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:
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.
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,
There are several other common notations for multiplication:
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.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.
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.
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.
Methods of multiplication were documented in the Egyptian, Greece, Babylonian, Indus valley, and Chinese civilizations.
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:
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.
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.
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.
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 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:
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:
In case m = n, the value of the product is the same as that of the single factor xm. If m > n, the product is the empty product, with the value 1.
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,
One can similarly replace m with negative infinity, and define:
provided both limits exist.
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
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.
For integers, fractions, and real and complex numbers, multiplication has certain properties:
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.
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.
So we have proven:
The identity (−1) · x = −x can also be proven using the distributive property:
The proof that (−1) · (−1) = 1 is now easy:
In the book Arithmetices principia, nova methodo exposita, Giuseppe Peano proposed a new system for multiplication based on his axioms for natural numbers.[2]
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.
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.
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
Multiplication also is an abelian group, since it follows the commutative property a×b=b×a.
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).
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
indicates that the base a to be multiplied by itself n times.