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.

Question

(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.

Answer 1

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.

Answer 2

So what’s the answer?

Answer 3

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.