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, 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+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.
, 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, 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+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.
Answers
GPT-4o mini
To explain why \( \frac{ad + bc}{bd} \) is a rational number, we can refer to 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 the reasoning:
- **Closure Property**: This property states that, for a given set of numbers (in this case, integers), performing an operation (addition or multiplication) on elements of that set always yields an element of the same set.
- In this case:
- \( a, b, c, \) and \( d \) are integers. Thus, their products \( ad \) and \( bc \) are also integers.
- The sum \( ad + bc \) is an integer (since the sum of integers is also an integer).
- The product \( bd \) is also an integer because \( b \) and \( d \) are both non-zero integers.
- Therefore, \( \frac{ad + bc}{bd} \) is the quotient of two integers (where \( bd \) is not zero), which by definition is a rational number.
This reasoning ensures that the expression \( \frac{ad + bc}{bd} \) meets the criteria of being a rational number since it represents a ratio of integers.
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 the reasoning:
- **Closure Property**: This property states that, for a given set of numbers (in this case, integers), performing an operation (addition or multiplication) on elements of that set always yields an element of the same set.
- In this case:
- \( a, b, c, \) and \( d \) are integers. Thus, their products \( ad \) and \( bc \) are also integers.
- The sum \( ad + bc \) is an integer (since the sum of integers is also an integer).
- The product \( bd \) is also an integer because \( b \) and \( d \) are both non-zero integers.
- Therefore, \( \frac{ad + bc}{bd} \) is the quotient of two integers (where \( bd \) is not zero), which by definition is a rational number.
This reasoning ensures that the expression \( \frac{ad + bc}{bd} \) meets the criteria of being a rational number since it represents a ratio of integers.