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

By the closure Property, ad + bc and bd are both quotients of integers, and so ad+bc/bd is a quotient of two integers
By the closure property, a quotient of imaginary numbers is a rational number
by the closure property, ad+bc/bd 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+bc/bd is a quotient of two integers

3 answers

To explain why ad+bcbd is a rational number when a, b, c, and d are integers with b and d being non-zero:

  1. Definition of Rational Numbers: A number is considered rational if it can be expressed as a quotient of two integers, where the denominator is not zero.

  2. Numerator ad+bc: Here ad+bc is the result of the addition of two products, ad and bc. Since a, b, c, and d are integers, both ad and bc are also integers (the product of integers is an integer). Thus, when we add these two integers, ad+bc is itself an integer.

  3. Denominator bd: Since both b and d are given to be non-zero integers, their product bd is also a non-zero integer. This is important because a rational number cannot have a zero denominator.

  4. Combining the Numerator and Denominator: Now, we have ad+bcbd. The numerator ad+bc is an integer, and the denominator bd is a non-zero integer. Therefore, we are looking at a quotient of two integers.

  5. Conclusion: Since the quotient of an integer (the numerator) by a non-zero integer (the denominator) satisfies the definition of a rational number, we conclude that ad+bcbd is indeed a rational number.

Your original statement seemed to mix concepts, particularly the mention of imaginary numbers. The reasoning based on closure properties of integers regarding addition and multiplication leading to integers is the correct approach here. Thus, the final understanding is that ad+bcbd is a rational number because it is a quotient of two integers, where the denominator is non-zero.

just choose one

The correct statement is:

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

This clearly indicates that ad+bcbd is a rational number because it is the quotient of an integer by a non-zero integer.