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. (1 point)
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, ad+bc/bd reduces to a whole number, which is a subset of rational numbers.
By the Closure Property, a quotient of imaginary numbers is a rational number
By the Closure Property, ad + be and bd are both integers, and so ad+bc/bd is a quotient of two integers.
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, ad+bc/bd reduces to a whole number, which is a subset of rational numbers.
By the Closure Property, a quotient of imaginary numbers is a rational number
By the Closure Property, ad + be and bd are both integers, and so ad+bc/bd is a quotient of two integers.
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.**
This is the most accurate answer because:
1. \( a, b, c, d \) are integers by the problem statement.
2. The product of integers is also an integer, so \( ad \) and \( bc \) are integers.
3. The sum of integers \( ad + bc \) is also an integer.
4. The product \( bd \) is an integer since both \( b \) and \( d \) are non-zero integers.
5. Therefore, \( \frac{ad + bc}{bd} \) is a quotient of two integers, which satisfies the definition of a rational number.
Thus, \( \frac{ad + bc}{bd} \) is indeed a rational number.
**By the Closure Property, ad + bc and bd are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.**
This is the most accurate answer because:
1. \( a, b, c, d \) are integers by the problem statement.
2. The product of integers is also an integer, so \( ad \) and \( bc \) are integers.
3. The sum of integers \( ad + bc \) is also an integer.
4. The product \( bd \) is an integer since both \( b \) and \( d \) are non-zero integers.
5. Therefore, \( \frac{ad + bc}{bd} \) is a quotient of two integers, which satisfies the definition of a rational number.
Thus, \( \frac{ad + bc}{bd} \) is indeed a rational number.