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Encyclopedia > Multiplication

In mathematics, multiplication is an elementary arithmetic operation. When one of the numbers is a whole number, multiplication is the repeated sum of the other number. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... The whole numbers are the nonnegative integers (0, 1, 2, 3, ...) The set of all whole numbers is represented by the symbol = {0, 1, 2, 3, ...} Algebraically, the elements of form a commutative monoid under addition (with identity element zero), and under multiplication (with identity element one). ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ...

displaystyle a times n = underbrace{a + cdots + a}_{n}

For example, 4 × 7 (verbally, "four times seven") is the same as 7 + 7 + 7 + 7.


Fractions are multiplied by separately multiplying their denominators and numerators: a/b × c/d = (ac)/(bd). For example, 2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2.


Multiplication can be defined for real and complex numbers, polynomials, matrices and other mathematical quantities as well; see product (mathematics). The inverse of multiplication is division. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... This article gives an overview of the various ways to perform matrix multiplication. ... In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...

Contents

Computation

The standard 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. Many mathematics curricula developed according to the 1989 standards of the NCTM do not teach standard arithmetic methods, instead guiding students to invent their own methods of computation. Though widely adopted by many school districts in nations such as the United States, they have encountered resistance from some parents and mathematicians, and some districts have since abandoned such curricula in favor of traditional mathematics. A multiplication algorithm is an algorithm (or method) to multiply two numbers. ... In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. ... It has been suggested that this article or section be merged into Egyptian mathematics. ... The National Council of Teachers of Mathematics (NCTM) was founded in 1920. ... Traditional mathematics is the term used for the style of mathematics instruction used for a period in the 20th century before the appearance of reform mathematics based on NCTM standards, so it is best defined by contrast with the alternatives. ...


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 twentieth 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. In mathematics, the common logarithm is the logarithm with base 10. ... A typical 10 inch student slide rule (Pickett N902-T simplex trig). ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s The 20th century lasted from 1901 to 2000 in the Gregorian calendar (often from (1900 to 1999 in common usage). ... For other uses, see Calculator (disambiguation). ... The Marchant Calculating Machine Co. ... This article is about the machine. ...


Historical algorithms

Methods of multiplication were documented in the Egyptian, Greece, Babylonian, Indus valley, and Chinese civilizations. Babylonia was a state in southern Mesopotamia, in modern Iraq, combining the territories of Sumer and Akkad. ... Excavated ruins of Mohenjo-daro. ...


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 2 × 21 = 42, 4 × 21 = 84, 8 × 21 = 168. The full product could then be found by adding the correct terms found in the doubling: (note 13 = 1 + 4 + 8) It has been suggested that this article or section be merged into Egyptian mathematics. ... The Moscow and Rhind Mathematical Papyri are two of our oldest mathematical texts and perhaps our best indication of what ancient Egyptian mathematics might have been like near 2000 BC. They are both written on papyrus. ...

13times 21 = (1times 21) + (4times 21) + (8 times 21) = 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 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. The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... ... Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus... In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. ...


Chinese

In the books, Chou Pei Suan Ching dated prior to 300 B.C., and the Nine Chapters on the Mathematical Art, multiplication calculations were written out in words, although the early Chinese mathematicians employed an abacus in hand calculations involving addition and multiplication. Wikipedia does not have an article with this exact name. ... Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 350s BC 340s BC 330s BC 320s BC 310s BC - 300s BC - 290s BC 280s BC 270s BC 260s BC 250s BC Years: 305 BC 304 BC 303 BC 302 BC 301 BC - 300 BC - 299 BC 298 BC... The Nine Chapters on the Mathematical Art (九章算術) is a Chinese mathematics book, probably composed in the 1st century AD, but perhaps as early as 200 BC. This book is the earliest surviving mathematical text from China that has come down to us by being copied by scribes and (centuries later... It has been suggested that Abax be merged into this article or section. ...


Indus Valley

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.
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.

The early Hindu mathematicians of the Indus valley region used a variety of intuitive tricks to perform multiplication. Most calculations were performed on small slate hand tablets, using chalk tables. One technique was that of lattice multiplication (or gelosia multiplication). Here a table was drawn up with the rows and columns labelled by the multiplicands. Each box of the table was divided diagonally into two, as a triangular lattice. The entries of the table held the partial products, written as decimal numbers. The product could then be formed by summing down the diagonals of the lattice. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... A multiplication algorithm is an algorithm (or method) to multiply two numbers. ... In mathematics, a lattice can be either of two things: In one usage, a lattice is a partially ordered set (poset) in which any two elements have a supremum and an infimum—see lattice (order). ...


Terminology

The two numbers being multiplied are formally called the multiplicand and the multiplier, respectively. (Because of the commutative property of multiplication, there is generally no need to distinguish between the two numbers, so they are more commonly referred to as the factors.) The result of the multiplication is referred to as the product. Mathematical meaning A map or binary operation is said to be commutative when, for any x in A and any y in B . ...


Some write the multiplier first, and say that 7 × 4 stands for 4 + 4 + 4 + 4 + 4 + 4 + 4, but this usage is less common. The difference was important in Roman numerals and similar systems where multiplication is transformation of symbols and their addition. For example, to multiply VII by XV one changes the VII to LXX (multiplying VII by X) plus XXV (V times V) plus X (II times V), but to multiply XV by VII one changes XV into LXXV (XV times V) plus XV plus XV (each XV times I). Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. ...


Notation

Multiplication can be denoted in several equivalent ways. All of the following mean, "5 multiplied by 2":

5×2 (see ×)
5·2
(5)2, 5(2), (5)(2), 5[2], [5]2, [5][2]
5*2
5.2

The asterisk (*) is often used on computers because it is a symbol on every keyboard, but it is rarely used when writing math by hand. This usage originated in the FORTRAN programming language. In its simplest form, multiplication is a quick way of adding identical numbers. ... An asterisk (*), is a typographical symbol or glyph. ... Fortran (previously FORTRAN[1]) is a general-purpose[2], procedural,[3] imperative programming language that is especially suited to numeric computation and scientific computing. ...


Frequently, multiplication is implied by juxtaposition rather than shown in a notation. This is standard in algebra, taking forms like This article is about the branch of mathematics. ...

5x and xy

This notation is potentially confusing if variables are permitted to have names longer than one letter, as in computer programming languages. The notation is not used with numbers alone: 52 never means 5 × 2.


If the terms are not written out individually, then the product may be written with an ellipsis to mark out the missing terms, as with other series operations (like sums). Thus, the product of all the natural numbers from 1 to 100 can be written 1 cdot 2 cdot ldots cdot 99 cdot 100. This can also be written with the ellipsis vertically placed in the middle of the line, as 1 cdot 2 cdot cdots cdot 99 cdot 100. This article is about the punctuation symbol. ...


Capital pi notation

The product of a series 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 (∏) is defined a n-ary product for this purpose, distinct from U+03A0 (Π), the letter. This is defined as: For other uses, see Pi (disambiguation) Pi (upper case Π, lower case Ï€ or Ï–) is the sixteenth letter of the Greek alphabet. ... The Greek alphabet is an alphabet that has been used to write the Greek language since about the 9th century BCE. It was the first alphabet in the narrow sense, that is, a writing system using a separate symbol for each vowel and consonant alike. ... In telecommunication, a n -ary code is a code that has n significant conditions, where n is a positive integer greater than 1. ...

 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}.

The subscript gives the symbol for a dummy variable (i in our case) and its lower value (m); the superscript gives its upper value. So for example: In computer programming, a free variable is a variable referred to in a function that is not a local variable or an argument of that function. ...

 prod_{i=2}^{6} left(1 + {1over i}right) = left(1 + {1over 2}right) cdot left(1 + {1over 3}right) cdot left(1 + {1over 4}right) cdot left(1 + {1over 5}right) cdot left(1 + {1over 6}right) = {7over 2}.

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. In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. ...


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 infinity symbol (∞). In the reals, 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: In mathematics, for a sequence of numbers a1, a2, a3, ... the infinite product is defined to be the limit of the partial products a1a2. ... In mathematics, for a sequence of numbers a1, a2, a3, ... the infinite product is defined to be the limit of the partial products a1a2. ... The infinity symbol ∞ in several typefaces. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...

 prod_{i=m}^{infty} x_{i} := lim_{ntoinfty} prod_{i=m}^{n} x_{i}.

One can similarly replace m with negative infinity, and

prod_{i=-infty}^infty x_i := left(lim_{ntoinfty}prod_{i=-n}^m x_iright) cdot left(lim_{ntoinfty}prod_{i=m+1}^n x_iright),

for some integer m, provided both limits exist.


Interpretation

Cartesian product

The definition of multiplication as repeated addition provides a way to arrive at a set-theoretic interpretation of multiplication of cardinal numbers. In the expression

displaystyle a cdot n = underbrace{a + cdots + a}_{n},

if the n copies of a are to be combined in disjoint union then clearly they must be made disjoint; an obvious way to do this is to use either a or n as the indexing set for the other. Then, the members of a cdot n, are exactly those of the Cartesian product a times n,. The properties of the multiplicative operation as applying to natural numbers then follow trivially from the corresponding properties of the Cartesian product. In mathematics, the Cartesian product is a direct product of sets. ...


Properties

For integers, fractions, real and complex numbers, multiplication has certain properties:

  • the order in which two numbers are multiplied does not matter. This is called the commutative property,
x · y = y · x.
(x · yz = x·(y · z).
Note from algebra: the parentheses mean that the operations inside the parentheses must be done before anything outside the parentheses is done.
x·(y + z) = x·y + x·z.
  • Also of interest is that any number times 1 is equal to itself, thus,
1 · x = x.
and this is called the identity property. In this regard the number 1 is known as the multiplicative identity.
  • The sum of zero numbers is zero.
This fact is directly received by means of the distributive property:
m · 0 = (m · 0) + mm = (m · 0) + (m · 1) − m = m · (0 + 1) − m = (m · 1) − m = mm = 0.
So,
m · 0 = 0
no matter what m is (as long as it is finite).
  • Multiplication with negative numbers also requires a little thought. First consider negative one (−1). For any positive integer m:
(−1)m = (−1) + (−1) +...+ (−1) = −m
This is an interesting fact that shows that any negative number is just negative one multiplied by a positive number. So multiplication with any integers can be represented by multiplication of whole numbers and (−1)'s.
All that remains is to explicitly define (−1)·(−1):
(−1)·(−1) = −(−1) = 1
However, from a formal viewpoint, multiplication between two negative numbers is (again) directly received by means of the distributive property, e.g:
(−1)·(−1)
=  (−1)·(−1) + (−2) + 2
=  (−1)·(−1) + (−1)·2 + 2
=  (−1)·(−1 + 2) + 2
=  (−1)·1 + 2
=  (−1) + 2
=  1
  • Multiplication by a positive number preserves order: if a > 0, then if b > c then a·b > a·c. Multiplication by a negative number reverses order: if a < 0, then if b > c then a·b < a·c.

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. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... For other senses of this word, see zero or 0. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... The reciprocal function: y = 1/x. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...


Multiplication with Peano's axioms

In the book Arithmetices principia, nova methodo exposita, Giuseppe Peano proposed a new system for multiplication based on his axioms for natural numbers. [1]
  • a×1=a
  • a×b'=(a×b)+a
Here, b' represents the successor of b, or the natural number which follows b. With his other nine axioms, it is possible to prove common rules of multiplication, such as the distributive or associative properties.

Giuseppe Peano Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory. ... A successor function is the label in the literature for what is actually an operation. ... In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of the natural numbers. ...

Multiplication with set theory

It is possible, though difficult, to create a recursive definition of multiplication with set theory. Such a system usually relies on the peano definition of multiplication.


Multiplication with group theory

It is easy to show that there is a group for multiplication- the non-zero rational numbers.[2] Multiplication with the non-zero numbers satisfies

  • Closure - For all a and b in the group, a×b is in the group.
  • Associativity - This is just the associative property! (a×b)×c=a×(b×c)
  • Identity - This follows straight from the peano definition. Anything multiplied by one is itself.
  • Inverse - All non-zero numbers have a multiplicative inverse.

Multiplication also is an abelian group, since it follows the commutative property. The reciprocal function: y = 1/x. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) such that a * b = b * a for all a and b in G. In other words, the order in which the binary operation is performed doesnt matter. ...


a×b=b×a


See also

The reciprocal function: y = 1/x. ... A multiplication algorithm is an algorithm (or method) to multiply two numbers. ... The Karatsuba multiplication algorithm, a technique for quickly multiplying large numbers, was discovered by A. Karatsuba and published together with Yu. ... Toom-Cook, sometimes known as Toom-3, is a multiplication algorithm, a method of multiplying two large integers. ... In mathematics, the Schönhage-Strassen algorithm is an asympotically fast method for multiplication of large integer numbers. ... In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. ... In digital design, a multiplier or multiplication ALU is a hardware circuit dedicated to multiplying two binary values. ... Booths multiplication algorithm is a multiplication algorithm that multiplies two signed binary numbers in twos complement notation. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... In computing, a fused multiply-add computes a multiply-accumulate FMA(A, B, C) = AB + C with a single rounding of floating point numbers. ... The multiply-accumulate operation computes a product and adds it to an accumulator. ... Wallace tree is an efficient hardware implementation of multiplication of two integers. ... Napiers bones are an abacus invented by John Napier for calculation of products and quotients of numbers. ... It has been suggested that this article or section be merged with Ancient Egyptian multiplication. ... In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. ... A typical 10 inch student slide rule (Pickett N902-T simplex trig). ...

Notes

  1. ^ [1]
  2. ^ [2]

References

  • Boyer, Carl B. (revised by Merzbach, Uta C.) (1991). History of Mathematics. John Wiley and Sons, Inc.. ISBN 0-471-54397-7. 

Carl Benjamin Boyer (April 28, 1906 - April 21, 1976) was a historian of mathematics. ...

External links

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