# Multiplication

In its simplest form, **multiplication** is a quick way of adding identical numbers. The result of multiplying numbers is called a *product*. The numbers being multiplied are called *coefficients* or *factors*, and individually as the *multiplicand* and *multiplicator*.

Table of contents |

2 Definition 3 Computation 4 In music 5 See also |

## Notation

Multiplication can be denoted in several equivalent ways. All of the following mean, "5 times 2":

The asterisk is often used on computers because it is a symbol on every keyboard, but it is never used when writing math by hand, and should only be used when there are no other alternatives. (This usage originated in the FORTRAN programming language.) Frequently, multiplication is implied rather than show in a notation. This is standard in algebra, taking forms like

- and

If the terms are not written out individually, then the product may be written with an ellipsis to mark out the missing terms, as with other series operations (like sums). Thus, the product of all the natural numbers from 1 to 100 can be written . This can also be written with the ellipsis vertically placed in the middle of the line, as .

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

*n*above by the infinity symbol (∞). The product of such a series is defined as the limit of the product of the first terms, as grows without bound. That is:

## Definition

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

This is just a shorthand for saying, "Add m to itself n times." Expanding the above to make its meaning more clear:

- 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

**associative property**. The associative property means that for any three numbers x, y, and z:

- (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

**identity property**

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)

## Computation

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.

## In music

In music and musical set theory, multiplication modulo 12 is a basic operation which may be performed on pitch or pitch class sets. Dealing with all twelve tones, or a tone row, there are only a few numbers which one may multiply a row by and still end up with twelve tones. Taking the prime or unaltered form as P_{0}, multiplication is indicated by M*x*, *x* being the multiplicator:

_{0}= M1

_{0}, I

_{0}= M11

_{0}, M7

_{0}=I(M5

_{0}). Thus, for the untransposed form of all:

M1 | M5 | M7 | M11 |

M5 | M1 | M11 | M7 |

M7 | M11 | M1 | M5 |

M11 | M7 | M5 | M1 |

Even numbers remain unchanged under M7 and all odd numbers become transposed by a tritone.

The chromatic scale may be mapped onto the circle of fourths with M5, and the circle of fifths with M7.