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How does the Closure Property prove that the sum of a rational and irrational number is irrational?(1 point) Responses Irration...Question
How does the Closure Property prove that the sum of a rational and irrational number is irrational?(1 point)
Responses
Irrational numbers can be added or subtracted.
Irrational numbers can be added or subtracted.
The sum or difference of two rational numbers is a rational number.
The sum or difference of two rational numbers is a rational number.
The sum of an irrational and rational number can’t be predicted.
The sum of an irrational and rational number can’t be predicted.
The sum or difference of a rational and irrational number is a rational number.
Responses
Irrational numbers can be added or subtracted.
Irrational numbers can be added or subtracted.
The sum or difference of two rational numbers is a rational number.
The sum or difference of two rational numbers is a rational number.
The sum of an irrational and rational number can’t be predicted.
The sum of an irrational and rational number can’t be predicted.
The sum or difference of a rational and irrational number is a rational number.
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Answered by
GPT-4o mini
The correct response is:
**The sum or difference of two rational numbers is a rational number.**
To explain further: If you have a rational number \( r \) (which can be expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \)) and an irrational number \( i \), then the sum \( r + i \) cannot be expressed in the same form as the rational number and remains irrational. This is because if \( r + i \) were rational, then you could subtract \( r \) to isolate \( i \), suggesting that \( i \) would be rational, which contradicts its definition. Hence, the conclusion is that the sum of a rational and an irrational number is irrational.
**The sum or difference of two rational numbers is a rational number.**
To explain further: If you have a rational number \( r \) (which can be expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \)) and an irrational number \( i \), then the sum \( r + i \) cannot be expressed in the same form as the rational number and remains irrational. This is because if \( r + i \) were rational, then you could subtract \( r \) to isolate \( i \), suggesting that \( i \) would be rational, which contradicts its definition. Hence, the conclusion is that the sum of a rational and an irrational number is irrational.
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