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
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.
By the Closure Property, ad+bc
and bd
are both quotients of integers, and so ad+bcbd
is a quotient of two integers.
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, ad+bc
and bd
are both integers, and so ad+bcbd
is a quotient of two integers.
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.
By the Closure Property, a quotient of imaginary numbers is a rational number.
, 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
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.
By the Closure Property, ad+bc
and bd
are both quotients of integers, and so ad+bcbd
is a quotient of two integers.
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, ad+bc
and bd
are both integers, and so ad+bcbd
is a quotient of two integers.
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.
By the Closure Property, a quotient of imaginary numbers is a rational number.
Answers
GPT-4o mini
To explain why \( \frac{ad + bc}{bd} \) is a rational number, let's analyze the responses provided. The correct reasoning ensures that both the numerator and denominator consist of integers, and thus the overall expression is a quotient of integers.
The best option is:
**"By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers."**
Here's the reasoning:
1. **Closure Property**: This property states that the sum or product of two integers is also an integer.
- \( a, b, c, \) and \( d \) are integers. Therefore, the products \( ad \) and \( bc \) are integers (since the product of integers is an integer).
- Consequently, their sum \( ad + bc \) is also an integer.
- Since \( b \) and \( d \) are non-zero integers, their product \( bd \) is also an integer.
2. **Quotient of Integers**: The expression \( \frac{ad + bc}{bd} \) consists of an integer in the numerator (\( ad + bc \)) and a non-zero integer in the denominator (\( bd \)). Therefore, the expression is a quotient of two integers.
Since the quotient of two integers (where the denominator is not zero) is defined to be a rational number, it follows that \( \frac{ad + bc}{bd} \) is indeed a rational number.
The other options either misstate the property or contain unnecessary complexity. Thus, in this context, the chosen response accurately captures the essence of the reasoning required.
The best option is:
**"By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers."**
Here's the reasoning:
1. **Closure Property**: This property states that the sum or product of two integers is also an integer.
- \( a, b, c, \) and \( d \) are integers. Therefore, the products \( ad \) and \( bc \) are integers (since the product of integers is an integer).
- Consequently, their sum \( ad + bc \) is also an integer.
- Since \( b \) and \( d \) are non-zero integers, their product \( bd \) is also an integer.
2. **Quotient of Integers**: The expression \( \frac{ad + bc}{bd} \) consists of an integer in the numerator (\( ad + bc \)) and a non-zero integer in the denominator (\( bd \)). Therefore, the expression is a quotient of two integers.
Since the quotient of two integers (where the denominator is not zero) is defined to be a rational number, it follows that \( \frac{ad + bc}{bd} \) is indeed a rational number.
The other options either misstate the property or contain unnecessary complexity. Thus, in this context, the chosen response accurately captures the essence of the reasoning required.