Question
in the proof that a/b + c/d where a, b, c, and d are integers and b and d are non-zero, explain why ad+bc/bd is a rational number.
A. By the closure property, ad+bc/bd reduces to a whole number, which is a subset of rational numbers.
B. By the closure property, ad + bc and bd are both quotients of integers, and so ad+bc/bd is a quotient of two integers.
C. By the closure property, a quotient of imaginary numbers is a rational number.
D. By the closure property, ad + bc and bd are both integers, and so ad+bc/bd is a quotient of two integers
A. By the closure property, ad+bc/bd reduces to a whole number, which is a subset of rational numbers.
B. By the closure property, ad + bc and bd are both quotients of integers, and so ad+bc/bd is a quotient of two integers.
C. By the closure property, a quotient of imaginary numbers is a rational number.
D. By the closure property, ad + bc and bd are both integers, and so ad+bc/bd is a quotient of two integers
Answers
Answered by
GPT-4o mini
The correct explanation for why \( \frac{ad + bc}{bd} \) is a rational number is found in option **D**: "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 a breakdown of why this is the right choice:
1. **Closure Property**: The closure property under addition and multiplication of integers states that the sum or product of any two integers is also an integer. Therefore, since \( a \), \( b \), \( c \), and \( d \) are integers, both \( ad \) and \( bc \) are integers, and thus \( ad + bc \) is also an integer.
2. **Denominator**: The denominator \( bd \) is the product of two non-zero integers (since \( b \) and \( d \) are non-zero), so \( bd \) is also a non-zero integer.
3. **Forming a Quotient**: Since \( ad + bc \) is an integer and \( bd \) is a non-zero integer, the expression \( \frac{ad + bc}{bd} \) is a quotient of two integers where the denominator is non-zero.
4. **Definition of Rational Numbers**: A rational number is defined as any number that can be expressed as a quotient \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
So, the conclusion is that \( \frac{ad + bc}{bd} \) is indeed a rational number, making option D the correct choice.
Here's a breakdown of why this is the right choice:
1. **Closure Property**: The closure property under addition and multiplication of integers states that the sum or product of any two integers is also an integer. Therefore, since \( a \), \( b \), \( c \), and \( d \) are integers, both \( ad \) and \( bc \) are integers, and thus \( ad + bc \) is also an integer.
2. **Denominator**: The denominator \( bd \) is the product of two non-zero integers (since \( b \) and \( d \) are non-zero), so \( bd \) is also a non-zero integer.
3. **Forming a Quotient**: Since \( ad + bc \) is an integer and \( bd \) is a non-zero integer, the expression \( \frac{ad + bc}{bd} \) is a quotient of two integers where the denominator is non-zero.
4. **Definition of Rational Numbers**: A rational number is defined as any number that can be expressed as a quotient \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
So, the conclusion is that \( \frac{ad + bc}{bd} \) is indeed a rational number, making option D the correct choice.