multiplication

 

fundamental operation in arithmetic and algebra. Multiplication by a whole number can be interpreted as successive addition. For example, a number N multiplied by 3 is N + N + N. In general, multiplying positive numbers N and M gives the area of the rectangle with sides N and M. The result of a multiplication is known as the product. Numbers that give a product when multiplied together are called factors of that product. The symbol of the operation is × or · and, in algebra, simple juxtaposition (e.g., xy means x×y or x·y). Like addition, multiplication, in arithmetic and elementary algebra, obeys the associative law, the commutative law, and, in combination with addition, the distributive law. Multiplication in abstract algebra, as between vectors or other mathematical objects, does not always obey these rules. Quantities with unlike units may sometimes be multiplied, resulting in such units as foot-pounds, gram-centimeters, and kilowatt-hours. See also division.

Multiplication is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:

Multiplication can also be viewed as counting objects arranged in a rectangle, or finding the area of rectangle whose sides have given lengths.

Multiplication is one of four main operations in elementary arithmetic, and most people learn basic multiplication algorithms in elementary school. The inverse of multiplication is division.

Notation and terminology

Multiplication is written using the multiplication sign "×" between the terms; that is, in infix notation. The result is expressed with an equals sign. For example,

2×3 = 6 (verbally, "two times three equals six")

3×4 = 12

2×3×5 = 30

2×2×2×2×2 = 32

There are several other common notations for multiplication:

• Multiplication is sometimes denoted by either a middle dot or a period:

  Failed to parse (unknown function\text): 5 \cdot 2 \quad\text{or}\quad 5\,.\,2

  The middle dot is standard in the United States, the United Kingdom, and other countries where the period is used as a decimal point. In countries that use a comma as a decimal point, the period is used for multiplication instead.
• The asterisk (e.g. 5 * 2) is often used with computers because it appears on every keyboard. This usage originated in the FORTRAN programming language.
• In algebra, multiplication involving variables is often written as a juxtaposition (e.g. xy for x times y or 5x for five times x). This notation can also be used for numbers that are surrounded by parentheses (e.g. 5(2) or (5)(2) for five times two).

The numbers to be multiplied are generally called the "factors" or "multiplicands". When thinking of multiplication as repeated addition, the number to be repeated is called the "multiplicand", while the number of repetitions is called the "multiplier". In algebra, a number that is multiplied by a variable or expression (i.e. the 3 in 3xy²) is called a coefficient.

The result of a multiplication is called a product, and is a multiple of each factor. For example 15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5.

Computation

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.

Historical algorithms

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

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)

13×21 = (1×21) + (4×21) + (8×21) = 273.

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.

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.

Indus Valley

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.

Products of sequences

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:

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:

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.

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:

One can similarly replace m with negative infinity, and

for some integer m, provided both limits exist.

Interpretation

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

Failed to parse (unknown function\displaystyle): \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 are exactly those of the Cartesian product . The properties of the multiplicative operation as applying to natural numbers then follow trivially from the corresponding properties of the Cartesian product.

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.

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.

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.

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 commutative property.

a×b=b×a

See also

Notes

1. ^ [1]
2. ^ [2]

References

• Boyer, Carl B. (revised by Merzbach, Uta C.) (1991). History of Mathematics. John Wiley and Sons, Inc.. ISBN 0-471-54397-7. 

External links

• Practicing and Learning Multiplication
• Multiplication and Arithmetic Operations In Various Number Systems at cut-the-knot

• Modern Chinese Multiplication Techniques on an Abacus
• Multiplication Worksheets and Puzzles
• Math Games for Multiplication

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