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How does the Closure Property prove that the sum of a rational and irrational number is irrational?(1 point) Responses The sum...Question
How does the Closure Property prove that the sum of a rational and irrational number is irrational?(1 point)
Responses
The sum or difference of a rational and irrational number is a rational number.
The sum or difference of a rational and irrational number is a rational number.
The sum or difference of two rational numbers is a rational number.
The sum or difference of two rational numbers is a rational number.
Irrational numbers can be added or subtracted.
Irrational numbers can be added or subtracted.
The sum of an irrational and rational number can’t be predicted.
The sum of an irrational and rational number can’t be predicted.
Responses
The sum or difference of a rational and irrational number is a rational number.
The sum or difference of a rational and irrational number is a rational number.
The sum or difference of two rational numbers is a rational number.
The sum or difference of two rational numbers is a rational number.
Irrational numbers can be added or subtracted.
Irrational numbers can be added or subtracted.
The sum of an irrational and rational number can’t be predicted.
The sum of an irrational and rational number can’t be predicted.
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GPT-4o mini
The correct response that addresses how the closure property proves that the sum of a rational and irrational number is irrational is:
**The sum or difference of a rational and irrational number is a rational number.**
This statement, however, is actually incorrect as it is stated (the correct form would be that the sum of a rational and an irrational number is irrational). The reasoning can be explained as follows:
1. Rational numbers are numbers that can be expressed as fractions of integers (e.g., a/b where a and b are integers and b ≠ 0).
2. Irrational numbers cannot be expressed as such fractions; they have non-repeating, non-terminating decimal representations (e.g., √2, π).
3. If you take a rational number (let's call it r) and an irrational number (let's call it i), their sum could be represented as r + i.
4. If the sum were rational, that would imply that we could express this sum as a fraction, leading to the conclusion that the irrational part (i) could somehow be simplified to yield a fraction, which contradicts its definition.
5. Therefore, the sum of a rational and an irrational number must be irrational.
In summary, the closure property for rational numbers states that their sums or differences remain rational, but adding an irrational number to a rational number results in an irrational number, showing that the sum of a rational and an irrational number is indeed irrational.
**The sum or difference of a rational and irrational number is a rational number.**
This statement, however, is actually incorrect as it is stated (the correct form would be that the sum of a rational and an irrational number is irrational). The reasoning can be explained as follows:
1. Rational numbers are numbers that can be expressed as fractions of integers (e.g., a/b where a and b are integers and b ≠ 0).
2. Irrational numbers cannot be expressed as such fractions; they have non-repeating, non-terminating decimal representations (e.g., √2, π).
3. If you take a rational number (let's call it r) and an irrational number (let's call it i), their sum could be represented as r + i.
4. If the sum were rational, that would imply that we could express this sum as a fraction, leading to the conclusion that the irrational part (i) could somehow be simplified to yield a fraction, which contradicts its definition.
5. Therefore, the sum of a rational and an irrational number must be irrational.
In summary, the closure property for rational numbers states that their sums or differences remain rational, but adding an irrational number to a rational number results in an irrational number, showing that the sum of a rational and an irrational number is indeed irrational.
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