Multiplication Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division. The multiplication of whole numbers may be thought as a repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the value of the other one, the multiplier. Normally, the multiplier is written first and multiplicand second (though this can vary by language). a × b = b + ⋯ + b ⏟ a displaystyle atimes b=underbrace b+cdots +b _ a For example, 4 multiplied by 3 (often written as 3 × 4 displaystyle 3times 4 and spoken as "3 times 4") can be calculated by adding 3 copies of 4 together: 3 × 4 = 4 + 4 + 4 = 12 displaystyle 3times 4=4+4+4=12 Here 3 and 4 are the "factors" and 12 is the "product". One of the main properties of multiplication is the commutative property: adding 3 copies of 4 gives the same result as adding 4 copies of 3: 4 × 3 = 3 + 3 + 3 + 3 = 12 displaystyle 4times 3=3+3+3+3=12 Thus the designation of multiplier and multiplicand does not affect the result of the multiplication. The multiplication of integers (including negative numbers), rational numbers (fractions) and real numbers is defined by a systematic generalization of this basic definition. Multiplication 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 given lengths. The area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for instance multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division. For example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication Multiplication by 3, followed by division by 3, yields the original number (since the division of a number other than 0 by itself equals 1). Multiplication Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter. A listing of the many different kinds of products that are used in mathematics is given in the product (mathematics) page.Contents1 Notation and terminology 2 Computation2.1 Historical algorithms2.1.1 Egyptians 2.1.2 Babylonians 2.1.3 Chinese2.2 Modern methods2.2.1 Grid Method3 Products of measurements 4 Products of sequences4.1 Capital Pi notation 4.2 Infinite products5 Properties 6 Axioms 7 Multiplication Multiplication with set theory 8 Multiplication Multiplication in group theory 9 Multiplication Multiplication of different kinds of numbers 10 Exponentiation 11 See also 12 Notes 13 References 14 External linksNotation and terminology See also: Multiplier (linguistics)The multiplication sign ×In arithmetic, multiplication is often written using the sign "×" between the terms; that is, in infix notation. For example, 2 × 3 = 6 displaystyle 2times 3=6 (verbally, "two times three equals six") 3 × 4 = 12 displaystyle 3times 4=12 2 × 3 × 5 = 6 × 5 = 30 displaystyle 2times 3times 5=6times 5=30 2 × 2 × 2 × 2 × 2 = 32 displaystyle 2times 2times 2times 2times 2=32 The sign is encoded in Unicode at U+00D7 × MULTIPLICATION SIGN (HTML × · ×). There are other mathematical notations for multiplication: Multiplication Multiplication is also denoted by dot signs, usually a middle-position dot (rarely period): 5 ⋅ 2 or 5 . 2 displaystyle 5cdot 2quad text or quad 5,.,2 The middle dot notation, encoded in Unicode as U+22C5 ⋅ dot operator, is standard in the United States, the United Kingdom, 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.[citation needed]In algebra, multiplication involving variables is often written as a juxtaposition (e.g., xy for x times y or 5x for five times x), also called implied multiplication. The notation can also be used for quantities that are surrounded by parentheses (e.g., 5(2) or (5)(2) 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. In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a cross product of two vectors, yielding a vector as the 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 due to the fact that most computers historically were limited to small character sets (such as ASCII ASCII and EBCDIC) that lacked a multiplication sign (such as ⋅ or ×), while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language. The numbers to be multiplied are generally called the "factors". The number to be multiplied is called the "multiplicand", while the number of times the multiplicand is to be multiplied comes from 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. Additionally, there are some sources in which 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 3xy2) is called a coefficient. The result of a multiplication is called a product. 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. Computation The 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. 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 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 Methods of multiplication were documented in the Egyptian, Greek, Indian and Chinese civilizations. The Ishango bone, dated to about 18,000 to 20,000 BC, hints at a knowledge of multiplication in the Upper Paleolithic Upper Paleolithic era in Central Africa. Egyptians Main article: Ancient Egyptian multiplication 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 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:13 × 21 = (1 + 4 + 8) × 21 = (1 × 21) + (4 × 21) + (8 × 21) = 21 + 84 + 168 = 273.Babylonians The Babylonians 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. Chinese38 × 76 = 2888In 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 Rod calculus involving place value addition, subtraction, multiplication and division. These place value decimal arithmetic algorithms were introduced by Al Khwarizmi Al Khwarizmi to Arab countries in the early 9th century. Modern methodsProduct 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 Brahmagupta gave rules for addition, subtraction, multiplication and division. Henry Burchard Fine, then 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 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.Grid Method Grid method multiplication or the box method, is used in primary schools in England and Wales & in some areas 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 like:  30 410 300 403 90 12and then add the entries. Computer Computer algorithms Main article: Multiplication Multiplication algorithm The classical method of multiplying two n-digit numbers requires n2 simple multiplications. Multiplication Multiplication algorithms have been designed that reduce the computation time considerably when multiplying large numbers. In particular for very large numbers, methods based on the discrete Fourier transform can reduce the number of simple multiplications to the order of n log(n) log log(n). Products of measurements Main article: Dimensional analysis One can only meaningfully add or subtract quantities of the same type but can multiply or divide quantities of different types. Four bags with three marbles each can be though 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 the measurements. The general theory is given by dimensional analysis. This analysis is routinely applied in physics but has also found applications in finance. A common example is multiplying speed by time gives distance, so50 kilometers per hour × 3 hours = 150 kilometers.Other examples: 2.5  meters × 4.5  meters = 11.25  square meters displaystyle 2.5 text meters times 4.5 text meters =11.25 text square meters 11  meters/second × 9  seconds = 99  meters displaystyle 11 text meters/second times 9 text seconds =99 text meters Products of sequences Capital Pi notation 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: ∏ i = 1 4 i = 1 ⋅ 2 ⋅ 3 ⋅ 4 , displaystyle prod _ i=1 ^ 4 i=1cdot 2cdot 3cdot 4, that is ∏ i = 1 4 i = 24. displaystyle prod _ i=1 ^ 4 i=24. The subscript gives the symbol for a dummy variable (i in this case), called the "index of multiplication" together with its lower bound (1), whereas the superscript (here 4) 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: ∏ i = 1 6 i = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 6 = 720 displaystyle prod _ i=1 ^ 6 i=1cdot 2cdot 3cdot 4cdot 5cdot 6=720 More generally, the notation is defined as ∏ i = m n x i = x m ⋅ x m + 1 ⋅ x m + 2 ⋅ ⋯ ⋅ x n − 1 ⋅ x n , displaystyle prod _ i=m ^ n x_ i =x_ m cdot x_ m+1 cdot x_ m+2 cdot ,,cdots ,,cdot x_ n-1 cdot x_ n , where m and n are integers or expressions that evaluate to integers. 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. Infinite products Main article: Infinite product One may also consider products of infinitely many terms; these are called infinite products. Notationally, we would replace n above by the lemniscate ∞. 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, ∏ i = m ∞ x i = lim n → ∞ ∏ i = m n x i . displaystyle prod _ i=m ^ infty x_ i =lim _ nto infty prod _ i=m ^ n x_ i . One can similarly replace m with negative infinity, and define: ∏ i = − ∞ ∞ x i = ( lim m → − ∞ ∏ i = m 0 x i ) ⋅ ( lim n → ∞ ∏ i = 1 n x i ) , displaystyle prod _ i=-infty ^ infty x_ i =left(lim _ mto -infty prod _ i=m ^ 0 x_ i right)cdot left(lim _ nto infty prod _ i=1 ^ n x_ i right), provided both limits exist. Properties Multiplication 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 a singular matrix where the determinant is 0. In this process, information is lost and cannot be regained.For the real and complex numbers, which includes for example natural numbers, integers and fractions, multiplication has certain properties:Commutative property The order in which two numbers are multiplied does not matter: x ⋅ y = y ⋅ x . displaystyle xcdot y=ycdot x. Associative property Expressions solely involving multiplication or addition are invariant with respect to order of operations: ( x ⋅ y ) ⋅ z = x ⋅ ( y ⋅ z ) displaystyle (xcdot y)cdot z=xcdot (ycdot z) Distributive property Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions: x ⋅ ( y + z ) = x ⋅ y + x ⋅ z displaystyle xcdot (y+z)=xcdot y+xcdot z Identity element The multiplicative identity is 1; anything multiplied by 1 is itself. This feature of 1 is known as the identity property: x ⋅ 1 = x displaystyle xcdot 1=x Property of 0 Any number multiplied by 0 is 0. This is known as the zero property of multiplication: x ⋅ 0 = 0 displaystyle xcdot 0=0 Negation −1 times any number is equal to the additive inverse of that number. ( − 1 ) ⋅ x = ( − x ) displaystyle (-1)cdot x=(-x) where ( − x ) + x = 0 displaystyle (-x)+x=0 –1 times –1 is 1. ( − 1 ) ⋅ ( − 1 ) = 1 displaystyle (-1)cdot (-1)=1 Inverse element Every number x, except 0, has a multiplicative inverse, 1 x displaystyle frac 1 x , such that x ⋅ ( 1 x ) = 1 displaystyle xcdot left( frac 1 x right)=1 .Order preservation Multiplication Multiplication by a positive number preserves order:For a > 0, if b > c then ab > ac.For a < 0, if b > c then ab < ac.The complex numbers do not have an ordering.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. Axioms Main article: Peano axioms 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: x × 0 = 0 displaystyle xtimes 0=0 x × S ( y ) = ( x × y ) + x displaystyle xtimes S(y)=(xtimes y)+x Here S(y) represents the successor of y, or 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 x × 1 = x × S ( 0 ) = ( x × 0 ) + x = 0 + x = x displaystyle xtimes 1=xtimes S(0)=(xtimes 0)+x=0+x=x 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 ( x p , x m ) × ( y p , y m ) = ( x p × y p + x m × y m , x p × y m + x m × y p ) displaystyle (x_ p ,,x_ m )times (y_ p ,,y_ m )=(x_ p times y_ p +x_ m times y_ m ,;x_ p times y_ m +x_ m times y_ p ) The rule that −1 × −1 = 1 can then be deduced from ( 0 , 1 ) × ( 0 , 1 ) = ( 0 × 0 + 1 × 1 , 0 × 1 + 1 × 0 ) = ( 1 , 0 ) displaystyle (0,1)times (0,1)=(0times 0+1times 1,,0times 1+1times 0)=(1,0) Multiplication Multiplication is extended in a similar way to rational numbers and then to real numbers. Multiplication Multiplication with set theory 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. Multiplication Multiplication in group theory 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 we have identity 1, as opposed to groups under addition where the identity is typically 0. Note that with the rationals, we must exclude zero 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 we have an abelian group, but that is not always the case. To see this, look at the set of invertible square matrices of a given dimension, over a given field. Now it is straightforward to verify closure, associativity, and inclusion of identity (the identity matrix) and inverses. However, matrix multiplication is not commutative, therefore this group is nonabelian. Another fact of note is that the integers under multiplication is not a group, even if we exclude zero. This is easily seen by the nonexistence of an inverse for all elements other than 1 and −1. Multiplication 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 a ⋅ displaystyle cdot b or ab. When referring to a group via the indication of the set and operation, the dot is used, e.g., our first example could be indicated by ( Q ∖ 0 , ⋅ ) displaystyle left(mathbb Q smallsetminus 0 ,cdot right) Multiplication 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 quaternions).Integers N × M displaystyle Ntimes M 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 N × ( − M ) = ( − N ) × M = − ( N × M ) displaystyle Ntimes (-M)=(-N)times M=-(Ntimes M) and ( − N ) × ( − M ) = N × M displaystyle (-N)times (-M)=Ntimes M The same sign rules apply to rational and real numbers.Rational numbers Generalization to fractions A B × C D displaystyle frac A B times frac C D is by multiplying the numerators and denominators respectively: A B × C D = ( A × C ) ( B × D ) displaystyle frac A B times frac C D = frac (Atimes C) (Btimes D) . This gives the area of a rectangle A B displaystyle frac A B high and C D displaystyle frac C D wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.Real numbers Real numbers and their products can be defined in terms of sequences of rational numbers.Complex numbers Considering complex numbers z 1 displaystyle z_ 1 and z 2 displaystyle z_ 2 as ordered pairs of real numbers ( a 1 , b 1 ) displaystyle (a_ 1 ,b_ 1 ) and ( a 2 , b 2 ) displaystyle (a_ 2 ,b_ 2 ) , the product z 1 × z 2 displaystyle z_ 1 times z_ 2 is ( a 1 × a 2 − b 1 × b 2 , a 1 × b 2 + a 2 × b 1 ) displaystyle (a_ 1 times a_ 2 -b_ 1 times b_ 2 ,a_ 1 times b_ 2 +a_ 2 times b_ 1 ) . This is the same as for reals, a 1 × a 2 displaystyle a_ 1 times a_ 2 , when the imaginary parts b 1 displaystyle b_ 1 and b 2 displaystyle b_ 2 are zero.Equivalently, denoting − 1 displaystyle sqrt -1 as i displaystyle i , we have z 1 × z 2 = ( a 1 + b 1 i ) ( a 2 + b 2 i ) = ( a 1 × a 2 ) + ( a 1 × b 2 i ) + ( b 1 × a 2 i ) + ( b 1 × b 2 i 2 ) = ( a 1 a 2 − b 1 b 2 ) + ( a 1 b 2 + b 1 a 2 ) i . displaystyle z_ 1 times z_ 2 =(a_ 1 +b_ 1 i)(a_ 2 +b_ 2 i)=(a_ 1 times a_ 2 )+(a_ 1 times b_ 2 i)+(b_ 1 times a_ 2 i)+(b_ 1 times b_ 2 i^ 2 )=(a_ 1 a_ 2 -b_ 1 b_ 2 )+(a_ 1 b_ 2 +b_ 1 a_ 2 )i. Further generalizations See Multiplication 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 (you can add and multiply polynomials, but polynomials are not numbers in any usual sense.)Division Often division, x y displaystyle frac x y , is the same as multiplication by an inverse, x ( 1 y ) displaystyle xleft( frac 1 y right) . Multiplication Multiplication for some types of "numbers" may have corresponding division, without inverses; in an integral domain x may have no inverse " 1 x displaystyle frac 1 x " but x y displaystyle frac x y may be defined. In a division ring there are inverses, but x y displaystyle frac x y may be ambiguous in non-commutative rings since x ( 1 y ) displaystyle xleft( frac 1 y right) need not be the same as ( 1 y ) x displaystyle left( frac 1 y right)x .Exponentiation Main article: Exponentiation 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 a n = a × a × ⋯ × a ⏟ n displaystyle a^ n =underbrace atimes atimes cdots times a _ n 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. See alsoKaratsuba algorithm, for large numbers Toom–Cook multiplication, for very large numbers Schönhage–Strassen algorithm, for huge numbersBooth's multiplication algorithm Floating point Fused multiply–add Multiply–accumulate Wallace treeMultiplicative inverse, reciprocal Factorial Genaille–Lucas rulers Napier's bones Peasant multiplication Product (mathematics), for generalizations Slide ruleNotes^ a b c Devlin, Keith (January 2011). "What Exactly is Multiplication?". Mathematical Association of America. Retrieved May 14, 2017. With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first)  ^ "小学校の掛け算の授業では、順序に意味があるらしい。" [In elementary school multiplication lessons, the order would appear to be meaningful] (in Japanese). September 30, 2009. Retrieved May 14, 2017.  ^ Khan Academy (2015-08-14), Intro to multiplication Multiplication and division Arithmetic Arithmetic Khan Academy, retrieved 2017-03-07  ^ Khan Academy (2012-09-06), Why aren't we using the multiplication sign? Introduction to algebra Algebra Algebra I Khan Academy, retrieved 2017-03-07  ^ Announcing the TI Programmable 88! (PDF). Texas Instruments. 1982. Archived (PDF) from the original on 2017-08-03. Retrieved 2017-08-03.  ^ Crewton Ramone. "Multiplicand and Multiplier". Crewton Ramone's House of Math. Retrieved 10 November 2015. . ^ "Google book search". Google Book Search.  ^ Fine, Henry B. (1907). The Number System of Algebra Algebra – Treated Theoretically and Historically (PDF) (2nd ed.). p. 90.  ^ "Peano arithmetic". PlanetMath. ReferencesBoyer, Carl B. (revised by Merzbach, Uta C.) (1991). History of Mathematics. John Wiley and Sons, Inc. ISBN 0-471-54397-7. CS1 maint: Multiple names: authors list (link)External linksv t eElementary arithmetic   Division (÷ or /)Authority controlGND: 41707