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.
For example, 4 multiplied by 3, often written as and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together:
Here, 3 (the multiplier) and 4 (the multiplicand) are the factors, and 12 is the product.
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 finding the
area of a rectangle whose sides have some given
lengths. The area of a rectangle does not depend on which side is measured first—a consequence of the commutative property.
The product of two measurements (or
physical quantities) is a new type of measurement, usually with a
derivedunit. 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
The inverse operation of multiplication is 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,
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.
To reduce confusion between the multiplication sign × and the common variable x, multiplication is also denoted by dot signs, usually a middle-position dot (rarely
The middle dot notation or dot operator, encoded in Unicode as U+22C5⋅DOT OPERATOR, is now standard in the United States and other countries where the period is used as a
decimal point. When the dot operator character is not accessible, the
interpunct (·) is used. In other countries that use a
comma as a decimal mark, either the period or a middle dot is used for multiplication.
Historically, in the United Kingdom and Ireland, the middle dot was sometimes used for the decimal to prevent it from disappearing in the ruled line, and the period/full stop was used for multiplication. However, since the
Ministry of Technology ruled to use the period as the decimal point in 1968, and the
International System of Units (SI) standard has since been widely adopted, this usage is now found only in the more traditional journals such as The Lancet.
algebra, multiplication involving
variables is often written as a
juxtaposition (e.g., for times or for five times ), also called implied multiplication. The notation can also be used for quantities that are surrounded by
parentheses (e.g., , or for five times two). This implicit usage of multiplication can cause ambiguity when the concatenated variables happen to match the name of another variable, when a variable name in front of a parenthesis can be confused with a function name, or in the correct determination of the
order of operations.
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 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
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.
This article needs attention from an expert in Mathematics. The specific problem is: defining multiplication is not straightforward and different proposals have been made over the centuries, with competing ideas (e.g. recursive vs. non-recursive definitions). See the
talk page for details.WikiProject Mathematics may be able to help recruit an expert.(September 2023)
The product of two numbers or the multiplication between two numbers can be defined for common special cases: integers, natural numbers, fractions, real numbers, complex numbers, and quaternions.
Product of two natural numbers
Placing several stones into a rectangular pattern with rows and columns gives
Product of two integers
An integer can be either zero, a positive, or a negative 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.)
A negative number multiplied by a negative number is positive,
A negative number multiplied by a positive number is negative,
A positive number multiplied by a negative number is negative,
A positive number multiplied by a positive number is positive.
Product of two fractions
Two fractions can be multiplied by multiplying their numerators and denominators:
Product of two real numbers
There are several equivalent ways for define formally the real numbers; 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 a and b 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
§ Product of two integers. The construction of the real numbers through
Cauchy sequences is often preferred in order to avoid consideration of the four possible sign configurations.
Product of two complex numbers
Two complex numbers can be multiplied by the distributive law and the fact that , as follows:
Geometric meaning of complex multiplication can be understood rewriting complex numbers in
from which one obtains
The geometric meaning is that the magnitudes are multiplied and the arguments are added.
Product of two quaternions
The product of two
quaternions can be found in the article on
quaternions. Note, in this case, that and are in general different.
Many common 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. The example below illustrates "long multiplication" (the "standard algorithm", "grade-school multiplication"):
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, since adding logarithms is equivalent to multiplying. The
slide rule allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early 20th 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 Egyptian method of multiplication of integers and fractions, which is documented in the
Rhind Mathematical 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 = 2 × 42 = 84, 8 × 21 = 2 × 84 = 168. The full product could then be found by adding the appropriate terms found in the doubling sequence:
Babylonians used a
sexagesimalpositional 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 numbern: 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.
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.
These place value decimal arithmetic algorithms were introduced to Arab countries by
Al Khwarizmi in the early 9th century and popularized in the Western world by
Fibonacci in the 13th century.
Grid method multiplication, or the box method, is used in primary schools in England and Wales and in some areas[which?] of the United States to help teach an understanding of how multiple digit multiplication works. An example of multiplying 34 by 13 would be to lay the numbers out in a grid as follows:
The classical method of multiplying two n-digit numbers requires n2 digit multiplications.
Multiplication algorithms have been designed that reduce the computation time considerably when multiplying large numbers. Methods based on the
discrete Fourier transform reduce the
computational complexity to O(n log n log log n). In 2016, the factor log log n was replaced by a function that increases much slower, though still not constant. In March 2019, David Harvey and Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of  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 2172912 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:
[4 bags] × [3 marbles per bag] = 12 marbles.
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
distance. For example:
50 kilometers per hour × 3 hours = 150 kilometers.
In this case, the hour units cancel out, leaving the product with only kilometer units.
Other examples of multiplication involving units include:
2.5 meters × 4.5 meters = 11.25 square meters
11 meters/seconds × 9 seconds = 99 meters
4.5 residents per house × 20 houses = 90 residents
The product of a sequence of factors can be written with the product symbol , which derives from the capital letter Π (pi) in the
Greek alphabet (much like the same way the
summation symbol is derived from the Greek letter Σ (sigma)). The meaning of this notation is given by
which results in
In such a notation, the
variablei represents a varying
integer, called the multiplication index, that runs from the lower value 1 indicated in the subscript to the upper value 4 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 m = n, the value of the product is the same as that of the single factor xm; if m > n, the product is an
empty product whose value is 1—regardless of the expression for the factors.
Properties of capital pi notation
If all factors are identical, a product of n factors is equivalent to
One may also consider products of infinitely many terms; these are called
infinite products. Notationally, this consists in replacing n above by the
infinity symbol ∞. The product of such an infinite sequence 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 define:
When multiplication is repeated, the resulting operation is known as exponentiation. For instance, the product of three factors of two (2×2×2) 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 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
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 x − y when x and y are treated as integers. Thus both (0,1) and (1,2) are equivalent to −1. The multiplication axiom for integers defined this way is
The rule that −1 × −1 = 1 can then be deduced from
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 group—even if zero is excluded. This is easily seen by the nonexistence of an inverse for all elements other than 1 and −1.
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 ab 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 .
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
is the sum of N copies of M 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
The same sign rules apply to rational and real numbers.
Generalization to fractions is by multiplying the numerators and denominators, respectively: . This gives the area of a rectangle high and wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.
Alternatively, in trigonometric form, if , then
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 that is not any of the above number systems is a
polynomial ring (polynomials can be added and multiplied, but polynomials are not numbers in any usual sense).
Often division, , is the same as multiplication by an inverse, . Multiplication for some types of "numbers" may have corresponding division, without inverses; in an
integral domainx may have no inverse "" but may be defined. In a
division ring there are inverses, but may be ambiguous in non-commutative rings since need not be the same as .
^Announcing the TI Programmable 88!(PDF).
Texas Instruments. 1982.
Archived(PDF) from the original on 2017-08-03. Retrieved 2017-08-03. Now, implied multiplication is recognized by the
AOS and the square root, logarithmic, and trigonometric functions can be followed by their arguments as when working with pencil and paper. (NB. The TI-88 only existed as a prototype and was never released to the public.)
^Pletser, Vladimir (2012-04-04). "Does the Ishango Bone Indicate Knowledge of the Base 12? An Interpretation of a Prehistoric Discovery, the First Mathematical Tool of Humankind".