In the proof that ab+cd

Question

, where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbd
is a rational number.(1 point)
Responses

By the Closure Property, ad+bc
and bd
are both quotients of integers, and so ad+bcbd
is a quotient of two integers.
By the Closure Property, A d plus b c and b d are both quotients of integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers.

By the Closure Property, a quotient of imaginary numbers is a rational number.
By the Closure Property, a quotient of imaginary numbers is a rational number.

By the Closure Property, ad+bc
and bd
are both integers, and so ad+bcbd
is a quotient of two integers.
By the Closure Property, A d plus b c and b d are both integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers.

By the Closure Property, ad+bcbd
reduces to a whole number, which is a subset of rational numbers.
By the Closure Property, the fraction with numerator A d plus b c and denominator b d reduces to a whole number, which is a subset of rational numbers.

Answer 1

To explain why the expression \(\frac{ad + bc}{bd}\) is a rational number, we can use the following reasoning:

Correct Response: By the Closure Property, \(ad + bc\) and \(bd\) are both integers, and so \(\frac{ad + bc}{bd}\) is a quotient of two integers.

Explanation:

1. Definitions:

   • An integer is any whole number (positive, negative, or zero).
   • A rational number is defined as any number that can be expressed as the quotient of two integers (where the denominator is non-zero).

2. Components:

   • In the expression \(\frac{ad + bc}{bd}\):
     • \(ad\) is the product of two integers (since \(a\) and \(d\) are integers), thus \(ad\) is an integer.
     • \(bc\) is also the product of two integers (since \(b\) and \(c\) are integers), thus \(bc\) is an integer.
     • Therefore, \(ad + bc\) (which is the sum of two integers) is also an integer.

3. Denominator:

   • \(bd\) is the product of two integers \(b\) and \(d\), both of which are non-zero. Thus, \(bd\) is also a non-zero integer.

4. Conclusion:

   • Since both \(ad + bc\) and \(bd\) are integers, and \(bd\) is non-zero, the expression \(\frac{ad + bc}{bd}\) represents a quotient of two integers where the denominator is non-zero.
   • Therefore, \(\frac{ad + bc}{bd}\) is indeed a rational number, fulfilling the definition of rationality.

This line of reasoning confirms that the conclusion is valid based on the properties of integers and the definition of rational numbers.