To understand how the closure property can be used to prove that the sum of a rational number and an irrational number is irrational, let's analyze the options you provided:
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The sum of an irrational number can't be predicted.
- This statement is vague and not directly related to the closure property. It does not contribute to understanding the relationship between rational and irrational numbers.
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The sum or difference of a rational and irrational number is a rational number.
- This statement is incorrect. The closure property does not support this statement. In fact, it is known that the sum (or difference) of a rational number and an irrational number is always irrational.
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Irrational numbers can be added or subtracted.
- While this statement is true, it does not specifically address the relationship between rational and irrational numbers with respect to their sum or difference.
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The sum or difference of two rational numbers is a rational number.
- This statement is true and illustrates the closure property of rational numbers. However, it doesn't directly address the case of adding a rational number to an irrational number.
Conclusion:
To demonstrate that the sum of a rational number \( r \) and an irrational number \( i \) is irrational using the closure property, you would argue as follows:
- Assume, for the sake of contradiction, that \( r + i \) is rational.
- If \( r + i \) is rational, then we can represent it as \( r + i = q \) for some rational number \( q \).
- This implies that \( i = q - r \).
- Since the difference of two rational numbers (both \( q \) and \( r \) are rational) is rational, \( i \) would also be rational.
- This contradicts the assumption that \( i \) is irrational.
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