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+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, 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, 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+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, 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, a quotient of imaginary numbers is a rational number.
Answers
GPT-4o mini
To explain why \( \frac{ad + bc}{bd} \) is a rational number, we can focus on the properties of integers and rational numbers.
The correct response 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 why this response is correct:
1. **Closure Property**: The Closure Property states that the sum and product of integers result in integers. Since \(a\), \(b\), \(c\), and \(d\) are all integers, both \(ad\) (the product of integers) and \(bc\) (the product of integers) are integers. Thus, their sum \(ad + bc\) is also an integer.
2. Since \(b\) and \(d\) are non-zero, the product \(bd\) is also a non-zero integer.
3. A rational number is defined as a quotient of two integers. In this case, \(\frac{ad + bc}{bd}\) represents a quotient where the numerator is an integer \(ad + bc\) and the denominator is a non-zero integer \(bd\).
Therefore, since both the numerator and the denominator are integers and the denominator is non-zero, \( \frac{ad + bc}{bd} \) is indeed a rational number.
The correct response 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 why this response is correct:
1. **Closure Property**: The Closure Property states that the sum and product of integers result in integers. Since \(a\), \(b\), \(c\), and \(d\) are all integers, both \(ad\) (the product of integers) and \(bc\) (the product of integers) are integers. Thus, their sum \(ad + bc\) is also an integer.
2. Since \(b\) and \(d\) are non-zero, the product \(bd\) is also a non-zero integer.
3. A rational number is defined as a quotient of two integers. In this case, \(\frac{ad + bc}{bd}\) represents a quotient where the numerator is an integer \(ad + bc\) and the denominator is a non-zero integer \(bd\).
Therefore, since both the numerator and the denominator are integers and the denominator is non-zero, \( \frac{ad + bc}{bd} \) is indeed a rational number.