To understand how the Closure Property relates to the sum of a rational and an irrational number, we need to remember a few definitions:
- Rational Numbers: Numbers that can be expressed as the quotient of two integers (e.g., 1/2, 3, -5, etc.).
- Irrational Numbers: Numbers that cannot be expressed as the quotient of two integers (e.g., √2, π, e, etc.).
Now, let's evaluate the responses provided in the question:
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The sum of an irrational and rational number can’t be predicted.
- This is vague and doesn’t align with the closure property.
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The sum or difference of a rational and irrational number is a rational number.
- This statement is incorrect. The sum or difference of a rational number and an irrational number is not a rational number; it is, in fact, an irrational number.
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The sum or difference of two rational numbers is a rational number.
- This is true but is not relevant to the sum of a rational and irrational number.
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Irrational numbers can be added or subtracted.
- While true, this statement does not address the specific sum of a rational and irrational number.
The best response to answer the question about the closure property and the sum of a rational and irrational number is not explicitly listed among the responses, but it should highlight that:
- If you take a rational number and an irrational number, their sum cannot be expressed as a fraction of integers (which is the definition of rational numbers). Therefore, the result must be irrational, demonstrating that the set of rational numbers is not closed under addition with irrational numbers.
If you have to choose from the responses provided, none directly address the closure property accurately, but choose the one that aligns closest, or note that none of the responses effectively explain the relationship correctly.