To prove that the sum of a rational number and an irrational number is irrational, we can use the definitions and properties of rational and irrational numbers along with the concept of closure.
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Definitions:
- A rational number is a number that can be expressed as the quotient of two integers, \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0\).
- An irrational number is a number that cannot be expressed as such a quotient; it cannot be represented as a fraction of integers.
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Assumption:
- Let \( r \) be a rational number and \( i \) be an irrational number.
- We want to examine the expression \( r + i \).
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Contradiction:
- To show that \( r + i \) is irrational, we can use a proof by contradiction. Assume that the sum \( r + i \) is rational.
- If \( r + i \) is rational, then we can express it as a quotient of two integers, say \( \frac{m}{n} \), where \( m \) and \( n \) are integers and \( n \neq 0\).
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Rearranging the Equation:
- From our assumption that \( r + i \) is rational, we can rearrange this: \[ i = (r + i) - r \]
- Since \( r \) is rational (by our initial assumption) and \( r + i \) is also assumed to be rational, then \( (r + i) - r \) must be the difference of two rational numbers.
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Closure Property of Rational Numbers:
- The closure property states that the sum or difference of two rational numbers is always a rational number. Hence, \( i \) would also have to be rational.
- However, this leads to a contradiction because we originally stated that \( i \) is irrational.
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Conclusion:
- Our initial assumption that \( r + i \) is rational must be false.
- Therefore, the conclusion is that the sum of a rational number and an irrational number must be irrational.
To summarize, the relevant statement in your options would be:
- "The sum or difference of a rational and an irrational number is a rational number" is incorrect.
- The accurate property being used here is that the sum of a rational and an irrational number is indeed irrational, supported by the definitions and working through a contradiction.