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n written as a juxtaposition e g x y displaystyle xy for x displaystyle x times y displaystyle y or 5 x displaystyle 5x for five times x displaystyle x also called implied multiplication the notation can also be used for quantities that are surrounded by parentheses e g 5 2 displaystyle 5 2 5 2 displaystyle 5 2 or 5 2 displaystyle 5 2 for five times two 12 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 13 14 in vector multiplication there is a distinction between the cross and the dot symbols the cross symbol generally denotes taking a cross product of two vectors yielding a vector as its result while the dot denotes taking the dot product of two vectors resulting in a scalar in computer programming the asterisk as in 5 2 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 or citation needed while the asterisk appeared on every keyboard 15 this usage originated in the fortran programming language 16 even modern compilers do not recognize or 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 17 18 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 19 in algebra a number that is the multiplier of a variable or expression e g the 3 in 3 x y 2 displaystyle 3xy 2 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 2 π displaystyle 2 times pi is a multiple of π displaystyle pi as is 5133 486 π displaystyle 5133 times 486 times pi 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 definitions edit this section 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 natural numbers integers rational numbers real numbers complex numbers and quaternions product of two natural numbers edit 3 by 4 is 12 the product of two natural numbers r s n displaystyle r s in mathbb n is defined as r s i 1 s r r r r s times j 1 r s s s s r times displaystyle r cdot s equiv sum _ i 1 s r underbrace r r cdots r _ s text times equiv sum _ j 1 r s underbrace s s cdots s _ r text times product of two integers edit 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 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 product of two fractions edit two fractions can be multiplied by multiplying their numerators and denominators z n z n z z n n displaystyle frac z n cdot frac z n frac z cdot z n cdot n which is defined when n n 0 displaystyle n n neq 0 product of two real numbers edit 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 π displaystyle pi is the least upper bound of 3 3 1 3 14 3 141 displaystyle 3 3 1 3 14 3 141 ldots 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 a sup x a x displaystyle a sup _ x in a x and b sup y b y displaystyle b sup _ y in b y then a b sup x a y b x y displaystyle a cdot b sup _ x in a y in b x cdot y 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 edit two complex numbers can be multiplied by the distributive law and the fact that i 2 1 displaystyle i 2 1 as follows a b i c d i a c a d i b i c b d i 2 a c b d a d b c i displaystyle begin aligned a b i cdot c d i a cdot c a cdot d i b i cdot c b cdot d cdot i 2 a cdot c b cdot d a cdot d b cdot c i end aligned a complex number in polar coordinates the geometric meaning of complex multiplication can be understood by rewriting complex numbers in polar coordinates a b i r cos φ i sin φ r e i φ displaystyle a b i r cdot cos varphi i sin varphi r cdot e i varphi furthermore c d i s cos ψ i sin ψ s e i ψ displaystyle c d i s cdot cos psi i sin psi s cdot e i psi from which one obtains a c b d a d b c i r s e i φ ψ displaystyle a cdot c b cdot d a cdot d b cdot c i r cdot s cdot e i varphi psi the geometric meaning is that the magnitudes are multiplied and the arguments are added product of two quaternions edit the product of two quaternions can be found in the article on quaternions note in this case that a b displaystyle a cdot b and b a displaystyle b cdot a are in general different computation edit main article multiplication algorithm the educated monkey a tin 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 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 23958233 5830 00000000 23 958 233 0 71874699 23 958 233 30 191665864 23 958 233 800 119791165 23 958 233 5 000 139676498390 139 676 498 390 in some countries such as germany 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 20 23958233 5830 119791165 191665864 71874699 00000000 139676498390 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 historical algorithms edit methods of multiplication were documented in the writings of ancient egyptian greek indian citation needed and chinese civilizations the ishango bone dated to about 18 000 to 20 000 bc may hint at a knowledge of multiplication in the upper paleolithic era in central africa but this is speculative 21 verification needed egyptians edit main article ancient egyptian multiplication 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 22 13 21 1 4 8 21 1 21 4 21 8 21 21 84 168 273 babylonians edit the babylonians used a sexagesimal positional number system analogous to the modern day decimal system 23 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 2 n 20 n followed by the multiples of 10 n 30 n 40 n and 50 n 23 then to compute any sexagesimal product say 53 n one only needed to add 50 n and 3 n computed from the table citation needed chinese edit see also chinese multiplication table 38 76 2888 in the mathematical text zhoubi suanjing dated prior to 300 bc and the nine chapters on the mathematical art multiplication calculations were written out in words although the early chinese mathematicians employed rod calculus involving place value addition subtraction multiplication and division the chinese were already using a decimal multiplication table by the end of the warring states period 24 modern methods edit 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 this is a variant of lattice multiplication 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 a 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 25 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 26 grid method edit 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 30 4 10 300 40 3 90 12 and then add the entries computer algorithms edit main article multiplication algorithm fast multiplication algorithms for large inputs the classical method of multiplying two n digit numbers requires n 2 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 27 in march 2019 david harvey and joris van der hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of o n log n displaystyle o n log n 28 the algorithm also based on the fast fourier transform is conjectured to be asymptotically optimal 29 the algorithm is not practically useful as it only becomes faster for multiplying extremely large numbers having more than 2 1729 12 bits 30 products of measurements edit main article dimensional analysis 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 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 speed by time gives 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 second 9 seconds 99 meters 4 5 residents per house 20 houses 90 residents product of a sequence edit capital pi notation edit further information iterated binary operation notation the product of a sequence of factors can be written with the product symbol displaystyle textstyle prod which derives from the capital letter π pi in the greek alphabet much like the same way the summation symbol displaystyle textstyle sum is derived from the greek letter σ sigma 31 32 the meaning of this notation is given by i 1 4 i 1 1 1 2 1 3 1 4 1 displaystyle prod _ i 1 4 i 1 1 1 2 1 3 1 4 1 which results in i 1 4 i 1 120 displaystyle prod _ i 1 4 i 1 120 in such a notation the variable i 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 includ...