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Multiplication

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.

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":

<math>5\times 2</math>
<math>5\cdot2</math>
<math>(5)2,\ 5(2),\ 5[2],\ [5]2</math>
<math>5*2</math>

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 specifically noted. This arises in Algebra and looks like:

  • 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 is a capital Pi from the Greek alphabet. This is defined as:

<math> \prod_{i=m}^{n} x_{i} = x_{m} \cdot x_{m+1} \cdot x_{m+2} \cdot ... \cdot x_{n-1} \cdot x_{n}. </math>
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:
<math> \prod_{i=2}^{6} 1 + {1\over i} = (1 + {1\over 2}) \cdot (1 + {1\over 3}) \cdot (1 + {1\over 4}) \cdot (1 + {1\over 5}) \cdot (1 + {1\over 6}) = {7\over 2}. </math>

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:

<math> \prod_{i=m}^{\infty} x_{i} := \lim_{n\to\infty} \prod_{i=m}^{n} x_{i}. </math>
One can similarly replace m with negative infinity, and
<math>\prod_{i=-\infty}^\infty x_i := \lim_{n\to\infty}\prod_{i=-n}^m x_i + \lim_{n\to\infty}\prod_{i=m+1}^n x_i,</math>
for some integer m, provided both limits exist.

Definition

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

<math>\sum_{k=1}^n m=mn.</math>

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 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

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 is also has what is called a distributive property 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? 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

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 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 an arbitrary quantity, 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.

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:



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