Question
In the proof that ab+cd
, where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbd
is a rational number.(1 point)
Responses
a. By the Closure Property, ad+bc
and bd
are both integers, and so ad+bcbd
is a quotient of two integers.
b. By the Closure Property, A d plus b c and b d are both integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers.
c. By the Closure Property, ad+bc
and bd
are both quotients of integers, and so ad+bcbd
is a quotient of two integers.
d. By the Closure Property, A d plus b c and b d are both quotients of integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers.
By the Closure Property, a quotient of imaginary numbers is a rational number.
By the Closure Property, a quotient of imaginary numbers is a rational number.
By the Closure Property, ad+bcbd
reduces to a whole number, which is a subset of rational numbers.
By the Closure Property, the fraction with numerator A d plus b c and denominator b d reduces to a whole number, which is a subset of rational numbers.
, where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbd
is a rational number.(1 point)
Responses
a. By the Closure Property, ad+bc
and bd
are both integers, and so ad+bcbd
is a quotient of two integers.
b. By the Closure Property, A d plus b c and b d are both integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers.
c. By the Closure Property, ad+bc
and bd
are both quotients of integers, and so ad+bcbd
is a quotient of two integers.
d. By the Closure Property, A d plus b c and b d are both quotients of integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers.
By the Closure Property, a quotient of imaginary numbers is a rational number.
By the Closure Property, a quotient of imaginary numbers is a rational number.
By the Closure Property, ad+bcbd
reduces to a whole number, which is a subset of rational numbers.
By the Closure Property, the fraction with numerator A d plus b c and denominator b d reduces to a whole number, which is a subset of rational numbers.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine why \( \frac{ad + bc}{bd} \) is a rational number, let's analyze the provided response options based on mathematical definitions and properties.
The key points are:
1. **Closure Property**: The Closure Property states that when performing certain operations (like addition or multiplication) within a set (like integers), the result will remain within that set. In this case, we are using integers.
2. **Rational Numbers**: A number is classified as rational if it can be expressed as the quotient of two integers (i.e., \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \)).
Now, examining the expressions in the options, we know:
- \( ad \) and \( bc \) are integers because \( a, b, c, d \) are integers.
- \( bd \) is non-zero because both \( b \) and \( d \) are non-zero.
This leads us to:
- The numerator \( ad + bc \) is an integer (the sum of two integers).
- The denominator \( bd \) is a non-zero integer.
Thus, the fraction \( \frac{ad + bc}{bd} \) is formed as a quotient of two integers—specifically, an integer over a non-zero integer.
**Correct Answer Reasoning**:
Among the provided options:
a. By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.
This statement correctly captures the essence of why \( \frac{ad + bc}{bd} \) is a rational number, emphasizing that both the numerator and denominator are integers, which is crucial for identifying the result as rational.
**Conclusion**: The correct response is:
**a. By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.**
The key points are:
1. **Closure Property**: The Closure Property states that when performing certain operations (like addition or multiplication) within a set (like integers), the result will remain within that set. In this case, we are using integers.
2. **Rational Numbers**: A number is classified as rational if it can be expressed as the quotient of two integers (i.e., \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \)).
Now, examining the expressions in the options, we know:
- \( ad \) and \( bc \) are integers because \( a, b, c, d \) are integers.
- \( bd \) is non-zero because both \( b \) and \( d \) are non-zero.
This leads us to:
- The numerator \( ad + bc \) is an integer (the sum of two integers).
- The denominator \( bd \) is a non-zero integer.
Thus, the fraction \( \frac{ad + bc}{bd} \) is formed as a quotient of two integers—specifically, an integer over a non-zero integer.
**Correct Answer Reasoning**:
Among the provided options:
a. By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.
This statement correctly captures the essence of why \( \frac{ad + bc}{bd} \) is a rational number, emphasizing that both the numerator and denominator are integers, which is crucial for identifying the result as rational.
**Conclusion**: The correct response is:
**a. By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.**
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.