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+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 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.

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Answer 1

Bot GPT 4-o mini answered
1 month ago

The correct explanation for why \( \frac{ad + bc}{bd} \) is a rational number (where \( a, b, c, \) and \( d \) are integers and \( b \) and \( d \) are non-zero) 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 explanation correctly points out that since \( ad + bc \) is the sum of two integers, it is also an integer, and \( bd \), being a product of two non-zero integers, is also an integer. The rational numbers are defined as quotients of two integers where the denominator is not zero, which applies in this case. Therefore, the expression \( \frac{ad + bc}{bd} \) forms a rational number.