Question
In the proof that; + f, where a, b,c, and d are integers and b and d are non-zero, explain why adtoc is a rational number.
(1 point)
• By the Closure Property, ad + be and bd are both integers, and so ad+be is a quotient of two integers.
• 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 + bc bd
is a quotient of two integers.
• By the Closure Property,
reduces to a whole number, which is a subset of rational numbers.
(1 point)
• By the Closure Property, ad + be and bd are both integers, and so ad+be is a quotient of two integers.
• 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 + bc bd
is a quotient of two integers.
• By the Closure Property,
reduces to a whole number, which is a subset of rational numbers.
Answers
GPT-4o mini
To clarify the proof that \( \frac{ad + bc}{bd} \) is a rational number, let's analyze it step by step.
1. **Understanding the elements**: We are given that \( a, b, c, \) and \( d \) are integers, where \( b \) and \( d \) are non-zero. The expression in question is \( \frac{ad + bc}{bd} \).
2. **Calculate the numerator**:
- The numerator is \( ad + bc \).
- Here, \( ad \) (the product of two integers) is an integer, and \( bc \) (the product of two integers) is also an integer.
- The sum \( ad + bc \) is therefore an integer, based on the Closure Property of integers, which states that the sum or product of two integers is an integer.
3. **Calculate the denominator**:
- The denominator is \( bd \). Since both \( b \) and \( d \) are non-zero integers, \( bd \) is also a non-zero integer.
4. **Form the quotient**:
- Now we examine the fraction \( \frac{ad + bc}{bd} \).
- The numerator \( ad + bc \) is an integer, and the denominator \( bd \) is a non-zero integer. Therefore, \( \frac{ad + bc}{bd} \) is a quotient of two integers.
5. **Conclusion**:
- According to the definition of rational numbers, a rational number is any number that can be expressed as the quotient \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
- Since \( ad + bc \) is an integer and \( bd \) is a non-zero integer, \( \frac{ad + bc}{bd} \) is a rational number.
Thus, we can conclude that \( \frac{ad + bc}{bd} \) is indeed a rational number following the outlined steps based on the Closure Property. The key here is that both parts of the fraction (numerator and denominator) fulfill the criteria for forming a rational number.
1. **Understanding the elements**: We are given that \( a, b, c, \) and \( d \) are integers, where \( b \) and \( d \) are non-zero. The expression in question is \( \frac{ad + bc}{bd} \).
2. **Calculate the numerator**:
- The numerator is \( ad + bc \).
- Here, \( ad \) (the product of two integers) is an integer, and \( bc \) (the product of two integers) is also an integer.
- The sum \( ad + bc \) is therefore an integer, based on the Closure Property of integers, which states that the sum or product of two integers is an integer.
3. **Calculate the denominator**:
- The denominator is \( bd \). Since both \( b \) and \( d \) are non-zero integers, \( bd \) is also a non-zero integer.
4. **Form the quotient**:
- Now we examine the fraction \( \frac{ad + bc}{bd} \).
- The numerator \( ad + bc \) is an integer, and the denominator \( bd \) is a non-zero integer. Therefore, \( \frac{ad + bc}{bd} \) is a quotient of two integers.
5. **Conclusion**:
- According to the definition of rational numbers, a rational number is any number that can be expressed as the quotient \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
- Since \( ad + bc \) is an integer and \( bd \) is a non-zero integer, \( \frac{ad + bc}{bd} \) is a rational number.
Thus, we can conclude that \( \frac{ad + bc}{bd} \) is indeed a rational number following the outlined steps based on the Closure Property. The key here is that both parts of the fraction (numerator and denominator) fulfill the criteria for forming a rational number.
So what’s the answer?
GPT-4o mini
The answer to the statement "why \( \frac{ad + bc}{bd} \) is a rational number" is:
**By the Closure Property, \( ad + bc \) is an integer, and \( bd \) is a non-zero integer. Therefore, \( \frac{ad + bc}{bd} \) is a quotient of two integers, which means it is a rational number.**
This explanation is clear and concise, directly addressing the properties that ensure the result is a rational number.
**By the Closure Property, \( ad + bc \) is an integer, and \( bd \) is a non-zero integer. Therefore, \( \frac{ad + bc}{bd} \) is a quotient of two integers, which means it is a rational number.**
This explanation is clear and concise, directly addressing the properties that ensure the result is a rational number.