# Multiplication

In its simplest form, multiplication is the sum of a list of identical numbers. For example, the product 7 × 4 is 7 + 7 + 7 + 7. The numbers being multiplied are called the multiplicand and multiplier or the factors.

## Notation

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

5$\times$2
5$\cdot$2
(5)2, 5(2), (5)(2), 5[2], [5]2, [5][2]
5*2

The asterisk (*) is often used on computers because it is a symbol on every keyboard, but it is rarely used when writing math by hand. This usage originated in the FORTRAN programming language. Frequently, multiplication is implied by Juxtaposition rather than shown in a notation. This is standard in algebra, taking forms like

5x</b> and <b>xy </b>

This is potentially confusing if variables are permitted to have names longer than one letter. The notation is not used with numbers alone: 52 never means 5 × 2.

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 $1\cdot 2\cdot \ldots \cdot 99\cdot 100$. This can also be written with the ellipsis vertically placed in the middle of the line, as $1\cdot 2\cdot \cdots \cdot 99\cdot 100$.

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:

$\prod _{{i=m}}^{{n}}x_{{i}}:=x_{{m}}\cdot x_{{m+1}}\cdot x_{{m+2}}\cdot \cdots \cdot x_{{n-1}}\cdot x_{{n}}.$

The subscript gives the symbol for a dummy variable ($i$ in our case) and its lower value ($m$); the superscript gives its upper value. So for example:

$\prod _{{i=2}}^{{6}}\left(1+{1 \over i}\right)=\left(1+{1 \over 2}\right)\cdot \left(1+{1 \over 3}\right)\cdot \left(1+{1 \over 4}\right)\cdot \left(1+{1 \over 5}\right)\cdot \left(1+{1 \over 6}\right)={7 \over 2}.$

One may also consider products of infinitely many terms; these are called infinite products. Notationally, we would replace n above by the infinity symbol (∞). The product of such a series is defined as the limit of the product of the first $n$ terms, as $n$ grows without bound. That is:

$\prod _{{i=m}}^{{\infty }}x_{{i}}:=\lim _{{n\to \infty }}\prod _{{i=m}}^{{n}}x_{{i}}.$

One can similarly replace $m$ with negative infinity, and

$\prod _{{i=-\infty }}^{\infty }x_{i}:=\left(\lim _{{n\to \infty }}\prod _{{i=-n}}^{m}x_{i}\right)\cdot \left(\lim _{{n\to \infty }}\prod _{{i=m+1}}^{n}x_{i}\right),$

for some integer $m$, provided both limits exist.

## Definition

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

$mn:=\sum _{{k=1}}^{n}m$

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

such that there are n m's added together. So for instance:

• 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 two 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.

Multiplication also has what is called the associative property. The associative property means that for any three numbers x, y, and z,

(x · y)z = x(y · z).

Note from algebra: the parentheses mean that the operations inside the parentheses must be done before anything outside the parentheses is done.

Multiplication also has what is called a distributive property with respect to the addition, because

x(y + z) = xy + xz.

Also of interest is that any number times 1 is equal to itself, thus,

1 · x = x.

and this is called the identity property

What about zero? Well, we have:

m · 0 = m + m + m +...+ m

where there are zero m's added together. The sum of zero m's is zero, so

m · 0 = 0

no matter what m is (as long as it is finite).

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

(−1)m = (−1) + (−1) +...+ (−1) = −m

This is an interesting fact that shows that any negative number is just negative one multiplied by a positive number. So multiplication with any integers can be represented by multiplication of whole numbers and (−1)'s. All that remains is to explicitly define (−1)(−1):

(−1)(−1) = −(−1) = 1

In this way, the multiplication of any two integers is defined. The definitions can be extended to larger and larger sets of numbers: first to vulgar fractions called the rational numbers, then to infinitely long decimals called real numbers, and then to the complex numbers.

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)

where x is a real number, and y is a natural number. Once multiplication has been defined for natural numbers, it can be extended to include integers, and then to real and complex numbers.

## Computation

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

Some algorithms are suitable for multiplying numbers using pencil and paper. Most, such as lattice multiplication, require a multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9); the peasant multiplication algorithm does not.