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The set of integers consists of zero, the natural numbers and their inverse (negatives). This is denoted by a boldface Z standing for the German word Zahlen. It means that Z is a subset of the sets of rational and real numbers and is countably infinite.
Addition is the inverse operation of subtraction and multiplication is the inverse operation of division. The word inverse means "opposite".
An inverse is another word for opposite. The inverse for adding is subtraction, multiplication is division, etc. If you are solving an equation, and have to get a variable alone, you must eliminate any other numbers with the variable, which means undoing the operation (x, +, -, /); so you perform the inverse. Example: x + 3 = 9. Subtract 3 on both sides to get x alone, because subtraction is the inverse of addition: x = 6. Example: 2x + 3 = 9. You must do the inverse of addition and subtraction before the inverse of multiplication and division. In this case, after subtracting 3 you have: 2x = 6. x is being multiplied by 2, so the inverse is division, and your answer is x = 3.
Mathematically, an inverse is an opposite, it is something that reverses what its inverse does, for example, addition and subtraction are inverse functions, as are multiplication and division. The inverse of a fraction is obtained by exchanging numerator and denominator; the inverse of a half is two.
The symbol for the set of integers is Z and it comes from the German word Zahlen, meaning numbers.