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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

1 Notation 2 In music 3 See also

Notation

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":

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:

Definition

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)

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 P₀, multiplication is indicated by Mx, x being the multiplicator:

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.

See also

• Peasant multiplication
• reciprocal
• tables of multiplication