To prove that the sum of a rational and an irrational number is irrational, we rely on the closure property of rational numbers.
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Closure Property of Rational Numbers: This property states that the sum or difference (and also the product) of two rational numbers is always a rational number.
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Definitions:
- A rational number is any number that can be expressed as the quotient of two integers (where the denominator is not zero).
- An irrational number is any real number that cannot be expressed as the quotient of two integers.
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Proof by Contradiction:
- Let's assume we have a rational number \( r \) and an irrational number \( i \).
- If we add them, \( s = r + i \).
- Suppose, for contradiction, that \( s \) is rational.
- Since \( r \) is rational and, under the closure property, the sum or difference of two rational numbers is rational, if \( s \) is rational, we can rearrange to express \( i = s - r \).
- Here, \( s - r \) (which is a rational number minus a rational number) must also be rational.
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Contradiction: This implies that we could express the irrational number \( i \) as a rational number, which contradicts the definition of an irrational number.
Thus, we conclude that the sum of a rational number and an irrational number must be irrational.
Correct Response: The sum of a rational and an irrational number cannot be predicted to be rational; it is irrational. The other responses are incorrect because they either contradict the established properties or state the opposite of what is true.