Multiplication

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This article is about multiplication in mathematics. For multiplication in music, see Multiplication (music). For the social networking website, see Multiply (website). For the record label, see Multiply Records.

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.

$\displaystyle a \times n = \underbrace{a + \cdots + a}_{n}$

For example, 7 × 4 (verbally, "seven times four") is the same as 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; see product (mathematics). The inverse of multiplication is division.

1 Computation
- 1.1 Historical algorithms
  - 1.1.1 Egyptians
  - 1.1.2 Babylonians
  - 1.1.3 Chinese
  - 1.1.4 Indus Valley
2 Terminology
3 Notation
- 3.1 Capital pi notation
- 3.2 Infinite products
4 Interpretation
- 4.1 Cartesian product
5 Properties
6 Multiplication with Peano's axioms
7 Multiplication with set theory
8 Multiplication with group theory
9 See also
10 Notes
11 References
12 External links

[edit] Computation

For methods of computing products, including those of very large numbers, see multiplication algorithm.

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. Many mathematics curricula developed according to the 1989 standards of the NCTM do not teach standard arithmetic methods, instead guiding students to invent their own methods of computation. Though widely adopted by many school districts in nations such as the United States, they have encountered resistance from some parents and mathematicians, and some districts have since abandoned such curricula in favor of traditional mathematics.

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 multiplied 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] Historical algorithms

Methods of multiplication were documented in the Egyptian, Greece, Babylonian, Indus valley, and Chinese civilizations.

[edit] Egyptians

Main article: Ancient Egyptian multiplication

The Egyptian method of multiplication of integers and fractions, documented in the Ahmes Papyrus, was by successive additions and doubling. For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining 2 × 21 = 42, 4 × 21 = 84, 8 × 21 = 168. The full product could then be found by adding the correct terms found in the doubling: (note 13 = 1 + 4 + 8)

$13\times 21 = (1\times 21) + (4\times 21) + (8 \times 21) = 273.$

[edit] Babylonians

The Babylonians used a sexagesimal positional number system, analogous to the modern day decimal system. Thus, Babylonian multiplication was very similar to modern decimal multiplication. Because of the relative difficulty of remembering 60 × 60 different products, Babylonian mathematicians employed multiplication tables. These tables consisted of a list of the first twenty multiples of a certain principal number n: n, 2n, ..., 20n; followed by the multiples of 10n: 30n 40n, and 50n. Then to compute any sexagesimal product, say 53n, one only needed to add 50n and 3n computed from the table.

[edit] Chinese

In the books, Chou Pei Suan Ching dated prior to 300 B.C., and the Nine Chapters on the Mathematical Art, multiplication calculations were written out in words, although the early Chinese mathematicians employed an abacus in hand calculations involving addition and multiplication.

[edit] Indus Valley

Product of 45 and 256. Note the order of the numerals in 45 is reversed down the left column. The carry step of the multiplication can be performed at the final stage of the calculation (in bold), returning the final product of 45 × 256 = 11520.

The early Hindu mathematicians of the Indus valley region used a variety of intuitive tricks to perform multiplication. Most calculations were performed on small slate hand tablets, using chalk tables. One technique was that of lattice multiplication (or gelosia multiplication). Here a table was drawn up with the rows and columns labelled by the multiplicands. Each box of the table was divided diagonally into two, as a triangular lattice. The entries of the table held the partial products, written as decimal numbers. The product could then be formed by summing down the diagonals of the lattice.

[edit] Terminology

The two numbers being multiplied are formally called the multiplicand and the multiplier, respectively. (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.

Some write the multiplier first, and say that 7 × 4 stands for 4 + 4 + 4 + 4 + 4 + 4 + 4, but this usage is less common. The difference was important in Roman numerals and similar systems where multiplication is transformation of symbols and their addition. For example, to multiply VII by XV one changes the VII to LXX (multiplying VII by X) plus XXV (V times V) plus X (II times V), but to multiply XV by VII one changes XV into LXXV (XV times V) plus XV plus XV (each XV times I).

[edit] Notation

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

5×2 (see ×)

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

The product of a series of terms 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

Main article: Infinite product

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 the expression

$\displaystyle a \cdot n = \underbrace{a + \cdots + a}_{n},$

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 either a or n as the indexing set for the other. Then, the members of $a \cdot n\,$ are exactly those of the Cartesian product $a \times n\,$ . 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) = x·y + x·z.

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 a·b > a·c. Multiplication by a negative number reverses order: if a < 0, then if b > c then a·b < a·c.

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] Multiplication with Peano's axioms

In the book Arithmetices principia, nova methodo exposita, Giuseppe Peano proposed a new system for multiplication based on his axioms for natural numbers. ^[1]

a×1=a
a×b'=(a×b)+a

Here, b' represents the successor of b, or the natural number which follows b. With his other nine axioms, it is possible to prove common rules of multiplication, such as the distributive or associative properties.

[edit] Multiplication with set theory

It is possible, though difficult, to create a recursive definition of multiplication with set theory. Such a system usually relies on the peano definition of multiplication.

[edit] Multiplication with group theory

It is easy to show that there is a group for multiplication- the non-zero rational numbers.^[2] Multiplication with the non-zero numbers satisfies

Closure - For all a and b in the group, a×b is in the group.
Associativity - This is just the associative property! (a×b)×c=a×(b×c)
Identity - This follows straight from the peano definition. Anything multiplied by one is itself.
Inverse - All non-zero numbers have a multiplicative inverse.

Multiplication also is an abelian group, since it follows the communative property.

a×b=b×a