There are many things that can be multiplied: numbers, vectors, matrices, functions, equations, sets, pegs...
As an abstract operation, multiplication is the same as addition. Both are binary (i.e., defined for two elements) operations satisfying the same set of axioms:
- There exists a unique special element called neutral such that the operation on any element and the neutral does not change the element.
- For every element there exists an inverse such that the operation on an element and its inverse is always neutral.
- The operation is associative: it does not matter how you apply the operation to three elements. You may apply it to the first two and then to the result and the third element. Or you may first apply the operation to the last two and then to the first and the result of the previous operation.
An operation may also be commutative
- The order of two elements in operation is not important.
Now, an operation must be given a symbol, and many are in use. A dot, a star, or an x normally denote multiplication, while a cross + denotes addition. For addition, the neutral element is known as zero 0. The inverse of a is denoted as -a and is often called negative a which is most often misleading. Minus a is by far more preferable.
For multiplication, the neutral element is known as unit 1. The inverse of a is called its reciprocal (or the multiplicative inverse) and denoted as a-1. This is how the axioms look like for addition and multiplication:
|1.||a + 0 = 0+a = a||a·1 = 1·a = a||unit element|
|2.||a + (-a) = (-a) + a = 0||a·a-1 = a-1·a = 1||inverse|
|3.||a + (b + c) = (a + b) + c||a·(b·c) = (a·b)·c||associativity|
|4.||a + b = b + a||a·b = b·a||commutativity|
Historical conventions apart, when there is only one operation defined on a set, there is absolutely no significance in preferring additive notations to multiplicative. When two operations are defined on the same set of elements that possess additional properties customs prevail. For example, two operations (I'll use "+" and "·" for convenience) may satisfy one or both of the distributive laws:
- a · (b + c) = (a · b) + (a · c)
- a + (b · c) = (a + b) · (a + c)
(Parentheses are used to disambiguate the order of operations.) When only one of the distributive laws holds we prefer notations that lead to the first law: addition is distributive with respect to multiplication. (You may want to play with a Java applet to see how that works.) When operations are defined over a set of numbers with the usual meaning for "+" and "·", indeed only the first law holds. Over other sets and with a differently defined operations, both of them may.