In the proof that ab+cd , 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 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, 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+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. By the Closure Property, ad+bc and bd are both quotients of integers, and so ad+bcbd is a quotient of two integers.

Question

GPT-4o mini · Answer

The correct explanation for why $ \frac{ad + bc}{bd} $ is a rational number involves understanding the properties of integers and rational numbers. To clarify, let's analyze the expression $ \frac{ad + bc}{bd} $: 1. **Identifying the Parts**: - In this case, $ ad + bc $ is a sum of products of integers. Since $ a, b, c, $ and $ d $ are all integers, both $ ad $ and $ bc $ are also integers. Therefore, their sum $ ad + bc $ is also an integer. 2. **Denominator**: - The product $ bd $ is also an integer because it is the product of two integers ($ b $ and $ d $), and since both $ b $ and $ d $ are non-zero, $ bd $ is non-zero. 3. **Quotients of Integers**: - A rational number is defined as any number that can be expressed as a quotient $ \frac{p}{q} $ where $ p $ and $ q $ are integers and $ q \neq 0 $. In this case, $ ad + bc $ is an integer, and $ bd $ is a non-zero integer. Thus, $ \frac{ad + bc}{bd} $ represents a quotient of two integers where the denominator is non-zero. 4. **Conclusion**: - Therefore, since both the numerator ($ ad + bc $) and denominator ($ bd $) are integers with $ bd \neq 0$, the expression $ \frac{ad + bc}{bd} $ is indeed a rational number. The most accurate response from your options is the second one: "By the Closure Property, $ ad + bc $ and $ bd $ are both integers, and so the fraction with numerator $ ad + bc $ and denominator $ bd $ is a quotient of two integers." This captures the essence of why the result is a rational number.