Information about Multiplication
Multiplication is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:
Multiplication can also be viewed as counting objects arranged in a rectangle, or finding the area of rectangle whose sides have given lengths.
Multiplication is one of four main operations in elementary arithmetic, and most people learn basic multiplication algorithms in elementary school. The inverse of multiplication is division.
Notation and terminologyMultiplication is written using the multiplication sign "×" between the terms; that is, in infix notation. The result is expressed with an equals sign. For example,
- (verbally, "two times three equals six")
There are several other common notations for multiplication:
- Multiplication is sometimes denoted by either a middle dot or a period:
The middle dot is standard in the United States, the United Kingdom, and other countries where the period is used as a decimal point. In countries that use a comma as a decimal point, the period is used for multiplication instead.
- The asterisk (e.g. 5 * 2) is often used with computers because it appears on every keyboard. This usage originated in the FORTRAN programming language.
- In algebra, multiplication involving variables is often written as a (e.g. xy for x times y or 5x for five times x). This notation can also be used for numbers that are surrounded by parentheses (e.g. 5(2) or (5)(2) for five times two).
The result of a multiplication is called a product, and 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.
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.
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.
Historical algorithmsMethods of multiplication were documented in the Egyptian, Greece, Babylonian, Indus valley, and Chinese civilizations.
BabyloniansThe 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.
ChineseIn 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.
Indus ValleyThe 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.
Products of sequences
Capital pi notationThe 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:
The subscript gives the symbol for a dummy variable ( in our case) and its lower value (); the superscript gives its upper value. So for example:
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.
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 terms, as grows without bound. That is:
One can similarly replace with negative infinity, and
for some integer , provided both limits exist.
Cartesian productThe definition of multiplication as repeated addition provides a way to arrive at a set-theoretic interpretation of multiplication of cardinal numbers. In the expression
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 are exactly those of the Cartesian product . The properties of the multiplicative operation as applying to natural numbers then follow trivially from the corresponding properties of the Cartesian product.
PropertiesFor 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.
- The associative property means that for any three numbers x, y, and z,
- (x · y)·z = 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.
- Multiplication also has what is called a distributive property with respect to the addition,
- 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) + m − m = (m · 0) + (m · 1) − m = m · (0 + 1) − m = (m · 1) − m = m − m = 0.
- 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
- Every number x, except zero, has a multiplicative inverse, 1/x, such that x·(1/x) = 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.
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. 
- 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.
Multiplication with set theoryIt 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 theoryIt is easy to show that there is a group for multiplication- the non-zero rational numbers. 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.
- Multiplicative inverse, the reciprocal
- Multiplication algorithm
- Karatsuba algorithm, method for large numbers
- Toom-Cook algorithm, method for very large numbers
- Schönhage-Strassen algorithm, method for huge numbers
- Multiplication table (times table)
- Multiplication ALU, how computers multiply
- Booth's multiplication algorithm
- Floating point
- Fused multiply-add
- Wallace tree
- Napier's bones
- Peasant multiplication
- Product (mathematics) - lists generalizations
- Slide rule
- Boyer, Carl B. (revised by Merzbach, Uta C.) (1991). History of Mathematics. John Wiley and Sons, Inc.. ISBN 0-471-54397-7.
In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values. There are two common types of operations: unary and binary.
- Practicing and Learning Multiplication
- Multiplication and Arithmetic Operations In Various Number Systems at cut-the-knot
- Modern Chinese Multiplication Techniques on an Abacus
- Multiplication Worksheets and Puzzles
- Math Games for Multiplication
..... Click the link for more information.rectangle is defined as a quadrilateral where all four of its angles are right angles.
From this definition, it follows that a rectangle has two pairs of parallel sides; that is, a rectangle is a parallelogram.
..... Click the link for more information.Area is a physical quantity expressing the size of a part of a surface. The term Surface area is the summation of the areas of the exposed sides of an object.
UnitsUnits for measuring surface area include:
- square metre = SI derived unit
..... Click the link for more information.Length is the long dimension of any object. The length of a thing is the distance between its ends, its linear extent as measured from end to end. This may be distinguished from height, which is vertical extent, and width or breadth
..... Click the link for more information.Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. Most people learn elementary arithmetic in elementary school.
..... Click the link for more information.In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will proceed through a well-defined series of successive states, eventually terminating in an
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- ''Main article Primary education
An elementary school is an institution where children receive the first stage of compulsory education known as elementary or primary education.
..... Click the link for more information.In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.
Specifically, if c times b equals a, written:
..... Click the link for more information.The multiplication sign is the symbol × (multiplication sign is the preferred Unicode name for the codepoint represented by that glyph). The symbol is similar to the letter x but is a more symmetric cross, and has different uses.
..... Click the link for more information.Infix notation is the common arithmetic and logical formula notation, in which operators are written infix-style between the operands they act on (e.g. 2 + 2). It is not as simple to parse by computers as prefix notation ( e.g. + 2 2 ) or postfix notation ( e.g.
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- The equal sign, equals sign, or "=" is a mathematical symbol used to indicate equality. It was invented in 1557 by Welshman Robert Recorde.
..... Click the link for more information.An interpunct · is a small dot used for interword separation in ancient Latin script, being perhaps the first consistent visual representation of word boundaries in written language. The dot is vertically centered, e.g.
..... Click the link for more information.Period and periodic may refer to:
- An interval of time that an event, chain of events, instance or happening, takes place within. It is measured between a start point and an end point and generally repeats (which is where the term period came to describe a
..... Click the link for more information.Motto
"In God We Trust" (since 1956)
"E Pluribus Unum" ("From Many, One"; Latin, traditional)
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"Dieu et mon droit"  (French)
"God and my right"
"God Save the Queen" 
..... Click the link for more information.This article requires authentication or verification by an expert.
Please assist in recruiting an expert or [ improve this article] yourself. See the talk page for details. This article has been tagged since June 2007.
..... Click the link for more information.Comma may refer to:
- Comma (punctuation), a punctuation mark (,)
- Comma (music), a kind of interval in music theory
- Comma (butterfly), a species of butterfly
- Comma (rhetoric), a short clause in Greek rhetoric
..... Click the link for more information.asterisk (*), is a typographical symbol or glyph. It is so called because it resembles a conventional image of a star (Latin astrum). Computer scientists and mathematicians often pronounce it as star (as, for example, in the A* search algorithm
..... Click the link for more information.Fortran
Paradigm: multi-paradigm: procedural, imperative, structured, object-oriented
Appeared in: 1957
Designed by: John W. Backus
Developer: John W.
..... Click the link for more information.Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Arabic mathematician, astronomer, astrologer and geographer,
..... Click the link for more information.variable (IPA pronunciation: [ˈvæɹiəbl]) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression.
..... Click the link for more information.Parenthesis may be:
- Parenthesis, either of the curved-bracket ( ) punctuation marks that together make a set of parentheses
- Parenthesis (rhetoric), parenthetical expression
..... Click the link for more information.coefficient is a constant multiplicative factor of a certain object. For example, the coefficient in 9x2 is 9.
The object can be such things as a variable, a vector, a function, etc.
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- Product (biology), something manufactured by an organelle
- Product (business), an item that ideally satisfies a market's want or need
- Product (chemistry), a substance found at the end of a chemical reaction
..... Click the link for more information.The word multiple can refer to:
- multiples of numbers
- dissociative identity disorder, for people with multiple personalities, sometimes called "multiples".
- multiple birth, because having twins is sometimes called having "multiples".
..... Click the link for more information.multiplication table is a mathematical table used to define a multiplication operation for an algebraic system.
The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic
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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.
..... Click the link for more information.In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10(x), or sometimes Log(x) with a capital L
..... Click the link for more information.slide rule (often nicknamed a "slipstick") is a mechanical analog computer, consisting of at least two finely divided scales (rules), most often a fixed outer pair and a movable inner one, with a sliding window called the cursor.
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