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. bd
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. bd
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.**
Here's a breakdown of the reasoning:
1. Since \(a\), \(b\), \(c\), and \(d\) are integers, the products \(ad\) and \(bc\) are also integers (as the product of integers is an integer).
2. Therefore, \(ad + bc\) is the sum of two integers, which is also an integer.
3. Similarly, since \(b\) and \(d\) are non-zero integers, their product \(bd\) is a non-zero integer.
4. By the definition of rational numbers, a number is considered rational if it can be expressed as a quotient of two integers, where the denominator is not zero. Since both \(ad + bc\) (an integer) and \(bd\) (a non-zero integer) fit this definition, \(\frac{ad + bc}{bd}\) is indeed a rational number.
So, the explanation provided in your first option correctly states the reasoning.
**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 the reasoning:
1. Since \(a\), \(b\), \(c\), and \(d\) are integers, the products \(ad\) and \(bc\) are also integers (as the product of integers is an integer).
2. Therefore, \(ad + bc\) is the sum of two integers, which is also an integer.
3. Similarly, since \(b\) and \(d\) are non-zero integers, their product \(bd\) is a non-zero integer.
4. By the definition of rational numbers, a number is considered rational if it can be expressed as a quotient of two integers, where the denominator is not zero. Since both \(ad + bc\) (an integer) and \(bd\) (a non-zero integer) fit this definition, \(\frac{ad + bc}{bd}\) is indeed a rational number.
So, the explanation provided in your first option correctly states the reasoning.