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How does the Closure Property prove that the sum of a rational and irrational number is irrational? (1 point)

1.Irrational numbers can be added or subtracted.

2. The sum of an irrational and rational number can't be predicted.

3.The sum or difference of a rational and irrational number is a rational number.

4. The sum or difference of two rational numbers is a rational number.

3 answers

The correct answer is:

2. The sum of an irrational and rational number can't be predicted.

This choice aligns with the idea that if you add a rational number (which can be expressed as a fraction) to an irrational number (which cannot be expressed as a fraction), the result will always be irrational. This is because if the result were rational, it would imply that the irrational number could be expressed as a rational number, which contradicts the definition of an irrational number.

Thus, the closure property implies that rational and irrational numbers behave in a way that leads to the sum being irrational.

So what what is the the answer

Apologies for any confusion in my previous response. The answer that correctly reflects the closure property regarding the sum of a rational and an irrational number is:

None of the provided options are strictly correct.

However, if you need to select the closest one based on standard reasoning, the fundamental principle you would rely on is:

  • The sum or difference of a rational and an irrational number is not a rational number (it is, in fact, irrational).

So, if you have to choose based on what's presented, none truly express this vital conclusion correctly, but keep in mind that the sum of a rational and an irrational number is irrational. If you need a definitive choice among your options, the second option reflects some ambiguity about predictability, even if it's not ideally phrased.

For educational clarity, I would suggest conveying that while the options may not capture the essence of the statement, the mathematical fact is clear:

  • The sum of a rational number and an irrational number is indeed irrational.