} + \sqrt{2} \) is rational but deduced that \( \sqrt{2} \) is rational, which it is not.

Therefore, our initial assumption that \( \frac{1}{2} + \sqrt{2} \) is rational must be false. Hence, we conclude that the sum \( \frac{1}{2} + \sqrt{2} \) is irrational.

This completes the proof.