Multiplication: Facts, Discussion Forum, and Encyclopedia Article

All Topics

Multiplication

	Email		Print
	Bookmark		Link

	Multiplication
	Home Discussion Definition
	Topics Multiplication Topic Home Multiplication of whole numbersNatural number In mathematics, a natural number is either a positive integer or a non-negative integer .... is the mathematical operationOperation (mathematics) In its simplest meaning in mathematics and logic, an operation combines two values to produce a third.... 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 rectangleRectangle In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles.... , or finding the areaArea Area is a physical quantity expressing the size of a part of a surface.... of rectangle whose sides have given lengthLength Length is the long dimension of any object.... s. Multiplication is one of four main operations in elementary arithmeticFacts About Elementary arithmetic Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplica... , and most people learn basic multiplication algorithmAlgorithm In mathematics and computing, an algorithm is a procedure for accomplishing some task which, given an initial state, will t... s in elementary schoolElementary school Summary An elementary school is an institution where children receive the first stage of compulsory education known as elementary or... . The inverse of multiplication is divisionDivision (mathematics) Overview In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplicat... . Multiplication is generalizedMultiplication In mathematics, multiplication is an elementary arithmetic operation.... to many kinds of numbers and to more abstract constructs such as matrices. Computation The standard methods for multiplying numbers using pencil and paper require a multiplication tableMultiplication table In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic sy... of memorized or consulted products of small numbers (typically any two numbers from 0 to 9), however one method, the peasant multiplicationAncient Egyptian multiplication Ancient Egyptian multiplication is an algorithm for multiplication that does not require the multiplication table, only the ... 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 mathematicsTraditional mathematics Traditional mathematics is the term used for the style of mathematics instruction used before the adoption of standards-base... . Multiplying numbers to more than a couple of decimal places by hand is tedious and error prone. Common logarithmCommon logarithm Summary In mathematics, the common logarithm is the logarithm with base 10.... s were invented to simplify such calculations. The slide ruleSlide rule The slide rule is a mechanical analog computer, consisting of at least two finely divided scales , most often a fixed outer... allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early twentieth century, mechanical calculatorCalculator A calculator is a device for performing calculations.... s, such as the Marchant, automated multiplication of up to 10 digit numbers. Modern electronic computerComputer A computer is a machine for manipulating data according to a list of instructions known as a program.... s and calculators have greatly reduced the need for multiplication by hand. Historical algorithms Methods of multiplication were documented in the EgyptianFacts About Ancient Egypt Ancient Egypt was a long-lived ancient civilization in north-eastern Africa.... , GreeceAncient Greece Ancient Greece is the period in Greek history which lasted for around one thousand years and ended with the rise of Christia... , BabyloniaBabylonia Babylonia, named for its capital city, Babylon, was an ancient state in the south part of Mesopotamia , combining the territ... n, Indus valleyIndus Valley Civilization The Indus Valley Civilisation was an ancient civilisation thriving along the Indus River and the Ghaggar-Hakra River in Pak... , and Chinese civilizations. 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 , , and . The full product could then be found by adding the appropriate terms found in the doubling sequence: . Babylonians The Babylonians used a sexagesimalSexagesimal Overview The sexagesimal is a numeral system with sixty as the base.... 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 tableMultiplication table In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic sy... s. 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 In the books, Chou Pei Suan ChingChou Pei Suan Ching The Chou Pei Suan Ching is one of the oldest and most famous Chinese mathematical texts.... 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 abacusAbacus An abacus is a calculation tool, often constructed as a wooden frame with beads sliding on wires.... in hand calculations involving addition and multiplication. Indus Valley 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 latticeLattice (mathematics) In mathematics, a lattice can be either of two things:... . 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. Modern method The modern method of multiplication based on the Hindu-Arabic numeral systemHindu-Arabic numeral system The Hindu-Arabic numeral system is a positional decimal numeral system documented from the 9th century.... was first described by BrahmaguptaBrahmagupta Brahmagupta was an Indian mathematician and astronomer.... . Brahmagupta gave rules for addition, subtraction, multiplication and division. Henry Burchard FineHenry Burchard Fine Henry Burchard Fine was an American university dean and mathematician.... , then professor of Mathematics at Princeton UniversityPrinceton University Princeton University is a coeducational private university located in Princeton, New Jersey.... , 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 and subtraction they performed quite as they are performed nowadays; multiplication they effected in many ways, ours among them, but division they did cumbrously. 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 ? (Pi)Pi (letter) Pi is the sixteenth letter of the Greek alphabet.... in the Greek alphabetGreek alphabet Summary The Greek alphabet is an alphabet that has been used to write the Greek language since about the 9th century BC.... . Unicode position U+220F (?) contains a glyph for denoting such a product, distinct from U+03A0 (?), the letter. The meaning of this notation is given by: The subscript gives the symbol for a dummy variableFree variables and bound variables Overview In computer programming, a free variable is a variable referred to in a function, which is not an argument of that function.... (i in this case), called the "index of multiplication" together with its lower bound (m), whereas the superscript (here n) 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: In case m = n, the value of the product is the same as that of the single factor x_m. If m > n, the product is the empty productEmpty product In mathematics, an empty product, or nullary product, is the result of multiplying no numbers.... , with the value 1. Infinite products One may also consider products of infinitely many terms; these are called infinite productInfinite product In mathematics, for a sequence of numbers a1, a2, a3, ...... s. Notationally, we would replace n above by the lemniscate (infinity symbol) 8. In the reals, the product of such a series is defined as the limitLimit (mathematics) In mathematics, the concept of a "limit" is used to describe the behavior of a function as its argument either gets "close" ... of the product of the first terms, as grows without bound. That is, by definition, One can similarly replace with negative infinity, and define: provided both limits exist. Interpretation Cartesian product The definition of multiplication as repeated additionAddition Addition is the mathematical operation of increasing one amount by another.... 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 productCartesian product In mathematics, the Cartesian product of two sets X and Y, denoted X Y, is the set of all possible ordered... . The properties of the multiplicative operation as applying to natural numbers then follow trivially from the corresponding properties of the Cartesian product. Properties For integers, fractions, real and complex numbers, multiplication has certain properties: Commutative property The order in which two numbers are multiplied does not matter: . Associative property Problems solely involving multiplication are invariant with respect to order of operationsOrder of operations In arithmetic and algebra, certain rules are used for the order in which the operations in expressions are to be evaluated.... : Distributive property Holds with respect to addition over multiplication. This identity is of prime importance in simplifying algebraic expressions: Identity elementIdentity element In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set.... of multiplication is 1; anything multiplied by one is itself. This is known as the identity property: Zero element Anything multiplied by zero is zero. This is known as the zero property of multiplication: Inverse property Every number , except zero, has a multiplicative inverseMultiplicative inverse In mathematics, the reciprocal, or multiplicative inverse, of a number x'' is the number which, when multiplied by '... , , such that . Order preservation Multiplication by a positive number preserves orderOrder theory Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ... : if , then if then . Multiplication by a negative number reverses order: if and then . Negative one times any number is equal to the negative of that number. Negative one times negative one is positive one. 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 quaternionQuaternion In mathematics, quaternions are a non-commutative extension of complex numbers.... s. Proofs Not all of these properties are independent; some are a consequence of the others. A property that can be proven from the others is the zero property of multiplication. It is proven by means of the distributive property. We assume all the usual properties of addition and subtraction, and −x means the same as 0 − x. So we have proven: The identity (−1) · x = −x can also be proven using the distributive property: The proof that (−1) · (−1) = 1 is now easy: Multiplication with Peano's axioms In the book Arithmetices principia, nova methodo exposita, Giuseppe PeanoGiuseppe Peano Giuseppe Peano was the leading Italian mathematician of his day, whose work is of exceptional philosophical value.... proposed a new system for multiplication based on his axioms for natural numbers. Here, represents the successorSuccessor ordinal When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation ... of b, or the natural number which follows b. With his other nine axiomsPeano axioms In mathematics, the Peano axioms are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of ... , it is possible to prove common rules of multiplication, such as the distributive or associative properties. 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 in group theory It 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 inverseMultiplicative inverse Summary In mathematics, the reciprocal, or multiplicative inverse, of a number x'' is the number which, when multiplied by '... . Multiplication also is an abelian groupFacts About Abelian group In mathematics, an abelian group, also called a commutative group, is a group such that a'' * b'' = b'' * a''... , since it follows the commutative property. a×b=b×a 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 quantuum mechanics, multiplication has been generalized to more complicated and abstract types of numbers, and to things that aren't numbers (such as matricesMatrix (mathematics) In mathematics, a matrix is a rectangular table of numbers or, more generally, a table consisting of abstract quantities tha... ) or don't look much like numbers (such as quaternionQuaternion In mathematics, quaternions are a non-commutative extension of complex numbers.... s). Integers is the sum of M copies of N 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 . Rationals Generalization to fractions is by multiplying the numerators and denominators respectively: . This gives the area of a rectangle high and wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers. Reals is the limit of the products of the corresponding terms in certain sequences of rationals that converge to and , respectively, and is significant in CalculusCalculus Calculus is a central branch of mathematics, developed from algebra and geometry.... . This gives the area of a rectangle high and wide. See aboveMultiplication Summary In mathematics, multiplication is an elementary arithmetic operation.... . ComplexComplex number In mathematics, a complex number is a number of the form ... Considering complex numbers and as ordered pairs or real numbers and , the product is . This is the same as for reals, , when the imaginary parts and are zero. Further generalizations See aboveMultiplication In mathematics, multiplication is an elementary arithmetic operation.... and Multiplicative GroupMultiplicative group In mathematics, multiplicative group in group theory may mean... , which for example includes matrix multiplication. A very general, and abstract, concept of multiplication is as the "multiplicatively denoted" (second) binary operation in a ringRing (mathematics) Summary In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed... . An example of a ring which is not any of the above number systems is polynomial ringPolynomial ring In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring.... s (you can add and multiply polynomials, but polynomials are not numbers in any usual sense.) Division Often division is the same as multiplication by an inverse, . Multiplication for some types of "numbers" may have corresponding division, without inverses; in an Integral domainIntegral domain In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 su... may have no inverse "" but may be defined. In a Division ringDivision ring In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero e... there are inverses but they are not commutative (since is not the same as , may be ambiguous). See also External links and at cut-the-knotCut-the-knot cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variet...