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Q: Is negative 2 a nonzero number?

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They are the positive and negative even numbers.

A mathematical element that when added to another numeral makes the same numeral

Not if the original number is positive, otherwise yes.

It means that the number is an integer, AND that it is not zero.

That sounds like a nonzero whole number, which could be either negative or positive, so -57, 896, 52, -99 etc., and infinitely many more, would qualify.

Taking the reciprocal (multiplicative inverse) does not affect the positive or negative status of an integer. So the reciprocal of a negative number is negative and the reciprocal of a positive number is positive. The reciprocals will be opposites (positive/negative) just as the original numbers were.

A nonzero whole number is a quantity which does not equal zero and number without fractions.

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).

The product is also a whole number. If the (number of positive factors) minus the (number of negative factors) is zero or an even number, then the product is positive. Otherwise the product is negative.

The sum of a number and its negative (additive inverse) is zero. For any nonzero value n, n + (-n) = 0

Actually the product of a nonzero rational number and another rational number will always be rational.The product of a nonzero rational number and an IRrational number will always be irrational. (You have to include the "nonzero" caveat because zero times an irrational number is zero, which is rational)

The product of two nonzero whole numbers will be a nonzero whole number.