To show that \(\frac{1}{2} + \sqrt{2}\) is irrational, we will use a proof by contradiction.
-
Assume the contrary: Suppose \(\frac{1}{2} + \sqrt{2}\) is rational. This means that we can express it as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\).
-
Re-arrange the equation: From the assumption, we have: \[ \sqrt{2} = \frac{a}{b} - \frac{1}{2} \]
-
Combine the fractions: \[ \sqrt{2} = \frac{2a - b}{2b} \]
-
Rational Expression: Here, \(\frac{2a - b}{2b}\) is a rational number because both \(2a - b\) and \(2b\) are integers and \(2b \neq 0\).
-
Contradiction: We know that \(\sqrt{2}\) is irrational. However, this leads to the conclusion that \(\sqrt{2}\) can be written as a rational number (since we expressed it as \(\frac{2a - b}{2b}\)). This is a contradiction.
-
Conclusion: Therefore, our initial assumption that \(\frac{1}{2} + \sqrt{2}\) is rational must be false, which means \(\frac{1}{2} + \sqrt{2}\) is indeed irrational.
In summary, the sum \(\frac{1}{2} + \sqrt{2}\) is irrational because it results in a rational expression implying that \(\sqrt{2}\) itself is rational, contradicting the well-known fact that \(\sqrt{2}\) is irrational.