Question
n 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, 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 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, 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 quotient of imaginary numbers is a rational number.
Answers
GPT-4o mini
The correct explanation for why \( \frac{ad + bc}{bd} \) is a rational number is:
**By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.**
### Explanation:
1. **Integer products:** Since \( a, b, c, \) and \( d \) are integers, the products \( ad \) and \( bc \) are also integers. Therefore, their sum \( ad + bc \) is an integer.
2. **Non-zero denominator:** Since \( b \) and \( d \) are both non-zero integers, their product \( bd \) is also a non-zero integer.
3. **Quotient of integers:** The expression \( \frac{ad + bc}{bd} \) is a fraction where both the numerator \( (ad + bc) \) and the denominator \( (bd) \) are integers, with the denominator not equal to zero.
4. **Rational numbers:** By definition, a rational number is any number that can be expressed as the quotient of two integers (where the denominator is not zero). Hence, \( \frac{ad + bc}{bd} \) fits this definition and is thus a rational number.
The other options you've listed are either incorrect or not suitable explanations.
**By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.**
### Explanation:
1. **Integer products:** Since \( a, b, c, \) and \( d \) are integers, the products \( ad \) and \( bc \) are also integers. Therefore, their sum \( ad + bc \) is an integer.
2. **Non-zero denominator:** Since \( b \) and \( d \) are both non-zero integers, their product \( bd \) is also a non-zero integer.
3. **Quotient of integers:** The expression \( \frac{ad + bc}{bd} \) is a fraction where both the numerator \( (ad + bc) \) and the denominator \( (bd) \) are integers, with the denominator not equal to zero.
4. **Rational numbers:** By definition, a rational number is any number that can be expressed as the quotient of two integers (where the denominator is not zero). Hence, \( \frac{ad + bc}{bd} \) fits this definition and is thus a rational number.
The other options you've listed are either incorrect or not suitable explanations.