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multiplication
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{{other uses}}{{refimprove|date=April 2012}}File:Multiply 4 bags 3 marbles.svg|thumb|right|Four bags with three marbles per bag gives twelve marbles (4 Ã— 3 = 12).]]File:Multiply scaling.svg|thumb|right|Multiplication can also be thought of as scaling. Here we see 2 being multiplied by 3 using scaling, giving 6 as a result.]](File:Multiplication as scaling integers.gif|thumb|Animation for the multiplication 2 Ã— 3 = 6.)(File:Multiplication scheme 4 by 5.jpg|thumb|right|4 Ã— 5 = 20. The large rectangle is composed of 20 squares, each having dimensions of 1 by 1.)(File:Multiply field fract.svg|thumb|right|Area of a cloth {{nowrap|1=4.5m Ã— 2.5m = 11.25m2}}; {{nowrap|1=4Â½ Ã— 2Â½ = 11Â¼}})Multiplication (often denoted by the cross symbol "Ã—", by a point "â‹…", by juxtaposition, or, on computers, by an asterisk "âˆ—") 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 secondWEB, Devlin, Keith,weblink What Exactly is Multiplication?, Keith Devlin, Mathematical Association of America, January 2011, With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first), May 14, 2017, (though this can vary by languageWEB,weblink å°å¦æ ¡ã®æŽ›ã‘ç®—ã®æŽˆæ¥ã§ã¯ã€é †åºã«æ„å‘³ãŒã‚ã‚‹ã‚‰ã—ã„ã€‚, Japanese, In elementary school multiplication lessons, the order would appear to be meaningful, September 30, 2009, May 14, 2017, ).
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atimes b = underbrace{b + cdots + b}_a
For example, 4 multiplied by 3 (often written as 3 times 4 and spoken as "3 times 4") can be calculated by adding 3 copies of 4 together:
3 times 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 times 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 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 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 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.Notation and terminology
{{See also|Multiplier (linguistics)}}thumb|right|The multiplication sign Ã—In arithmetic, multiplication is often written using the sign "Ã—" between the terms; that is, in infix notation.{{Citation |last=Khan Academy |title=Intro to multiplication {{!}} Multiplication and division {{!}} Arithmetic {{!}} Khan Academy |date=2015-08-14 |url=https://www.youtube.com/watch?v=RNxwasijbAo |accessdate=2017-03-07}} For example,
2times 3 = 6 (verbally, "two times three equals six")
3times 4 = 12
2times 3times 5 = 6times 5 = 30
2times 2times 2times 2times 2 = 32
The sign is encoded in Unicode at {{unichar|D7|MULTIPLICATION SIGN|nlink=Multiplication sign|html=}}.There are other mathematical notations for multiplication: - Multiplication is also denoted by dot signs,{{Citation |last=Khan Academy |title=Why aren't we using the multiplication sign? {{!}} Introduction to algebra {{!}} Algebra I {{!}} Khan Academy |date=2012-09-06 |url=https://www.youtube.com/watch?v=vDaIKB19TvY |accessdate=2017-03-07}} usually a middle-position dot (rarely period):
5 cdot 2 quadtext{or}quad 5,.,2
The middle dot notation, encoded in Unicode as {{unichar|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 (Â·) 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|date=August 2011}}
- {{anchor|Implicit|Explicit}}In algebra, multiplication involving variables is often written as a (wikt:juxtaposition|juxtaposition) (e.g., xy for x times y or 5x for five times x), also called implied multiplication.BOOK, Announcing the TI Programmable 88!, Texas Instruments, 1982,weblink 2017-08-03, no,weblink" title="web.archive.org/web/20170803091337weblink">weblink 2017-08-03, 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.
Computation
(file:×¦×¢×¦×•×¢ ×ž×›× ×™ ×ž×©× ×ª 1918 ×œ×—×™×©×•×‘×™ ×œ×•×— ×”×›×¤×œ The Educated Monkey.jpg|200px|right|thumb|The Educated Monkey - a tin toy dated 1918, used as a multiplication â€œcalculatorâ€. For example: set the monkeyâ€™s feet to 4 and 9, and get the product - 36 - in its hands.)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 era in Central Africa.Egyptians
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 {{nowrap|1=2 Ã— 21 = 42}}, {{nowrap|1=4 Ã— 21 = 2 Ã— 42 = 84}}, {{nowrap|1=8 Ã— 21 = 2 Ã— 84 = 168}}. The full product could then be found by adding the appropriate terms found in the doubling sequence:
13 Ã— 21 = (1 + 4 + 8) Ã— 21 = (1 Ã— 21) + (4 Ã— 21) + (8 Ã— 21) = 21 + 84 + 168 = 273.
Babylonians
The 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 {{nowrap|60 Ã— 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.Chinese
(File:Multiplication algorithm.GIF|thumb|right|250px|{{nowrap|1=38 Ã— 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. These place value decimal arithmetic algorithms were introduced by Al Khwarizmi to Arab countries in the early 9th century.Modern methods
missing image!
- Gelosia multiplication 45 256.png -
right|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 {{nowrap|1=45 Ã— 256 = 11520}}. This is a variant of Lattice multiplication.
The modern method of multiplication based on the Hinduâ€“Arabic numeral system was first described by Brahmagupta. Brahmagupta gave rules for addition, subtraction, multiplication and division. Henry Burchard Fine, then professor of Mathematics at Princeton University, wrote the following:
- Gelosia multiplication 45 256.png -
right|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 {{nowrap|1=45 Ã— 256 = 11520}}. This is a variant of Lattice 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.BOOK, Fine, Henry B., Henry Burchard Fine, The Number System of Algebra â€“ Treated Theoretically and Historically, 2nd, 1907, 90,weblink
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:
{|class="wikitable" border=1 cellspacing=0 cellpadding=15 style="text-align: center;"
! scope="col" width="40pt" | ! scope="col" width="120pt" | 30! scope="col" width="40pt" | 4! scope="row" | 10|300|40Computer algorithms
The classical method of multiplying two n-digit numbers requires n2 simple multiplications. 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
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 thought of as:
[4 bags] Ã— [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 Ã— 3 hours = 150 kilometers.
Other examples:
2.5 text{ meters} times 4.5 text{ meters} = 11.25 text{ square meters}
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 Î (Pi) in the Greek alphabet. Unicode position U+220F (âˆ) contains a glyph for denoting such a product, distinct from U+03A0 (Î ), the letter. The meaning of this notation is given by:
prod_{i=1}^4 i = 1cdot 2cdot 3cdot 4,
that is
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:
prod_{i=1}^6 i = 1cdot 2cdot 3cdot 4cdot 5 cdot 6 = 720
More generally, the notation is defined as
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 {{nowrap|1=m = n}}, the value of the product is the same as that of the single factor xm. If {{nowrap|m > n}}, the product is the empty product, with the value 1.Infinite products
One may also consider products of infinitely many terms; these are called infinite products. Notationally, we would replace n above by the lemniscate âˆž. 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,
prod_{i=m}^infty x_i = lim_{ntoinfty} prod_{i=m}^n x_i.
One can similarly replace m with negative infinity, and define:
prod_{i=-infty}^infty x_i = left(lim_{mto-infty}prod_{i=m}^0 x_iright) cdot left(lim_{ntoinfty} prod_{i=1}^n x_iright),
provided both limits exist.Properties
Image:Multiplication chart.svg|thumb|right|Multiplication of numbers 0â€“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 determinantdeterminantFor 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:xcdot y = ycdot x.
- Associative property
- Expressions solely involving multiplication or addition are invariant with respect to order of operations:(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: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:xcdot 1 = x
- Property of 0
- Any number multiplied by 0 is 0. This is known as the zero property of multiplication:xcdot 0 = 0