Multiplication

For other uses, see Multiplication (disambiguation).

3 × 4 = 12, so twelve dots can be arranged in three rows of four (or four columns of three).

Multiplication of whole numbers is the mathematical operation of adding together multiple copies of the same number. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an 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 For example, four multiplied by three is twelve, since three sets of four make twelve:

$4 + 4 + 4 = 12.\!\,$

Multiplication can also be viewed as counting objects arranged in a rectangle, or finding the area of rectangle whose sides have given lengths. In Geometry, a rectangle is defined as a Quadrilateral where all four of its angles are Right angles A rectangle with vertices ABCD would be denoted as Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. 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

Multiplication is one of four main operations in elementary arithmetic, and most people learn basic multiplication algorithms in elementary school. Elementary arithmetic is the most basic kind of Mathematics: it concerns the operations of Addition, Subtraction, Multiplication, and division In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation See also Primary education An elementary school is an institution where children receive the first stage of Compulsory education known as elementary The inverse of multiplication is division. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication.

Multiplication is generalized to many kinds of numbers and to more abstract constructs such as matrices.

Notation and terminology

Multiplication is written using the multiplication sign "×" between the terms; that is, in infix notation. The multiplication sign is the symbol × ( multiplication sign is the preferred Unicode name for the Codepoint represented by that Glyph Infix notation is the common arithmetic and logical formula notation in which Operators are written Infix -style between the Operands they act on (e The result is expressed with an equals sign. History The "=" symbol that is now universally accepted by mathematics for equality was first recorded by Welsh mathematician Robert Recorde in The For example,

$2\times 3 = 6$ (verbally, "two times three equals six")

$3\times 4 = 12$

$2\times 3\times 5 = 30$

$2\times 2\times 2\times 2\times 2 = 32$

There are several other common notations for multiplication:

Multiplication is sometimes denoted by either a middle dot or a period:

$5 \cdot 2 \quad\text{or}\quad 5\,.\,2$

The middle dot is standard in the United States, the United Kingdom, and other countries where the period is used as a decimal point. An interpunct ( ·) is a small dot used for Interword separation in ancient Latin script, being perhaps the first consistent visual representation of word boundaries A full stop or period (sometimes stop, full point, decimal point, or dot) is the Punctuation mark commonly placed at the The United States of America —commonly referred to as the The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom, the UK or Britain,is a Sovereign state located In a positional Numeral system, the decimal separator is a Symbol used to mark the boundary between the integral and the fractional In some countries that use a comma as a decimal point, the period is used for multiplication instead. A comma ( , is a Punctuation mark It has the same shape as an Apostrophe or single closing Quotation mark in many typefaces but it differs
The asterisk (as in 5*2) is often used in programming languages because it appears on every keyboard and is easier to see on older monitors. An asterisk ( *) (Latin asteriscum "little star" from Greek ἀστερίσκος) is a Typographical symbol or Glyph A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. This usage originated in the FORTRAN programming language. Fortran (previously FORTRAN) is a general-purpose, procedural, imperative Programming language that is especially suited to
In algebra, multiplication involving variables is often written as a juxtaposition (e. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. g. $x y$ for $x$ times $y$ or $5 x$ 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).

In matrix multiplication, there is actually a distinction between the cross and the dot symbols. In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix The cross symbol generally denotes a vector multiplication, while the dot denotes a scalar multiplication. A like convention distinguishes between the cross product and the dot product of two vectors. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added

The numbers to be multiplied are generally called the "factors" or "multiplicands". When thinking of multiplication as repeated addition, the number to be multiplied is called the "multiplicand", while the number of multiples is called the "multiplier". In algebra, a number that is the multiplier of a variable or expression (e. g. the $3$ in $3 x y 2$ ) is called a coefficient. In Mathematics, a coefficient is a Constant multiplicative factor of a certain object

The result of a multiplication is called a product, and is a multiple of each factor that is an integer. In Mathematics, a product is the Result of multiplying, or an expression that identifies factors to be multiplied In Mathematics, a multiple of an Integer is the product of that integer with another integer For example 15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5.

Computation

For methods of computing products, including those of very large numbers, see multiplication algorithm. A multiplication algorithm is an Algorithm (or method to multiply two numbers

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. Ancient Egyptian multiplication is a systematic method for multiplying two numbers that does not require the Multiplication table, only the ability to multiply and divide 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. The National Council of Teachers of Mathematics (NCTM was founded in 1920. 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. Traditional mathematics (sometimes classical math education) is a term used to describe the predominant methods of Mathematics education in the United States

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 common logarithm is the Logarithm with base 10 It is also known as the decadic logarithm, named after its base The slide rule allowed numbers to be quickly multiplied to about three places of accuracy. The slide rule, also known as a slipstick, is a mechanical Analog computer. Beginning in the early twentieth century, mechanical calculators, such as the Marchant, automated multiplication of up to 10 digit numbers. The twentieth century of the Common Era began on A calculator is device for performing mathematical calculations distinguished from a Computer by having a limited problem solving ability and an interface optimized for interactive The Marchant Calculating Machine Co was founded in 1911 by Rodney and Alfred Marchant in Oakland California. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand. A computer is a Machine that manipulates data according to a list of instructions.

Historical algorithms

Methods of multiplication were documented in the Egyptian, Greece, Babylonian, Indus valley, and Chinese civilizations. Ancient Egypt was an Ancient Civilization in eastern North Africa, concentrated along the lower reaches of the Nile River in what is now The term ancient Greece refers to the period of Greek history lasting from the Greek Dark Ages ca Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin Chinese civilization originated in various city-states along the Yellow River ( valley in the Neolithic era

Egyptians

Main article: Ancient Egyptian multiplication

The Egyptian method of multiplication of integers and fractions, documented in the Ahmes Papyrus, was by successive additions and doubling. Ancient Egyptian multiplication is a systematic method for multiplying two numbers that does not require the Multiplication table, only the ability to multiply and divide The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining $1\times 21 = 21$ , $2\times 21 = 42$ , $4\times 21 = 84$ and $8\times 21 = 168$ . The full product could then be found by adding the appropriate terms found in the doubling sequence:

$13\times 21 = (1 + 4 + 8)\times 21 = (1\times 21) + (4\times 21) + (8\times 21) = 21 + 84 + 168 = 273$ .

Babylonians

The Babylonians used a sexagesimal positional number system, analogous to the modern day decimal system. Sexagesimal ( base-sixty) is a Numeral system with sixty as the base. A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a This article gives a mathematical definition For a more accessible article see Decimal. 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.

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. The Zhou Bi Suan Jing (周髀算经 The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven is one of the oldest and most famous Chinese Events By place Egypt Pyrrhus, the King of Epirus, is taken as a hostage to Egypt after the Battle of Ipsus The Nine Chapters on the Mathematical Art ( is a Chinese Mathematics book composed by several generations of scholars from the 10th&ndash2nd century BC and An abacus, also called a counting frame, is a calculating tool used primarily by Asians for performing arithmetic processes

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.

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). A multiplication algorithm is an Algorithm (or method to multiply two numbers 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. In Mathematics, the term lattice can mean A Partially ordered set (poset in which any two elements have a Supremum and an Infimum 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 system was first described by Brahmagupta. The Hindu-Arabic numeral system is a Positional Decimal Numeral system first documented in the ninth century Brahmagupta ( (598–668 was an Indian mathematician and astronomer. Brahmagupta gave rules for addition, subtraction, multiplication and division. Henry Burchard Fine, then professor of Mathematics at Princeton University, 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. Henry Burchard Fine (1858-1928 was an American university dean and mathematician. Princeton University is a private Coeducational research university located in Princeton, New Jersey. 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. ^[1]

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 Π (Pi) in the Greek alphabet. Pi (uppercase &Pi, lower case &pi) is the sixteenth letter of the Greek alphabet. The Greek alphabet (Ελληνικό αλφάβητο is a set of twenty-four letters that has been used to write the Greek language since the late 9th or early 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:

$\prod_{i=m}^{n} x_{i} = x_{m} \cdot x_{m+1} \cdot x_{m+2} \cdot \,\,\cdots\,\, \cdot x_{n-1} \cdot x_{n}.$

The subscript gives the symbol for a dummy variable (i in this case), called the "index of multiplication" together with its lower bound (m), whereas the superscript (here n) gives its upper bound. In Mathematics, and in other disciplines involving Formal languages including Mathematical logic and Computer science, a free variable is a 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:

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

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 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. In Mathematics, for a Sequence of numbers a 1 a 2 a 3. the infinite product In Mathematics, for a Sequence of numbers a 1 a 2 a 3. the infinite product Notationally, we would replace n above by the lemniscate (infinity symbol) ∞. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness 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. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" That is, by definition,

$\prod_{i=m}^{\infty} x_{i} = \lim_{n\to\infty} \prod_{i=m}^{n} x_{i}.$

One can similarly replace $m$ with negative infinity, and define:

$\prod_{i=-\infty}^\infty x_i = \left(\lim_{m\to-\infty}\prod_{i=m}^0 x_i\right) \cdot \left(\lim_{n\to\infty}\prod_{i=1}^n x_i\right),$

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. Addition is the mathematical process of putting things together This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. 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 $a \times n\,$ . Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. 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:; $x\cdot y = y\cdot x$ . In Mathematics, commutativity is the ability to change the order of something without changing the end result

Associative property: Problems solely involving multiplication are invariant with respect to order of operations:; $(x\cdot y)\cdot z = x\cdot(y\cdot z)$

Distributive property: Holds with respect to addition over multiplication. In Mathematics, associativity is a property that a Binary operation can have In Algebra and Computer programming, when a number or expression is both preceded and followed by a Binary operation, a rule is required for which operation In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law This identity is of prime importance in simplifying algebraic expressions:; $x\cdot(y + z) = x\cdot y + x\cdot z$

Identity element: of multiplication is 1; anything multiplied by one is itself. 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 This is known as the identity property:; $x\cdot 1 = x$

Zero element: Anything multiplied by zero is zero. This is known as the zero property of multiplication:; $x\cdot 0 = 0$

Inverse property: Every number $x$ , except zero, has a multiplicative inverse, $\frac{1}{x}$ , such that $x\cdot\left(\frac{1}{x}\right) = 1$ . In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which

Order preservation: Multiplication by a positive number preserves order: if $a > 0$ , then if $b > c$ then $a b > a c$ . Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying Multiplication by a negative number reverses order: if $a < 0$ and $b > c$ then $a b < a c$ .

Negative one times any number is equal to the negative of that number.

$(-1)\cdot x = (-x)$

Negative one times negative one is positive one.

$(-1)\cdot (-1) = 1$

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. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician

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. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law We assume all the usual properties of addition and subtraction, and −x means the same as 0 − x.

$\begin{align} & {} \qquad x\cdot 0 \\ & {} = (x\cdot 0) + x - x \\ & {} = (x\cdot 0) + (x\cdot 1) - x \\ & {} = x\cdot (0 + 1) - x \\ & {} = (x\cdot 1) - x \\ & {} = x - x \\ & {}= 0 \end{align}$

So we have proven:

$x\cdot 0 = 0$

The identity (−1) · x = −x can also be proven using the distributive property:

$\begin{align} & {} \qquad(-1)\cdot x \\ & {} = (-1)\cdot x + x - x \\ & {} = (-1)\cdot x + 1\cdot x - x \\ & {} = (-1 + 1)\cdot x - x \\ & {} = 0\cdot x - x \\ & {} = 0 - x \\ & {} = -x \end{align}$

The proof that (−1) · (−1) = 1 is now easy:

$\begin{align} & {} \qquad (-1)\cdot (-1) \\ & {} = -(-1) \\ & {} = 1 \end{align}$

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. Giuseppe Peano ( August 27, 1858 &ndash April 20, 1932) was an Italian Mathematician, whose work was of exceptional ^[2]

$a\times 1=a$
$a\times b'=(a\times b)+a$

Here,

b'

represents the successor of b, or the natural number which follows b. When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation S to get the next higher one With his other nine axioms, it is possible to prove common rules of multiplication, such as the distributive or associative properties. In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural

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. ^[3] 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. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which

Multiplication also is an abelian group, since it follows the commutative property. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

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 matrices) or don't look much like numbers (such as quaternions). In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician

Integers $N\times M$ 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 $(N\times -M) = - (N\times M)$ .

Rationals Generalization to fractions $\frac{A}{B}\times \frac{C}{D}$ is by multiplying the numerators and denominators respectively: $\frac{A}{B}\times \frac{C}{D} = \frac{(A\times B)}{(C\times D)}$ . In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions This gives the area of a rectangle $\frac{A}{B}$ high and $\frac{C}{D}$ wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.

Reals $(x)(y)$ is the limit of the products of the corresponding terms in certain sequences of rationals that converge to $x$ and $y$ , respectively, and is significant in Calculus. In Mathematics, the real numbers may be described informally in several different ways Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives This gives the area of a rectangle $x$ high and $y$ wide. See above.

Complex Considering complex numbers $z 1$ and $z 2$ as ordered pairs or real numbers $(a 1, b 1)$ and $(a 2, b 2)$ , the product $z_1\times z_2$ is $(a_1\times a_2 - b_1\times b2, a_1\times b_2 + a_2\times b_1)$ . Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This is the same as for reals, $a_1\times a_2$ , when the imaginary parts $b 1$ and $b 2$ are zero.

Further generalizations See above and Multiplicative Group, which for example includes matrix multiplication. In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak A very general, and abstract, concept of multiplication is as the "multiplicatively denoted" (second) binary operation in a ring. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real An example of a ring which is not any of the above number systems is polynomial rings (you can add and multiply polynomials, but polynomials are not numbers in any usual sense. In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables )

Division Often division $\frac{x}{y}$ is the same as multiplication by an inverse, $x\left(\frac{1}{y}\right)$ . Multiplication for some types of "numbers" may have corresponding division, without inverses; in an Integral domain $x$ may have no inverse " $\frac{1}{x}$ " but $\frac{x}{y}$ may be defined. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In a Division ring there are inverses but they are not commutative (since $\left(\frac{1}{x}\right)\left(\frac{1}{y}\right)$ is not the same as $\left(\frac{1}{y}\right)\left(\frac{1}{x}\right)$ , $\frac{x}{y}$ may be ambiguous). In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible

Notes

^ Henry B. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which 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 Anatolii Alexeevich Karatsuba in 1960 and published in The Schönhage-Strassen algorithm is an asymptotically fast Multiplication algorithm for large Integers It was developed by Arnold Schönhage and Volker In digital design, a multiplier or multiplication ALU is a hardware circuit dedicated to multiplying two binary values Booth's multiplication algorithm is a Multiplication algorithm that multiplies two signed binary numbers in two's complement notation. In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. In computing especially Digital signal processing, multiply-accumulate is a common operation that computes the product of two numbers and adds that product to an accumulator In computing especially Digital signal processing, multiply-accumulate is a common operation that computes the product of two numbers and adds that product to an accumulator A Wallace tree is an efficient hardware implementation of a digital circuit that multiplies two integers Napier's bones is an Abacus created by John Napier for Calculation of products and quotients of numbers that was based on Arab mathematics and Ancient Egyptian multiplication is a systematic method for multiplying two numbers that does not require the Multiplication table, only the ability to multiply and divide In Mathematics, a product is the Result of multiplying, or an expression that identifies factors to be multiplied The slide rule, also known as a slipstick, is a mechanical Analog computer. Fine. The Number System of Algebra – Treated Theoretically and Historically, (2nd edition, with corrections, 1907), page 90, http://www.archive.org/download/numbersystemofal00fineuoft/numbersystemofal00fineuoft.pdf
^ PlanetMath: Peano arithmetic
^ The Dog School of Mathematics Presents

References

Boyer, Carl B. (revised by Merzbach, Uta C. Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he ) (1991). History of Mathematics. John Wiley and Sons, Inc. . ISBN 0-471-54397-7.

External links

Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics.

Dictionary

-noun

(uncountable, arithmetic, mathematics) The process of computing the sum of a number with itself a specified number of times, or any other analogous binary operation that combines other mathematical objects.(countable, arithmetic) A calculation involving multiplication.

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