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A multiplication by a negative number can be seen as a change of direction of the vector of magnitude equal to the absolute value of the product of the factors. When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. The sign of the product is determined by the following rules:
The numbers being multiplied are multiplicands, multipliers, or factors. Multiplication can be expressed as "five times three equals fifteen", "five times three is fifteen" or "fifteen is the product of five and three". Multiplication is represented using the multiplication sign (×), the asterisk (*), parentheses (), or a dot (⋅).
Multiplication by a positive number preserves the order: For a > 0, if b > c, then ab > ac. Multiplication by a negative number reverses the order: For a < 0, if b > c, then ab < ac. The complex numbers do not have an ordering that is compatible with both addition and multiplication. [30]
Arithmetic is the fundamental branch of mathematics that studies numbers and their operations. In particular, it deals with numerical calculations using the arithmetic operations of addition, subtraction, multiplication, and division. [1]
You are defining that multiplication of negative numbers follows the rules of a ring. If we had that a times b is 0 if either a or b is negative that would also be consistent with the rules for the multiplication for non-negative numbers. It is because we want the rules for negative numbers to be nicer than that that they are defined the way ...
Multiplying by a number is the same as dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication by its reciprocal yields the original number (since the product of the number and its reciprocal is 1).
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