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, 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.
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, a quotient of imaginary numbers is a rational number.

1 answer

The correct response to explain why ad+bcbd is a rational number is:

By the Closure Property, ad+bc and bd are both integers, and so ad+bcbd is a quotient of two integers.

This is valid because:

  • a,b,c, and d are all integers, and since the product and sum of integers are also integers, both ad and bc are integers.
  • Since b and d are non-zero, bd is also a non-zero integer.
  • Therefore, ad+bcbd is a quotient of integers where the denominator is not zero, confirming that it is a rational number.