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)

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/bd are both quotients of integers, and so the fraction with numerator Ad/bc and denominator bd 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, 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.

GPT-4o mini · Answer

To demonstrate why $\frac{ad + bc}{bd}$ is a rational number, we can analyze the components involved in the expression: 1. **Definition of Rational Numbers**: A number is called rational if it can be expressed as the quotient of two integers, where the denominator is not zero. Specifically, it takes the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. 2. **Given Variables**: In your case, $a$, $b$, $c$, and $d$ are all integers, with $b$ and $d$ being nonzero integers. 3. **Numerator Analysis**: The numerator $ad + bc$ can be analyzed: - $ad$ is the product of two integers $a$ and $d$. Therefore, it is also an integer. - $bc$ is the product of two integers $b$ and $c$. Thus, $bc$ is also an integer. - The sum $ad + bc$ of two integers is itself an integer by the Closure Property of integers. 4. **Denominator Analysis**: Now looking at the denominator $bd$: - Since both $b$ and $d$ are nonzero integers, their product $bd$ is also a nonzero integer. 5. **Forming the Fraction**: Now that we have established that $ad + bc$ is an integer and $bd$ is a nonzero integer, the fraction $\frac{ad + bc}{bd}$ is indeed the quotient of two integers, with the denominator being nonzero. 6. **Conclusion**: Therefore, since $\frac{ad + bc}{bd}$ meets the criteria for being a rational number—being the quotient of an integer $p = ad + bc$ and a nonzero integer $q = bd$—we conclude that $\frac{ad + bc}{bd}$ is a rational number. Thus, your reasoning can be summarized as: Because both the numerator $ad + bc$ and the denominator $bd$ are integers (with $bd$ being non-zero), the expression $\frac{ad + bc}{bd}$ satisfies the definition of a rational number.