Probably because that's more or less the definition of "rational number": a number that can be expressed as a ratio of integers.
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It is an incomplete definition of a rational number.
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).
Yes. Rational numbers are always the quotient of two integers. Integers are always real, and you cannot divide a real number by another real number and get an imaginary number. So, true.
Yes.
They are called a rational number.