To prove that \( \frac{1}{2} + \sqrt{2} \) is irrational, we'll start by assuming the opposite: that it is rational.
Assume \( \frac{1}{2} + \sqrt{2} = r \), where \( r \) is a rational number. We can now rearrange this equation:
\[ \sqrt{2} = r - \frac{1}{2} \]
The right side of this equation, \( r - \frac{1}{2} \), is rational because it is the difference of two rational numbers (since \( r \) is rational and \( \frac{1}{2} \) is rational).
Now, if \( \sqrt{2} \) is rational, then it can be expressed as a ratio of two integers. However, it is known that \( \sqrt{2} \) is irrational. This leads to a contradiction: assuming that \( \frac{1}{2} + \sqrt{2} \) is rational implies that an irrational number \( \sqrt{2} \) is rational.
Therefore, our initial assumption must be false. Thus, \( \frac{1}{2} + \sqrt{2} \) is irrational.
In summary, the sum \( \frac{1}{2} + \sqrt{2} \) must be irrational because it is the sum of a rational number, \( \frac{1}{2} \), and an irrational number, \( \sqrt{2} \); the sum of a rational and an irrational number is always irrational.
1 answer