Multiplication

3 × 4 = 12, so twelve dots can be arranged in three rows of four (or four columns of three).

Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic (the others being addition, subtraction and division).

Multiplication is defined for whole numbers in terms of repeated addition; for example, 4 multiplied by 3 (often said as "4 times 3") can be calculated by adding 3 copies of 4 together:

$4 + 4 + 4 = 12.\!\,$

Multiplication of rational numbers (fractions) and real numbers is defined by systematic generalization of this basic idea.

Multiplication can also be visualised as counting objects arranged in a rectangle (for whole numbers) or as finding the area of a rectangle whose sides have given lengths (for numbers generally). The inverse of multiplication is division: as 4 times 3 is equal to 12, so 12 divided by 3 is equal to 4.

Multiplication is generalized further to other types of numbers (such as complex numbers) and to more abstract constructs such as matrices.

Notation and terminology

The multiplication sign.

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,

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

$3\times 4 = 12$

$2\times 3\times 5 = 6\times 5 = 30$

$2\times 2\times 2\times 2\times 2 = 32$

There are several other common notations for multiplication:

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

$5 \cdot 2 \quad\text{or}\quad 5\,.\,2$

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 some countries that use a comma as a decimal point, the period is used for multiplication instead.
The asterisk (as in 5*2) is often used in programming languages because it appears on every keyboard and is easier to see on older monitors. 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).

In matrix multiplication, there is actually a distinction between the cross and the dot symbols. The cross symbol generally denotes a vector multiplication, while the dot denotes a scalar multiplication. A like convention distinguishes between the cross product and the dot product of two vectors.

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

The result of a multiplication is called a product, and is a multiple of each factor that is an integer. 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.

The "Rule of Nine"

A method exists to determine if the result is incorrect. The origin of the rule is unknown. For example to determine if 257 × 17 = 1049, you would add the digits in each multiplier until you arrive at a one digit result. So 257 is 2 + 5 + 7 = 14, as this is a two digit number, the operation is performed again until a one digit number is arrived at, so you add 1 + 4 = 5; the same operation is done to the second multiplier, to arrive at the number 8 (1 + 7). Those two numbers are multiplied together 5 x 8 = 40, the numbers of the result are added together to arrive at the number 4 (4 + 0). Finally the numbers of the proposed result are also added together, so 1 + 0 + 4 + 9 = 14, which is a two digit number, so the operation is performed again and 1 + 4 = 5. If the two results are not equal to the same number, then the multiplication result is incorrect. 257 x 17 is really 4369. Adding 4 + 3 + 6 + 9 = 22, and 2 + 2 = 4

Frequently the operation is performed using a large X next to the multiplication and to put result of the addition of the digits in the first multiplier in the space at the northernmost point, the second multiplier is put on the bottom. The result of multiplying these two numbers and adding the digits at the right. The result of adding the numbers of the result on the bottom at the left, thus:

  257  \5/
 x 17  5\4
 ----  /8\   This result is definitely wrong as 4 and 5 are not equal.
 1049

  257  \5/
 x 17  4\4
 ----  /8\  This result is not incorrect.  
 4369

This method does not guarantee the result is correct even if the numbers match, but it will eliminate many incorrect results.

In mathematical terms, this works because adding the digits preserves the number's value modulo 9, and within the context of modulo multiplication, if a≡b and x≡y modulo m, then, ax≡by modulo m.

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 1 × 21 = 21, 2 × 21 = 42, 4 × 21 = 84, 8 × 21 = 168. The full product could then be found by adding the appropriate terms found in the doubling sequence:

13 × 21 = (1 + 4 + 8) × 21 = (1 × 21) + (4 × 21) + (8 × 21) = 21 + 84 + 168 = 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

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.

Modern method

The modern method of multiplication based on the Hindu-Arabic numeral system was first described by Brahmagupta. Brahmagupta gave rules for addition, subtraction, multiplication and division. Henry Burchard Fine, then professor of Mathematics at Princeton University, wrote the following:

The Indians are the inventors not only of the positional decimal system itself, but of most of the processes involved in elementary reckoning with the system. Addition and subtraction they performed quite as they are performed nowadays; multiplication they effected in many ways, ours among them, but division they did cumbrously.^[1]

Products of sequences

Capital pi notation

The product of a sequence 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 (∏) contains a glyph for denoting such a product, distinct from U+03A0 (Π), the letter. The meaning of this notation is given by:

$\prod_{i=m}^{n} x_{i} = x_{m} \cdot x_{m+1} \cdot x_{m+2} \cdot \,\,\cdots\,\, \cdot x_{n-1} \cdot x_{n}.$

The subscript gives the symbol for a dummy variable (i in this case), called the "index of multiplication" together with its lower bound (m), whereas the superscript (here n) gives its upper bound. The lower and upper bound are expressions denoting integers. The factors of the product are obtained by taking the expression following the product operator, with successive integer values substituted for the index of multiplication, starting from the lower bound and incremented by 1 up to and including the upper bound. 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.

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 lemniscate (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, by definition,

$\prod_{i=m}^{\infty} x_{i} = \lim_{n\to\infty} \prod_{i=m}^{n} x_{i}.$

One can similarly replace m with negative infinity, and define:

$\prod_{i=-\infty}^\infty x_i = \left(\lim_{m\to-\infty}\prod_{i=m}^0 x_i\right) \cdot \left(\lim_{n\to\infty}\prod_{i=1}^n x_i\right),$

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

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

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.

Properties

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

Commutative property

The order in which two numbers are multiplied does not matter:

$x\cdot y = y\cdot x$ .

Associative property

Expressions solely involving multiplication are invariant with respect to order of operations:

$(x\cdot y)\cdot z = x\cdot(y\cdot z)$

Distributive property

Holds with respect to addition over multiplication. This identity is of prime importance in simplifying algebraic expressions:

$x\cdot(y + z) = x\cdot y + x\cdot z$

Identity element

The multiplicative identity is 1; anything multiplied by one is itself. This is known as the identity property:

$x\cdot 1 = x$

Zero element

Anything multiplied by zero is zero. This is known as the zero property of multiplication:

$x\cdot 0 = 0$

Inverse property: Every number x, except zero, has a multiplicative inverse, $\frac{1}{x}$ , such that $x\cdot\left(\frac{1}{x}\right) = 1$ .

Order preservation: 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 and b > c then ab < ac.

Negative one times any number is equal to the opposite of that number.

$(-1)\cdot x = (-x)$

Negative one times negative one is positive one.

$(-1)\cdot (-1) = 1$

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.

Proofs

Not all of these properties are independent; some are a consequence of the others. A property that can be proven from the others is the zero property of multiplication. It is proven by means of the distributive property. We assume all the usual properties of addition and subtraction, and −x means the same as 0 − x.

$\begin{align} & {} \qquad x\cdot 0 \\ & {} = (x\cdot 0) + x - x \\ & {} = (x\cdot 0) + (x\cdot 1) - x \\ & {} = x\cdot (0 + 1) - x \\ & {} = (x\cdot 1) - x \\ & {} = x - x \\ & {}= 0 \end{align}$

So we have proven:

$x\cdot 0 = 0$

The identity (−1) · x = −x can also be proven using the distributive property:

$\begin{align} & {} \qquad(-1)\cdot x \\ & {} = (-1)\cdot x + x - x \\ & {} = (-1)\cdot x + 1\cdot x - x \\ & {} = (-1 + 1)\cdot x - x \\ & {} = 0\cdot x - x \\ & {} = 0 - x \\ & {} = -x \end{align}$

The proof that (−1) · (−1) = 1 is now easy:

$\begin{align} & {} \qquad (-1)\cdot (-1) \\ & {} = -(-1) \\ & {} = 1 \end{align}$

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.^[2]

$a\times 1=a$

$a\times b'=(a\times b)+a$

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 in group theory

It is easy to show that there is a group for multiplication- the non-zero rational numbers.^[3] 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.

Multiplication of different kinds of numbers

Numbers can count (3 apples), order (the 3rd apple), or measure (3.5 feet high); as the history of mathematics has progressed from counting on our fingers to modelling quantum mechanics, multiplication has been generalized to more complicated and abstract types of numbers, and to things that are not numbers (such as matrices) or do not look much like numbers (such as quaternions).

Integers: $N\times M$ is the sum of M copies of N when N and M are positive whole numbers. This gives the number of things in an array N wide and M high. Generalization to negative numbers can be done by $N\times (-M) = (-N)\times M = - (N\times M)$ and $(-N)\times (-M) = N\times M$ . The same sign rules apply to rational and real numbers.

Rationals: Generalization to fractions $\frac{A}{B}\times \frac{C}{D}$ is by multiplying the numerators and denominators respectively: $\frac{A}{B}\times \frac{C}{D} = \frac{(A\times C)}{(B\times D)}$ . This gives the area of a rectangle $\frac{A}{B}$ high and $\frac{C}{D}$ wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.

Reals: $(x)(y)$ is the limit of the products of the corresponding terms in certain sequences of rationals that converge to $x$ and $y$ , respectively, and is significant in calculus. This gives the area of a rectangle $x$ high and $y$ wide. See Products of sequences, above.

Complex numbers: Considering complex numbers $z_1$ and $z_2$ as ordered pairs of real numbers $(a_1, b_1)$ and $(a_2, b_2)$ , the product $z_1\times z_2$ is $(a_1\times a_2 - b_1\times b_2, a_1\times b_2 + a_2\times b_1)$ . This is the same as for reals, $a_1\times a_2$ , when the imaginary parts $b_1$ and $b_2$ are zero.

Further generalizations: See Multiplication in group theory, above, and Multiplicative Group, which for example includes matrix multiplication. A very general, and abstract, concept of multiplication is as the "multiplicatively denoted" (second) binary operation in a ring. An example of a ring which is not any of the above number systems is a polynomial ring (you can add and multiply polynomials, but polynomials are not numbers in any usual sense.)

Division: Often division, $\frac{x}{y}$ , is the same as multiplication by an inverse, $x\left(\frac{1}{y}\right)$ . Multiplication for some types of "numbers" may have corresponding division, without inverses; in an integral domain $x$ may have no inverse " $\frac{1}{x}$ " but $\frac{x}{y}$ may be defined. In a division ring there are inverses but they are not commutative (since $\left(\frac{1}{x}\right)\left(\frac{1}{y}\right)$ is not the same as $\left(\frac{1}{y}\right)\left(\frac{1}{x}\right)$ , $\frac{x}{y}$ may be ambiguous).

Exponentiation

Main article: Exponentiation

When multiplication is repeated, the resulting operation is known as exponentiation. For instance, the product 2×2×2 of three factors of two is "two raised to the third power", and is denoted by 2³, a two with a superscript three. In this example, the number two is the base, and three is the exponent. In general, the exponent (or superscript) indicates how many times to multiply base by itself, so that the expression

$a^n = \underbrace{a\times a \times \cdots \times a}_n$

indicates that the base a to be multiplied by itself n times.

Notes

↑ Henry B. Fine. The Number System of Algebra – Treated Theoretically and Historically, (2nd edition, with corrections, 1907), page 90, http://www.archive.org/download/numbersystemofal00fineuoft/numbersystemofal00fineuoft.pdf
↑ PlanetMath: Peano arithmetic
↑ The Dog School of Mathematics Presents

References

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