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, 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.
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, 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.
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 determine why \( \frac{ad + bc}{bd} \) is a rational number, we will analyze the expressions involved.
1. **Understanding Rational Numbers**: A rational number is defined as any number that can be expressed as the quotient of two integers, where the denominator is not zero.
2. **Components of the Expression**:
- Here, \( a, b, c, \) and \( d \) are all integers, and it is given that \( b \) and \( d \) are non-zero.
- The expression \( ad + bc \) is the sum of two products of integers, which is also an integer.
- The denominator \( bd \) is the product of two non-zero integers, which is also a non-zero integer.
3. **Forming the Quotient**:
- Thus, \( \frac{ad + bc}{bd} \) is a fraction where:
- The numerator \( ad + bc \) is an integer.
- The denominator \( bd \) is a non-zero integer.
Since both the numerator and the denominator are integers, and the denominator is not zero, according to the definition of rational numbers, \( \frac{ad + bc}{bd} \) is a rational number.
Now, looking at the options provided, the correct explanation is:
**By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.**
This option clearly states that since both components are integers (with the denominator being non-zero), the resulting fraction is a rational number.
1. **Understanding Rational Numbers**: A rational number is defined as any number that can be expressed as the quotient of two integers, where the denominator is not zero.
2. **Components of the Expression**:
- Here, \( a, b, c, \) and \( d \) are all integers, and it is given that \( b \) and \( d \) are non-zero.
- The expression \( ad + bc \) is the sum of two products of integers, which is also an integer.
- The denominator \( bd \) is the product of two non-zero integers, which is also a non-zero integer.
3. **Forming the Quotient**:
- Thus, \( \frac{ad + bc}{bd} \) is a fraction where:
- The numerator \( ad + bc \) is an integer.
- The denominator \( bd \) is a non-zero integer.
Since both the numerator and the denominator are integers, and the denominator is not zero, according to the definition of rational numbers, \( \frac{ad + bc}{bd} \) is a rational number.
Now, looking at the options provided, the correct explanation is:
**By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.**
This option clearly states that since both components are integers (with the denominator being non-zero), the resulting fraction is a rational number.