In the proof that ab+cd

Question

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, 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+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, ad+bcbd
 reduces to a whole number, which is a subset of rational numbers.

GPT-4o mini · Answer

To explain why $ \frac{ad + bc}{bd} $ is a rational number, we can use the properties of integers and their closure under addition and multiplication, along with the definition of rational numbers. The correct response would be: **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 the reasoning: 1. **Integers and Operations**: Since $ a, b, c, $ and $ d $ are integers, both $ ad $ (the product of two integers $ a $ and $ d $) and $ bc $ (the product of two integers $ b $ and $ c $) are also integers. Therefore, $ ad + bc $ (the sum of two integers) is an integer. 2. **Non-Zero Denominator**: The denominator $ bd $ is also an integer because it is the product of two integers $ b $ (which is non-zero) and $ d $ (which is also non-zero). 3. **Quotient of Integers**: The expression $ \frac{ad + bc}{bd} $ is the ratio of two integers: the numerator $ ad + bc $ (an integer) and the denominator $ bd $ (also an integer but non-zero). 4. **Definition of Rational Numbers**: By definition, a rational number is any number that can be expressed as the quotient of two integers, where the denominator is not zero. Since both parts of our expression fit this definition, we can conclude that $ \frac{ad + bc}{bd} $ must be a rational number. Thus, the statement that $ ad + bc $ and $ bd $ are both integers, making $ \frac{ad + bc}{bd} $ a valid quotient of two integers, is the correct explanation.

Answer

witch one

GPT-4o mini · Answer

The most accurate response from your provided options is:

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

This option clearly states that both the numerator and denominator are integers, which is essential for concluding that the entire expression is a rational number.