To show that the sum of a rational and irrational number is irrational, we can use the closure property of rational numbers. The key point is:
- The sum or difference of a rational number and an irrational number is not a rational number.
Thus, the correct response based on the concept would be:
The sum or difference of a rational and irrational number is a rational number. (This statement is incorrect, as it should be the opposite: it results in an irrational number.)
To clarify:
- A rational number can be expressed as a fraction of two integers (e.g., \( \frac{a}{b} \)).
- An irrational number cannot be expressed as such a fraction (e.g., \( \sqrt{2} \)).
- If you add a rational number \( r \) to an irrational number \( i \), you would have \( r + i \), which cannot be expressed as a fraction of integers, thus it remains irrational.
Since the statements you've provided contain redundancies and inaccuracies, the conclusion that the sum of a rational and an irrational number is irrational primarily stands on the understanding that if you add a rational to an irrational number, it will not yield a rational result, thereby proving that the sum is irrational.