Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a product. Multiplication is often denoted by the cross symbol, , by the mid-line dot operator, , by juxtaposition, or, in programming languages, by an asterisk, .
The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the quantity of the other one, the multiplier. Both numbers can be referred to as factors. This is to be distinguished from terms, which are added.
Whether the first factor is the multiplier or the multiplicand may be ambiguous or depend upon context. For example, the expression can be phrased as "3 times 4", or "Three groups of four", notating the multiplication symbol with the word of, and evaluated as , where 3 is the multiplier, but also as "3 multiplied by 4", in which case 3 becomes the multiplicand. One of the main properties of multiplication is the commutative property, which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3. Thus, the designation of multiplier and multiplicand does not affect the result of the multiplication.
Systematic generalizations of this basic definition define the multiplication of integers (including negative numbers), rational numbers (fractions), and real numbers.
Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers) or as calculating the area of a rectangle with sides of given lengths. The area of a rectangle remains the same regardless of which side is measured firstâÂÂa result of the commutative property.
The product of two measurements (or physical quantities) is a new type of measurement (or new quantity), usually with a derived unit of measurement. For example, multiplying the lengths (in meters or feet) of the two sides of a rectangle gives its area (in square meters or square feet). Such a product is the subject of dimensional analysis.
The inverse operation of multiplication is considered division. For example, since 4 multiplied by 3 equals 12, 12 divided by 3 equals 4. Indeed, multiplication by 3, followed by division by 3, yields the original number. The division of a number other than 0 by itself equals 1.
Several mathematical concepts expand upon the fundamental idea of multiplication. The product of a sequence, vector multiplication, complex numbers, and matrices are all examples where this can be seen. These more advanced constructs tend to affect the basic properties in their own ways, such as becoming noncommutative in matrices and some forms of vector multiplication or changing the sign of complex numbers.
In arithmetic, multiplication is often written using the multiplication sign (either or ) between the factors (that is, in infix notation). For example,
There are other mathematical notations for multiplication:
In computer programming, the asterisk (as in <code>5*2</code>) is still the most common notation. This is because most computers historically were limited to small character sets (such as ASCII and EBCDIC) that lacked a multiplication sign (such as <code>ÃÂ</code> or <code>â </code>), while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language. (Even modern compilers do not recognize <code>ÃÂ</code> or <code>â </code> as multiplication operators.)
The numbers to be multiplied are generally called the factors (as in factorization). The number to be multiplied is the multiplicand, and the number by which it is multiplied is the multiplier. Usually, the multiplier is placed first, and the multiplicand is placed second; however, sometimes the first factor is considered the multiplicand and the second the multiplier. Also, as the result of multiplication does not depend on the order of the factors, the distinction between multiplicand and multiplier is useful only at a very elementary level and in some multiplication algorithms, such as the long multiplication. Therefore, in some sources, the term multiplicand is regarded as a synonym for factor. In algebra, a number that is the multiplier of a variable or expression (e.g., the 3 in ) is called a coefficient.
The result of a multiplication is called a product. When one factor is an integer, the product is a multiple of the other or of the product of the others. Thus, is a multiple of , as is . A product of integers 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.
The product of two numbers or the multiplication between two numbers can be defined for common special cases: natural numbers, integers, rational numbers, real numbers, complex numbers, and quaternions.
The product of two natural numbers is defined as:
An integer can be either zero, a nonzero natural number, or minus a nonzero natural number. The product of zero and another integer is always zero. The product of two nonzero integers is determined by the product of their positive amounts, combined with the sign derived from the following rule:
(This rule is a consequence of the distributivity of multiplication over addition, and is not an additional rule.)
In words:
Two fractions can be multiplied by multiplying their numerators and denominators:
There are several equivalent ways to define the real numbers formally; see Construction of the real numbers. The definition of multiplication is a part of all these definitions.
A fundamental aspect of these definitions is that every real number can be approximated to any accuracy by rational numbers. A standard way for expressing this is that every real number is the least upper bound of a set of rational numbers. In particular, every positive real number is the least upper bound of the truncations of its infinite decimal representation; for example, is the least upper bound of
A fundamental property of real numbers is that rational approximations are compatible with arithmetic operations, and, in particular, with multiplication. This means that, if and are positive real numbers such that and then In particular, the product of two positive real numbers is the least upper bound of the term-by-term products of the sequences of their decimal representations.
As changing the signs transforms least upper bounds into greatest lower bounds, the simplest way to deal with a multiplication involving one or two negative numbers, is to use the rule of signs described above in . The construction of the real numbers through Cauchy sequences is often preferred in order to avoid consideration of the four possible sign configurations.
Two complex numbers can be multiplied by the distributive law and the fact that , as follows:
The geometric meaning of complex multiplication can be understood by rewriting complex numbers in polar coordinates:
Furthermore,
from which one obtains
The geometric meaning is that the magnitudes are multiplied and the arguments are added.
The product of two quaternions can be found in the article on quaternions. Note, in this case, that <matH>a \cdot b</math> and The algorithm, also based on the fast Fourier transform, is conjectured to be asymptotically optimal. The algorithm is not practically useful, as it only becomes faster for multiplying extremely large numbers (having more than bits).
One can only meaningfully add or subtract quantities of the same type, but quantities of different types can be multiplied or divided without problems. For example, four bags with three marbles each can be thought of as:
When two measurements are multiplied together, the product is of a type depending on the types of measurements. The general theory is given by dimensional analysis. This analysis is routinely applied in physics, but it also has applications in finance and other applied fields.
A common example in physics is the fact that multiplying speed by time gives distance. For example:
In this case, the hour units cancel out, leaving the product with only kilometer units.
Other examples of multiplication involving units include:
The product of a sequence of factors can be written with the product symbol , which derives from the capital letter à(pi) in the Greek alphabet (much like the same way the summation symbol is derived from the Greek letter ã (sigma)). The meaning of this notation is given by
which results in
In such a notation, the variable represents a varying integer, called the multiplication index, that runs from the lower value indicated in the subscript to the upper value given by the superscript. The product is obtained by multiplying together all factors obtained by substituting the multiplication index for an integer between the lower and the upper values (the bounds included) in the expression that follows the product operator.
More generally, the notation is defined as
where m and n are integers or expressions that evaluate to integers. In the case where , the value of the product is the same as that of the single factor x<sub>m</sub>; if , the product is an empty product whose value is 1âÂÂregardless of the expression for the factors.
By definition,
If all factors are identical, a product of factors is equivalent to exponentiation:
Associativity and commutativity of multiplication imply
if is a non-negative integer, or if all are positive real numbers, and
if all are non-negative integers, or if is a positive real number.
One may also consider products of infinitely many factors; these are called infinite products. Notationally, this consists in replacing n above by the infinity symbol âÂÂ. The product of such an infinite sequence is defined as the limit of the product of the first n factors, as n grows without bound. That is,
One can similarly replace m with negative infinity, and define:
provided both limits exist.
When multiplication is repeated, the resulting operation is known as exponentiation. For instance, the product of three factors of two (2ÃÂ2ÃÂ2) is "two raised to the third power", and is denoted by 2<sup>3</sup>, 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 the base appears in the expression, so that the expression
indicates that n copies of the base a are to be multiplied together. This notation can be used whenever multiplication is known to be power associative.
For real and complex numbers, which includes, for example, natural numbers, integers, and fractions, 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. Hurwitz's theorem shows that for the hypercomplex numbers of dimension 8 or greater, including the octonions, sedenions, and trigintaduonions, multiplication is generally not associative.
In the book Arithmetices principia, nova methodo exposita, Giuseppe Peano proposed axioms for arithmetic based on his axioms for natural numbers. Peano arithmetic has two axioms for multiplication:
Here S(y) represents the successor of y; i.e., the natural number that follows y. The various properties like associativity can be proved from these and the other axioms of Peano arithmetic, including induction. For instance, S(0), denoted by 1, is a multiplicative identity because
The axioms for integers typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (x,y) as equivalent to when x and y are treated as integers. Thus both (0,1) and (1,2) are equivalent to âÂÂ1. The multiplication axiom for integers defined this way is
The rule that âÂÂ1 àâÂÂ1 = 1 can then be deduced from
Multiplication is extended in a similar way to rational numbers and then to real numbers.
The product of non-negative integers can be defined with set theory using cardinal numbers or the Peano axioms. See below how to extend this to multiplying arbitrary integers, and then arbitrary rational numbers. The product of real numbers is defined in terms of products of rational numbers; see construction of the real numbers.
There are many sets that, under the operation of multiplication, satisfy the axioms that define group structure. These axioms are closure, associativity, and the inclusion of an identity element and inverses.
A simple example is the set of non-zero rational numbers. Here identity 1 is had, as opposed to groups under addition where the identity is typically 0. Note that with the rationals, zero must be excluded because, under multiplication, it does not have an inverse: there is no rational number that can be multiplied by zero to result in 1. In this example, an abelian group is had, but that is not always the case.
To see this, consider the set of invertible square matrices of a given dimension over a given field. Here, it is straightforward to verify closure, associativity, and inclusion of identity (the identity matrix) and inverses. However, matrix multiplication is not commutative, which shows that this group is non-abelian.
Another fact worth noticing is that the integers under multiplication do not form a groupâÂÂeven if zero is excluded. This is easily seen by the nonexistence of an inverse for all elements other than 1 and âÂÂ1.
Multiplication in group theory is typically notated either by a dot or by juxtaposition (the omission of an operation symbol between elements). So multiplying element a by element b could be notated as a b or ab. When referring to a group via the indication of the set and operation, the dot is used. For example, our first example could be indicated by .
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).