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Arithmetic operations  

Multiplication (often denoted by the cross symbol ×, by the midline dot operator ⋅, by juxtaposition, or, on computers, by an asterisk *) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a product.
The multiplication of whole numbers may be thought of 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 quantity of the other one, the multiplier. Both numbers can be referred to as factors.
For example, 4 multiplied by 3, often written as and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together:
Here, 3 (the multiplier) and 4 (the multiplicand) are the factors, and 12 is the product.
One of the main properties of multiplication is the commutative property, which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3:
Thus the designation of multiplier and multiplicand does not affect the result of the multiplication.^{ [1] }
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 some given lengths. The area of a rectangle does not depend on which side is measured first—a consequence of the commutative property.
The product of two measurements is a new type of measurement. For example, multiplying the lengths of the two sides of a rectangle gives its area. Such products is the subject of dimensional analysis.
The inverse operation of multiplication is division. For example, since 4 multiplied by 3 equals 12, 12 divided by 3 equals 4. Indeed, multiplication by 3, followed by division by 3, yields the original number. 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 some of these more abstract constructs, the order in which the operands are multiplied together matters. A listing of the many different kinds of products used in mathematics is given in Product (mathematics).
In arithmetic, multiplication is often written using the sign "" between the terms (i.e., in infix notation).^{ [2] } For example,
The sign is encoded in Unicode at U+00D7× MULTIPLICATION SIGN (HTML ×
·×
).
There are other mathematical notations for multiplication:
In computer programming, the asterisk (as in 5*2
) is still the most common notation. This is due to the fact that most computers historically were limited to small character sets (such as ASCII and EBCDIC) that lacked a multiplication sign (such as ⋅
or ×
), while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language.
The numbers to be multiplied are generally called the "factors". The number to be multiplied is the "multiplicand", and the number by which it is multiplied is the "multiplier". Usually the multiplier is placed first and the multiplicand is placed second;^{ [1] } however sometimes the first factor is the multiplicand and the second the multiplier.^{ [5] } Also as the result of a multiplication does not depend on the order of the factors, the distinction between "multiplicand" and "multiplier" is useful only at a very elementary level and in some multiplication algorithms, such as the long multiplication. Therefore, in some sources, the term "multiplicand" is regarded as a synonym for "factor".^{ [6] } In algebra, a number that is the multiplier of a variable or expression (e.g., the 3 in 3xy^{2}) is called a coefficient.
The result of a multiplication is called a product. A product of integers 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 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. The example below illustrates "long multiplication" (the "standard algorithm", "gradeschool multiplication"):
23958233 × 5830 ——————————————— 00000000 ( = 23,958,233 × 0) 71874699 ( = 23,958,233 × 30) 191665864 ( = 23,958,233 × 800) + 119791165 ( = 23,958,233 × 5,000) ——————————————— 139676498390 ( = 139,676,498,390 )
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, since adding logarithms is equivalent to multiplying. 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.
Methods of multiplication were documented in the writings of ancient Egyptian, Greek, Indian and Chinese civilizations.
The Ishango bone, dated to about 18,000 to 20,000 BC, may hint at a knowledge of multiplication in the Upper Paleolithic era in Central Africa, but this is speculative.
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 = 2 × 42 = 84, 8 × 21 = 2 × 84 = 168. The full product could then be found by adding the appropriate terms found in the doubling sequence:
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 numbern: 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.
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. The Chinese were already using a decimal multiplication table by the end of the Warring States period.^{ [7] }
The modern method of multiplication based on the Hindu–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:
These place value decimal arithmetic algorithms were introduced to Arab countries by Al Khwarizmi in the early 9th century, and popularized in the Western world by Fibonacci in the 13th century.
Grid method multiplication or the box method, is used in primary schools in England and Wales and 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:
30  4  

10  300  40 
3  90  12 
and then add the entries.
The classical method of multiplying two ndigit numbers requires n^{2} digit multiplications. Multiplication algorithms have been designed that reduce the computation time considerably when multiplying large numbers. Methods based on the discrete Fourier transform reduce the computational complexity to O(n log n log log n). Recently, the factor log log n has been replaced by a function that increases much slower although it is still not constant (as it can be hoped).^{ [9] }
In March 2019, David Harvey and Joris van der Hoeven submitted an article presenting an integer multiplication algorithm with a claimed complexity of ^{ [10] } The algorithm, also based on the fast Fourier transform, is conjectured to be asymptotically optimal.^{ [11] } The algorithm is not considered practically useful, as its advantages only appear when multiplying extremely large numbers (having more than 2^{172912} bits).^{ [12] }
One can only meaningfully add or subtract quantities of the same type, but quantities of different types can be multiplied or divided without problem. For example, four bags with three marbles each can be thought of as:^{ [1] }
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 applications found in finance and other applied fields.
A common example in physics is the fact that multiplying speed by time gives distance. For example:
In this case, the hour units cancel out, leaving the product with only kilometer units.
Other examples of multiplication involving units include:
The product of a sequence of factors can be written with the product symbol, which derives from the capital letter (pi) in the Greek alphabet (much like the same way the capital letter (sigma) is used in the context of summation).^{ [13] }^{ [14] }^{ [15] } Unicode position U+220F∏ contains a glyph for denoting such a product, distinct from U+03A0Π, the letter. The meaning of this notation is given by:
that is
The subscript gives the symbol for a bound 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. For example:
More generally, the notation is defined as
where m and n are integers or expressions that evaluate to integers. In case where m = n, the value of the product is the same as that of the single factor x_{m}; if m > n, the product is an empty product whose value is 1—regardless of the expression for the factors.
By definition,
If all factors are identical, a product of n factors is equivalent to an exponentiation:
Associativity and commutativity of multiplication imply
if a is a nonnegative integer, or if all are positive real numbers, and
if all are nonnegative integers, or if x is a positive real number.
One may also consider products of infinitely many terms; these are called infinite products. Notationally, this consists in replacing n above by the Infinity symbol ∞. The product of such an infinite sequence is defined as the limit of the product of the first n terms, as n grows without bound. That is,
One can similarly replace m with negative infinity, and define:
provided both limits exist.
For the real and complex numbers, which includes for example natural numbers, integers, and fractions, multiplication has certain properties:
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 the book Arithmetices principia, nova methodo exposita , Giuseppe Peano proposed axioms for arithmetic based on his axioms for natural numbers.^{ [16] } Peano arithmetic has two axioms for multiplication:
Here S(y) represents the successor of y, or the natural number that followsy. The various properties like associativity can be proved from these and the other axioms of Peano arithmetic including induction. For instance S(0), denoted by 1, is a multiplicative identity because
The axioms for integers typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (x,y) as equivalent to x − y when x and y are treated as integers. Thus both (0,1) and (1,2) are equivalent to −1. The multiplication axiom for integers defined this way is
The rule that −1 × −1 = 1 can then be deduced from
Multiplication is extended in a similar way to rational numbers and then to real numbers.
The product of nonnegative integers can be defined with set theory using cardinal numbers or the Peano axioms. See below how to extend this to multiplying arbitrary integers, and then arbitrary rational numbers. The product of real numbers is defined in terms of products of rational numbers, see construction of the real numbers.
There are many sets that, under the operation of multiplication, satisfy the axioms that define group structure. These axioms are closure, associativity, and the inclusion of an identity element and inverses.
A simple example is the set of nonzero rational numbers. Here we have identity 1, as opposed to groups under addition where the identity is typically 0. Note that with the rationals, we must exclude zero because under multiplication, it does not have an inverse: there is no rational number that can be multiplied by zero to result in 1. In this example we have an abelian group, but that is not always the case.
To see this, consider the set of invertible square matrices of a given dimension over a given field. Here, it is straightforward to verify closure, associativity, and inclusion of identity (the identity matrix) and inverses. However, matrix multiplication is not commutative, which shows that this group is nonabelian.
Another fact worth noticing is that the integers under multiplication is not a group—even if we exclude zero. This is easily seen by the nonexistence of an inverse for all elements other than 1 and −1.
Multiplication in group theory is typically notated either by a dot, or by juxtaposition (the omission of an operation symbol between elements). So multiplying element a by element b could be notated as ab or ab. When referring to a group via the indication of the set and operation, the dot is used. For example, our first example could be indicated by .
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 quantum mechanics, multiplication has been generalized to more complicated and abstract types of numbers, and to things that are not numbers (such as matrices) or do not look much like numbers (such as quaternions).
When multiplication is repeated, the resulting operation is known as exponentiation . For instance, the product of three factors of two (2×2×2) is "two raised to the third power", and is denoted by 2^{3}, a two with a superscript three. In this example, the number two is the base, and three is the exponent. In general, the exponent (or superscript) indicates how many times the base appears in the expression, so that the expression
indicates that n copies of the base a are to be multiplied together. This notation can be used whenever multiplication is known to be power associative.
With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first)
In mathematics, the factorial of a nonnegative integer n, denoted by n!, is the product of all positive integers less than or equal to n:
In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the nonpositive integers. For any positive integer n,
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g. since 1000 = 10 × 10 × 10 = 10^{3}, the "logarithm base 10" of 1000 is 3, or log_{10} (1000) = 3. The logarithm of x to base b is denoted as log_{b} (x), or without parentheses, log_{b} x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.
In mathematics, a product is the result of multiplication, or an expression that identifies factors to be multiplied. For example, 30 is the product of 6 and 5, and is the product of and .
In mathematics, a square root of a number x is a number y such that y^{2} = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x. For example, 4 and −4 are square roots of 16, because 4^{2} = (−4)^{2} = 16. Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by where the symbol is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by because 3^{2} = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case 9.
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. This integral domain is a particular case of a commutative ring of quadratic integers. It does not have a total ordering that respects arithmetic.
In mathematics, the padic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two padic numbers are considered to be close when their difference is divisible by a high power of p: the higher the power, the closer they are. This property enables padic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.
A multiplication algorithm is an algorithm to multiply two numbers. Depending on the size of the numbers, different algorithms are used. Efficient multiplication algorithms have existed since the advent of the decimal system.
In mathematics, factorization or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15, and (x – 2)(x + 2) is a factorization of the polynomial x^{2} – 4.
Exponentiation is a mathematical operation, written as b^{n}, involving two numbers, the baseb and the exponent or powern, and pronounced as "b raised to the power of n". When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b^{n} is the product of multiplying n bases:
In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative ring with unity. Subalgebras called reduced incidence algebras give a natural construction of various types of generating functions used in combinatorics and number theory.
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x^{−1}, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth, and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse.
In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra (four) and iteration.
Mental calculation consists of arithmetical calculations using only the human brain, with no help from any supplies or devices such as a calculator. People use mental calculation when computing tools are not available, when it is faster than other means of calculation, or even in a competitive context. Mental calculation often involves the use of specific techniques devised for specific types of problems. People with unusually high ability to perform mental calculations are called mental calculators or lightning calculators.
Modular exponentiation is a type of exponentiation performed over a modulus. It is useful in computer science, especially in the field of publickey cryptography.
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, onehalf, eightfifths, threequarters. A common, vulgar, or simple fraction consists of a numerator displayed above a line, and a nonzero denominator, displayed below that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.
Methods of computing square roots are numerical analysis algorithms for finding the principal, or nonnegative, square root of a real number. Arithmetically, it means given S, a procedure for finding a number which when multiplied by itself, yields S; algebraically, it means a procedure for finding the nonnegative root of the equation x^{2}  S = 0; geometrically, it means given the area of a square, a procedure for constructing a side of the square.
In mathematics, particularly in the area of number theory, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In the standard notation of modular arithmetic this congruence is written as
A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic and the infinitely repeated portion is not zero. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating. For example, the decimal representation of 1/3 becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is 3227/555, whose decimal becomes periodic at the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... At present, there is no single universally accepted notation or phrasing for repeating decimals.
In mathematics, a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a nonzero denominator q. For example, −3/7 is a rational number, as is every integer. The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface Q ; it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient", and first appeared in Bourbaki's Algèbre.