Multiplication can be denoted several different ways, and for all real numbers the different notations are equivalent. All of the following mean, "5 times 2":

- 5x
- xy

This should only be done with variables that have one letter; with variable that have multiple letters (it is possible, and often less confusing, to write out the name of a variable rather than using a single letter, i.e. using "mass" instead of "m") it can become confusing as to where one variable ends and the other begins. It is also confusing to use this notation with just numbers because 52 could then mean fifty-two or five times two.

If the terms are not written out individually, then the product may be written with an ellipsis to mark out the missing terms. Thus, the product of all the natural numbers from 1 to 100 is 1 · 2 · ... · 99 · 100.

Alternatively, the product can be with the product symbol, which derives from the capital letter Π (Pi) in the Greek alphabet. This is defined as:

As for what multiplication means, the product of two whole numbers n and m is:

- m×n = m + m + m +...+ m

- 5×2 = 5 + 5 = 10
- 2×5 = 2 + 2 + 2 + 2 + 2 = 10
- 4×3 = 4 + 4 + 4 = 12
- m×6 = m + m + m + m + m + m

Using this definition, it is easy to prove some interesting properties of multiplication. As the first two examples above hint at, the order in which to numbers are multiplied does not matter. This is called the **commutative property** and it turns out to be true in general that for any two numbers x and y:

- x·y = y·x

- (x·y)z = x(y·z)

Multiplication is also has what is called a **distributive property** because:

- x(y + z) = xy + xz

- 1·x = x

What about zero? The initial definition above is little help because 1 is greater than zero. It is actually easier to define multiplication by zero using the second definition. So:

- m·0 = m + m + m +...+ m

- m·0 = 0

Multiplication with negative numbers also requires a little thought. First consider negative 1. For any positive integer m:

- (-1)m = (-1) + (-1) +...+ (-1) = -m

- (-1)(-1) = -(-1) = 1

Students are sometimes mystified when told that the result of multiplying no numbers is 1.

A formal recursive definition of multiplication can be given by the rules:

- x.0 = 0
- x.y = x + x.(y-1)

For fast ways to compute products of large numbers, see multiplication algorithms.

To multiply numbers using pencil and paper, you need to have a multiplication table (either in your head or on paper). You also need to know a "multiplication algorithm" (a way to multiply numbers) such as lattice multiplication.

See also: