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 given that \( a, b, c, \) and \( d \) are integers and \( b \) and \( d \) are non-zero, we can use properties of integers and rational numbers. The correct response would be:
**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. **Integer Operations**: Since \( a, b, c, \) and \( d \) are integers, the products \( ad \) and \( bc \) are also integers. Therefore, the sum \( ad + bc \) is an integer.
2. **Non-zero Denominator**: Since \( b \) and \( d \) are non-zero integers, their product \( bd \) is also a non-zero integer.
3. **Quotient of Integers**: A rational number is defined as any number that can be expressed as the quotient \( \frac{p}{q} \) of two integers \( p \) and \( q \), where \( q \neq 0 \). In this case, \( p = ad + bc \) (an integer) and \( q = bd \) (a non-zero integer).
Thus, the fraction \( \frac{ad + bc}{bd} \) is indeed a rational number, as it is formed by the quotient of two integers where the denominator is non-zero.
**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. **Integer Operations**: Since \( a, b, c, \) and \( d \) are integers, the products \( ad \) and \( bc \) are also integers. Therefore, the sum \( ad + bc \) is an integer.
2. **Non-zero Denominator**: Since \( b \) and \( d \) are non-zero integers, their product \( bd \) is also a non-zero integer.
3. **Quotient of Integers**: A rational number is defined as any number that can be expressed as the quotient \( \frac{p}{q} \) of two integers \( p \) and \( q \), where \( q \neq 0 \). In this case, \( p = ad + bc \) (an integer) and \( q = bd \) (a non-zero integer).
Thus, the fraction \( \frac{ad + bc}{bd} \) is indeed a rational number, as it is formed by the quotient of two integers where the denominator is non-zero.