The correct response that addresses how the closure property proves that the sum of a rational and irrational number is irrational is:
The sum or difference of a rational and irrational number is a rational number.
This statement, however, is actually incorrect as it is stated (the correct form would be that the sum of a rational and an irrational number is irrational). The reasoning can be explained as follows:
- Rational numbers are numbers that can be expressed as fractions of integers (e.g., a/b where a and b are integers and b ≠ 0).
- Irrational numbers cannot be expressed as such fractions; they have non-repeating, non-terminating decimal representations (e.g., √2, π).
- If you take a rational number (let's call it r) and an irrational number (let's call it i), their sum could be represented as r + i.
- If the sum were rational, that would imply that we could express this sum as a fraction, leading to the conclusion that the irrational part (i) could somehow be simplified to yield a fraction, which contradicts its definition.
- Therefore, the sum of a rational and an irrational number must be irrational.
In summary, the closure property for rational numbers states that their sums or differences remain rational, but adding an irrational number to a rational number results in an irrational number, showing that the sum of a rational and an irrational number is indeed irrational.