To explain why
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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.
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Numerator
: Here is the result of the addition of two products, and . Since , , , and are integers, both and are also integers (the product of integers is an integer). Thus, when we add these two integers, is itself an integer. -
Denominator
: Since both and are given to be non-zero integers, their product is also a non-zero integer. This is important because a rational number cannot have a zero denominator. -
Combining the Numerator and Denominator: Now, we have
. The numerator is an integer, and the denominator is a non-zero integer. Therefore, we are looking at a quotient of two integers. -
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
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