It's any real number distinct from 0.
For instance, in the expression x/y, where x and y are real numbers, y needs to be a nonzero real number. This is because otherwise the expression x/y is undefined (viz. x/0).
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A rational number is always the result of dividing an integer when the divisor is nonzero.
Yes! it is.
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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.