Multiplication

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This article is about multiplication in mathematics. For multiplication in music, see multiplication (music).

In mathematics, multiplication is an elementary arithmetic operation. When one of the numbers is a whole number, multiplication is the repeated sum of the other number.

${{n \times a = } \atop {\ }} {{\underbrace{a + \cdots + a}} \atop n}$

For example, 4 × 7 is 7 + 7 + 7 + 7.

Fractions are multiplied by separately multiplying their denominators and numerators: a/b × c/d = (ac)/(bd). For example, 2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2.

Multiplication can be defined for real and complex numbers, polynomials, matrices and other mathematical quantities as well. The inverse of multiplication is division.

1 Computation
2 Terminology
3 Notation
- 3.1 Capital pi notation
- 3.2 Infinite products
4 Interpretation
- 4.1 Cartesian product
5 Properties
6 See also
7 External links

[edit] Computation

For several ways to compute products, including very large numbers, see multiplication algorithms.

The standard methods for multiplying numbers using pencil and paper require a multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9), however one method, the peasant multiplication algorithm, does not.

Multiplying numbers to more than a couple of decimal places by hand is tedious and error prone. Common logarithms were invented to simplify such calculations. The slide rule allowed numbers to be quickly multplied to about three places of accuracy. Beginning in the early twentieth century, mechanical calculators, such as the Marchant, automated multiplication of up to 10 digit numbers. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand.

[edit] Terminology

The two numbers being multiplied are formally called the multiplier and the multiplicand (the number multiplied by the multiplier). The difference was important in those numeral systems, such as Roman numerals, in which multiplication is transformation of symbols and their addition. For example, a person multiplying VII by XV changes the VII to LXX (multiplying VII by X) plus XXV (V times V) plus X (II times V) when XV is the multiplier, and XV into LXXV (XV times V) plus XV plus XV (each XV times I) when it is VII. With Arabic numerals, the relations between the terms are all memorized.

Because of the commutative property of multiplication, there is generally no need to distinguish between the two numbers so they are more commonly referred to as the factors. The result of the multiplication is referred to as the product.

[edit] Notation

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

5×2

5·2

(5)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 and xy

This notation is potentially confusing if variables are permitted to have names longer than one letter, as in computer programming languages. 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 $1 \cdot 2 \cdot \cdots \cdot 99 \cdot 100$ .

[edit] Capital pi notation

Alternatively, a product can be written with the product symbol, which derives from the capital letter Π (Pi) in the Greek alphabet. Unicode position U+220F (∏) is defined a n-ary product for this purpose, distinct from U+03A0 (Π), the letter. 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:

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

In case m = n, the value of the product is the same as that of the single factor x_m. If m > n, the product is the empty product, with the value 1.

[edit] Infinite products

One may also consider products of infinitely many terms; these are called infinite products. Notationally, we would replace n above by the infinity symbol (∞). In the reals, 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.

[edit] Interpretation

[edit] Cartesian product

The definition of multiplication as repeated addition provides a way to arrive at a set-theoretic interpretation of multiplication of cardinal numbers. In ${{n \cdot a = } \atop {\ }} {{\underbrace{a + \cdots + a}} \atop n}$ , if the n copies of a are to be combined in disjoint union then clearly they must be made disjoint; an obvious way to do this is to use n as the indexing set. Then, the members of $n \cdot a\,$ are exactly those of the Cartesian product $n \times a\,$ . The properties of the multiplicative operation as applying to natural numbers then follow trivially from the corresponding properties of the Cartesian product.

[edit] Properties

For integers, fractions, real and complex numbers, multiplication has certain properties:

the order in which two numbers are multiplied does not matter. This is called the commutative property,

x · y = y · x.

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,

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. In this regard the number 1 is known as the multiplicative identity.

The sum of zero numbers is zero.

This fact is directly received by means of the distributive property:

m · 0 = (m · 0) + m − m = (m · 0) + (m · 1) − m = m · (0 + 1) − m = (m · 1) − m = m − m = 0.

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 one (-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

However, from a formal viewpoint, multiplication between two negative numbers is (again) directly received by means of the distributive property, e.g:

(−1) × (−1)
	= (−1) × (−1) + (−2) + 2
	= (−1) × (−1) + (−1) × 2 + 2
	= (−1) × (−1 + 2) + 2
	= (−1) × 1 + 2
	= (−1) + 2
	= 1

Every number x, except zero, has a multiplicative inverse, 1/x, such that x × 1/x = 1.

Multiplication by a positive number preserves order: if a > 0, then if b > c then ab > ac. Multiplication by a negative number reverses order: if a < 0, then if b > c then ab < ac.

Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for matrices and quaternions.

[edit] See also

Peasant multiplication
Multiplicative inverse, the reciprocal
Multiplication table (times table)
Napier's bones
Product (mathematics) - lists generalizations
Slide rule
Schönhage-Strassen algorithm

[edit] External links

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Categories: Elementary arithmetic | Arithmetic | Binary operations | Mathematical notation

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