Four bags with three marbles per bag gives twelve marbles (4 × 3 = 12).Multiplication can also be thought of asscaling. Here, 2 is being multiplied by 3 using scaling, giving 6 as a result.Animation for the multiplication 2 × 3 = 64 × 5 = 20. The large rectangle is made up of 20 squares, each 1 unit by 1 unit.Area of a cloth4.5m × 2.5m = 11.25m2;41/2 × 21/2 = 111/4
Multiplication is one of the four elementary mathematical operations ofarithmetic, with the other ones beingaddition,subtraction, anddivision. The result of a multiplication operation is called aproduct. Multiplication is often denoted by the cross symbol,×, by the mid-line dot operator,·, by juxtaposition, or, on computers, 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, themultiplicand, as the quantity of the other one, themultiplier; both numbers can be referred to asfactors. This is to be distinguished fromterms, which are added.
For example, the expression, phrased as "3 times 4" or "3 multiplied by 4", can be evaluated by adding 3 copies of 4 together:
Here, 3 (themultiplier) and 4 (themultiplicand) are thefactors, and 12 is theproduct.
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.[1][2]
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 (orphysical quantities) is a new type of measurement (or new quantity), usually with a derivedunit 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 ofdimensional analysis.
The inverse operation of multiplication isdivision. 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.
To reduce confusion between the multiplication sign × and the common variablex, multiplication is also denoted by dot signs,[4] usually a middle-position dot (rarelyperiod):.
The middle dot notation ordot operator, encoded in Unicode asU+22C5⋅DOT OPERATOR, is now standard in the United States and other countries . When the dot operator character is not accessible, theinterpunct (·) is used. In other countries that use acomma as a decimal mark, either the period or a middle dot is used for multiplication.[citation needed]
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 theMinistry of Technology ruled to use the period as the decimal point in 1968,[5] and theInternational System of Units (SI) standard has since been widely adopted, this usage is now found only in the more traditional journals such asThe Lancet.[6]
Inalgebra, multiplication involvingvariables is often written as ajuxtaposition (e.g., for times or for five times), also calledimplied multiplication.[7] The notation can also be used for quantities that are surrounded byparentheses (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 theorder of operations.[8][9]
Invector multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking across product of twovectors, yielding a vector as its result, while the dot denotes taking thedot product of two vectors, resulting in ascalar.
Incomputer programming, theasterisk (as in5*2) is still the most common notation. This is because most computers historically were limited to smallcharacter sets (such asASCII andEBCDIC) that lacked a multiplication sign (such as⋅ or×), while the asterisk appeared on every keyboard.[citation needed] This usage originated in theFORTRAN programming language.[10]
The numbers to be multiplied are generally called the "factors" (as infactorization). 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;[11][12] 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 somemultiplication algorithms, such as thelong multiplication. Therefore, in some sources, the term "multiplicand" is regarded as a synonym for "factor".[13]In algebra, a number that is the multiplier of a variable or expression (e.g., the 3 in) is called acoefficient.
The result of a multiplication is called aproduct. When one factor is an integer, the product is amultiple 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 sectionneeds 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 thetalk 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: natural numbers, integers, rational numbers, real numbers, complex numbers, and quaternions.
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 theirpositive amounts, combined with the sign derived from the following rule:
×
+
−
+
+
−
−
−
+
(This rule is a consequence of thedistributivity of multiplication over addition, and is not anadditional rule.)
In words:
A positive number multiplied by a positive number is positive (product of natural numbers),
A positive number multiplied by a negative number is negative,
A negative number multiplied by a positive number is negative,
A negative number multiplied by a negative number is positive.
There are several equivalent ways to define formally the real numbers; seeConstruction 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 byrational numbers. A standard way for expressing this is that every real number is theleast upper bound of a set of rational numbers. In particular, every positive real number is the least upper bound of thetruncations of its infinitedecimal representation; for example, is the least upper bound of
A fundamental property of real numbers is that rational approximations are compatible witharithmetic operations, and, in particular, with multiplication. This means that, ifa andb 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 thesequences 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 throughCauchy sequences is often preferred in order to avoid consideration of the four possible sign configurations.
The Educated Monkey—atin toy dated 1918, used as a multiplication "calculator".For example: set the monkey's feet to 4 and 9, and get the product—36—in its hands.
Many common methods for multiplying numbers using pencil and paper require amultiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9). However, one method, thepeasant multiplication algorithm, does not. The example below illustrates "long multiplication" (the "standard algorithm", "grade-school multiplication"):
In some countries such asGermany, the multiplication above is depicted similarly but with the original problem written on a single line and computation starting with the first digit of the multiplier:[14]
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. Theslide rule allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early 20th century, mechanicalcalculators, such as theMarchant, automated multiplication of up to 10-digit numbers. Modern electroniccomputers and calculators have greatly reduced the need for multiplication by hand.
The Egyptian method of multiplication of integers and fractions, which is documented in theRhind 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, obtaining2 × 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:[16]
TheBabylonians used asexagesimalpositional number system, analogous to the modern-daydecimal system. Thus, Babylonian multiplication was very similar to modern decimal multiplication. Because of the relative difficulty of remembering60 × 60 different products, Babylonian mathematicians employedmultiplication tables. These tables consisted of a list of the first twenty multiples of a certainprincipal 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.[citation needed]
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 of45 × 256 = 11520. This is a variant ofLattice multiplication.
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.[18]
These place value decimal arithmetic algorithms were introduced to Arab countries byAl Khwarizmi in the early 9th century and popularized in the Western world byFibonacci in the 13th century.[19]
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 twon-digit numbers requiresn2 digit multiplications.Multiplication algorithms have been designed that reduce the computation time considerably when multiplying large numbers. Methods based on thediscrete Fourier transform reduce thecomputational complexity toO(n logn log logn). In 2016, the factorlog logn was replaced by a function that increases much slower, though still not constant.[20] In March 2019, David Harvey and Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of[21] The algorithm, also based on the fast Fourier transform, is conjectured to be asymptotically optimal.[22] The algorithm is not practically useful, as it only becomes faster for multiplying extremely large numbers (having more than2172912 bits).[23]
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:[1]
[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 bydimensional 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 multiplyingspeed bytime givesdistance. 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 theGreek alphabet (much like the same way thesummation symbol is derived from the Greek letter Σ (sigma)).[24][25] The meaning of this notation is given by
which results in
In such a notation, thevariablei represents a varyinginteger, called the multiplication index, that runs from the lower value1 indicated in the subscript to the upper value4 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
wherem andn are integers or expressions that evaluate to integers. In the case wherem =n, the value of the product is the same as that of the single factorxm; ifm >n, the product is anempty product whose value is 1—regardless of the expression for the factors.
One may also consider products of infinitely many factors; these are calledinfinite products. Notationally, this consists in replacingn above by theinfinity symbol ∞. The product of such an infinite sequence is defined as thelimit of the product of the firstn factors, asn grows without bound. That is,
One can similarly replacem with negative infinity, and define:
When multiplication is repeated, the resulting operation is known asexponentiation. 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 asuperscript three. In this example, the number two is thebase, and three is theexponent.[26] In general, the exponent (or superscript) indicates how many times the base appears in the expression, so that the expression
indicates thatn copies of the basea are to be multiplied together. This notation can be used whenever multiplication is known to bepower associative.
Multiplication of numbers 0–10. Line labels = multiplicand.X axis = multiplier.Y axis = product. Extension of this pattern into other quadrants gives the reason why a negative number times a negative number yields a positive number. Note also how multiplication by zero causes a reduction in dimensionality, as does multiplication by asingular matrix where thedeterminant is 0. In this process, information is lost and cannot be regained.
HereS(y) represents thesuccessor ofy; i.e., the natural number that followsy. The various properties like associativity can be proved from these and the other axioms of Peano arithmetic, includinginduction. For instance,S(0), denoted by 1, is a multiplicative identity because
The axioms forintegers typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (x,y) as equivalent tox −y whenx andy 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
The product of non-negative integers can be defined with set theory usingcardinal numbers or thePeano axioms. Seebelow 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; seeconstruction of the real numbers.[32]
There are many sets that, under the operation of multiplication, satisfy the axioms that definegroup structure. These axioms are closure, associativity, and the inclusion of an identity element and inverses.
A simple example is the set of non-zerorational 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, anabelian 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 givenfield. Here, it is straightforward to verify closure, associativity, and inclusion of identity (theidentity 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 elementa by elementb could be notated asab orab. 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.[33]
Numbers cancount (3 apples),order (the 3rd apple), ormeasure (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 asmatrices) or do not look much like numbers (such asquaternions).
Integers
is the sum ofN copies ofM whenN andM are positive whole numbers. This gives the number of things in an arrayN wide andM high. Generalization to negative numbers can be done by
and
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.[27]
Considering complex numbers and as ordered pairs of real numbers and, the product is. This is the same as for reals when theimaginary parts and are zero.
Alternatively, in trigonometric form, if, then[27]
Further generalizations
SeeMultiplication in group theory, above, andmultiplicative group, which for example includes matrix multiplication. A very general, and abstract, concept of multiplication is as the "multiplicatively denoted" (second) binary operation in aring. An example of a ring that is not any of the number systems above is apolynomial ring (polynomials can be added and multiplied, but polynomials are not numbers in any usual sense).
Division
Often division,, is the same as multiplication by an inverse,. Multiplication for some types of "numbers" may have corresponding division, without inverses; in anintegral domainx may have no inverse "" but may be defined. In adivision ring there are inverses, but may be ambiguous in non-commutative rings since need not be the same as.[citation needed]
^Announcing the TI Programmable 88!(PDF).Texas Instruments. 1982.Archived(PDF) from the original on 2017-08-03. Retrieved2017-08-03.Now, implied multiplication is recognized by theAOS 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".arXiv:1204.1019 [math.HO].
^Cawagas, Raoul E.; Carrascal, Alexander S.; Bautista, Lincoln A.; Maria, John P. Sta.; Urrutia, Jackie D.; Nobles, Bernadeth (2009). "The Subalgebra Structure of the Cayley-Dickson Algebra of Dimension 32 (trigintaduonion)".arXiv:0907.2047v3 [math.RA].
