To understand how the closure property applies to proving that the sum of a rational number and an irrational number is irrational, we first need to clarify what rational and irrational numbers are:
- A rational number can be expressed as the fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\).
- An irrational number cannot be expressed as a fraction of integers; it has a non-repeating, non-terminating decimal expansion.
Now, let's analyze the provided statements:
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Irrational numbers can be added or subtracted.
- This statement is true, but it doesn't directly prove anything about the sum of a rational and irrational number.
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The sum or difference of a rational and an irrational number is a rational number.
- This statement is actually false. If this were true, it would contradict the definition of irrational numbers. Therefore, it cannot be used in the proof.
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The sum or difference of two rational numbers is a rational number.
- This statement is true due to the closure property of rational numbers, but it does not pertain directly to the sum of a rational and an irrational number.
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The sum of an irrational and rational number can’t be predicted.
- This statement is vague. While it touches on the uncertainty regarding the outcome of adding a rational and an irrational number, it's not a formal proof.
To prove that the sum of a rational number and an irrational number is irrational, we typically use the following reasoning:
- Let \( r \) be a rational number and \( x \) be an irrational number.
- If the sum \( r + x \) were to be rational, we could isolate \( x \) as follows: \( x = (r + x) - r\).
- Since \( r \) is rational, and if the sum \( r + x \) is also rational, then \((r + x) - r\) would also be rational (as the difference of rational numbers is rational).
- This would mean that \( x \) would have to be rational, which contradicts our assumption that \( x \) is irrational.
Thus, the conclusion is that the sum \( r + x \) must be irrational.
In summary, the only statements that would logically lead you toward proving the irrationality of the sum would be the properties of rational and irrational numbers, with 2 being crucially important to note as false in proving the original statement.