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Q: Is the quotient of an integer divided by a nonzero integer always a rational number?

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Any integer divided by a non-zero integer is rational.

Yes, it is.

Because that is how a rational number is defined!

Because that is how a rational number is defined!

Yes, always. That is the definition of a rational number.

Yes, by definition.

I had this name question for homework :| no

No.

The quotient of two nonzero integers is the definition of a rational number. There are nonzero numbers other than integers (imaginary, rational non-integers) that the quotient of would not be a rational number. If the two nonzero numbers are rational themselves, then the quotient will be rational. (For example, 4 divided by 2 is 2: all of those numbers are rational).

No.

Usually not.

Yes, as long as the two nonzero numbers are themselves rational. (Since a rational number is any number that can be expressed as the quotient of two rational numbers, or any number that can be written as a fraction using only rational numbers.) If one of the nonzero numbers is not rational, the quotient will most likely be irrational.

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