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.
Contents 1 Notation and terminology 2 Computation 2.1 Historical algorithms 2.1.1 Egyptians 2.1.2 Babylonians 2.1.3 Chinese 2.2 Modern methods 2.2.1 Grid Method 2.3
3 Products of measurements 4 Products of sequences 4.1 Capital Pi notation 4.2 Infinite products 5 Properties
6 Axioms
7
Notation and terminology[edit] See also: Multiplier (linguistics) The multiplication sign × In arithmetic, multiplication is often written using the sign "×" between the terms; that is, in infix notation.[3] 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:
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.[5] 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
13 × 21 = (1 + 4 + 8) × 21 = (1 × 21) + (4 × 21) + (8 × 21) = 21 + 84 + 168 = 273. Babylonians[edit]
The
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
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.
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.
Grid Method[edit]
30 4 10 300 40 3 90 12 and then add the entries.
[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, so 50 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[edit] Capital Pi notation[edit] 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[edit] 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[edit]
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
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[edit] 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.[9] 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)
⋅ 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)
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
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) .
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[edit] 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 also[edit] Dimensional analysis
Karatsuba algorithm, for large numbers Toom–Cook multiplication, for very large numbers Schönhage–Strassen algorithm, for huge numbers
Booth's multiplication algorithm Floating point Fused multiply–add Multiply–accumulate Wallace tree Multiplicative inverse, reciprocal Factorial Genaille–Lucas rulers Napier's bones Peasant multiplication Product (mathematics), for generalizations Slide rule Notes[edit] ^ 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
References[edit] Boyer, 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 links[edit]
v t e Elementary arithmetic
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