Question
In the proof that ad +bc
why
- is a rational number. (1 point)
+ d, where a, b, c, and d are integers and b and d are non-zero, explain
By the Closure Property, ad + bc and bd are both quotients of integers, and so ad+be is a quotient of two integers.
By the Closure Property, ad + bc and bd are both integers, and so ad-bc 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 reduces to a whole number, which is a subset of rational
numbers.
why
- is a rational number. (1 point)
+ d, where a, b, c, and d are integers and b and d are non-zero, explain
By the Closure Property, ad + bc and bd are both quotients of integers, and so ad+be is a quotient of two integers.
By the Closure Property, ad + bc and bd are both integers, and so ad-bc 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 reduces to a whole number, which is a subset of rational
numbers.
Answers
Answered by
GPT-4o mini
To understand why \( \frac{ad + bc}{bd} \) (assuming this is what you mean by "ad + bc") is a rational number given that \( a, b, c, \) and \( d \) are integers, and \( b \) and \( d \) are non-zero integers, we can invoke the Closure Property of rational numbers.
### Explanation:
1. **Definition of Rational Numbers**:
Rational numbers are defined as numbers that can be expressed as the quotient of two integers, where the denominator is not zero. That is, a number \( \frac{p}{q} \) is rational if \( p \) and \( q \) are integers, and \( q \neq 0 \).
2. **Calculate the Numerator**:
- The expression \( ad + bc \) is formed from the products of integers \( a, b, c, \) and \( d \).
- Since the product of integers is also an integer, both \( ad \) and \( bc \) are integers (because all combinations of integers yield integers).
- Thus, \( ad + bc \) is an integer as well (the sum of two integers is also an integer).
3. **Calculate the Denominator**:
- The product \( bd \) is also an integer (again, the product of integers).
- Since both \( b \) and \( d \) are non-zero, \( bd \neq 0 \).
4. **Form the Quotient**:
- Now, consider the expression \( \frac{ad + bc}{bd} \).
- Here, the numerator \( ad + bc \) is an integer, and the denominator \( bd \) is a non-zero integer.
5. **Conclusion by Closure Property**:
- According to the Closure Property of rational numbers, since \( ad + bc \) is an integer and \( bd \) is a non-zero integer, the quotient \( \frac{ad + bc}{bd} \) must be a rational number.
### Correction of the Original Statement:
The proof you provided has some inaccuracies and confusing parts. The essence of the proof relies on establishing that both the numerator (which is \( ad + bc \)) and the denominator (which is \( bd \)) meet the criteria for forming a rational number.
In summary, \( \frac{ad + bc}{bd} \) is indeed a rational number because both the numerator is an integer and the denominator is a non-zero integer, confirming that the quotient of these two integers is a rational number.
### Explanation:
1. **Definition of Rational Numbers**:
Rational numbers are defined as numbers that can be expressed as the quotient of two integers, where the denominator is not zero. That is, a number \( \frac{p}{q} \) is rational if \( p \) and \( q \) are integers, and \( q \neq 0 \).
2. **Calculate the Numerator**:
- The expression \( ad + bc \) is formed from the products of integers \( a, b, c, \) and \( d \).
- Since the product of integers is also an integer, both \( ad \) and \( bc \) are integers (because all combinations of integers yield integers).
- Thus, \( ad + bc \) is an integer as well (the sum of two integers is also an integer).
3. **Calculate the Denominator**:
- The product \( bd \) is also an integer (again, the product of integers).
- Since both \( b \) and \( d \) are non-zero, \( bd \neq 0 \).
4. **Form the Quotient**:
- Now, consider the expression \( \frac{ad + bc}{bd} \).
- Here, the numerator \( ad + bc \) is an integer, and the denominator \( bd \) is a non-zero integer.
5. **Conclusion by Closure Property**:
- According to the Closure Property of rational numbers, since \( ad + bc \) is an integer and \( bd \) is a non-zero integer, the quotient \( \frac{ad + bc}{bd} \) must be a rational number.
### Correction of the Original Statement:
The proof you provided has some inaccuracies and confusing parts. The essence of the proof relies on establishing that both the numerator (which is \( ad + bc \)) and the denominator (which is \( bd \)) meet the criteria for forming a rational number.
In summary, \( \frac{ad + bc}{bd} \) is indeed a rational number because both the numerator is an integer and the denominator is a non-zero integer, confirming that the quotient of these two integers is a rational number.