The correct response is that the sum or difference of a rational and irrational number is a rational number.
Explanation:
To understand why the sum of a rational and an irrational number is always irrational, we can use the definition of rational and irrational numbers:
- Rational numbers are numbers that can be expressed as the quotient of two integers (a/b, where a and b are integers and b ≠ 0).
- Irrational numbers cannot be expressed as such a quotient and have non-repeating, non-terminating decimal expansions.
Let's say you have a rational number \( r \) and an irrational number \( i \).
If the sum \( r + i \) were to be rational, then it could be expressed in the form of a/b. This would mean that:
\[ i = (r + i) - r \]
Since \( r \) is rational and \( r + i \) is rational (as assumed), then \( (r + i) - r \) would also have to be rational (as the difference of two rational numbers is rational).
However, this leads to a contradiction because it suggests that \( i \) is rational, which is not the case. Therefore, the original assumption that \( r + i \) is rational must be false. Hence, the sum of a rational number and an irrational number is irrational.
Thus, the correct response indicates that it's not the closure property that supports the statement about the sum of rational and irrational numbers being irrational, but rather the contradiction that arises from assuming their sum is rational.