Multiplication

Multiplication (often denoted by the cross symbol "×", or by the absence of symbol) is the third basic mathematical operation of arithmetic, the others being addition, subtraction and division (the division is the fourth one, because it requires multiplication to be defined).

The multiplication of two whole numbers is equivalent to the addition of one of them with itself as many times as the value of the other one; for example, 3 multiplied by 4 (often said as "3 times 4") can be calculated by adding 3 copies of 4 together:

3\times 4=4+4+4=12

Here 3 and 4 are the "factors" and 12 is the "product".

One of the main properties of multiplication is that the result does not depend on the place of the factor that is repeatedly added to itself (commutative property). 3 multiplied by 4 can also be calculated by adding 4 copies of 3 together:

3\times 4=3+3+3+3=12

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 given lengths. The area of a rectangle does not depend on which side is measured first, which illustrates the commutative property.

In general, multiplying two measurements gives a new type, depending on the measurements. For instance:

2.5{\mbox{ meters}}\times 4.5{\mbox{ meters}}=11.25{\mbox{ square meters}}

11{\mbox{ meters/second}}\times 9{\mbox{ seconds}}=99{\mbox{ meters}}

The inverse operation of the multiplication is the division. For example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number (since 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 these more abstract constructs, the order that the operands are multiplied sometimes does matter.

Notation and terminology

In arithmetics, multiplication is often written using the multiplication sign "×" between the terms; that is, in infix notation. For example,

2\times 3=6

(verbally, "two times three equals six")

3\times 4=12

2\times 3\times 5=6\times 5=30

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

There are other mathematical notations for multiplication:

Multiplication is sometimes denoted by dot signs, either a middle-position dot or a period:

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

The middle dot notation, encoded in Unicode as U+22C5 ⋅ DOT OPERATOR, is standard in the United States, the United Kingdom, and other countries where the period is used as a decimal point. When the dot operator character is not accessible, the interpunct (·) is used. In other countries that use a comma as a decimal point, either the period or a middle dot is used for multiplication.^{[citation needed]}

In algebra, multiplication involving variables is often written as a juxtaposition (e.g., xy for x times y or 5x for five times x). This notation can also be used for quantities 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. The cross symbol generally denotes a vector multiplication,^{[clarification needed]} while the dot denotes a scalar multiplication. A similar convention distinguishes between the cross product and the dot product of two vectors.

In computer programming, the asterisk (as in 5*2) is the standard notation: it belongs to most character sets and appears on every keyboard. This usage originated in the FORTRAN programming language.

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 addends is called the "multiplier". In algebra, a number that is the multiplier of a variable or expression (e.g., the 3 in 3xy²) is called a coefficient.

The result of a multiplication is called a product, and it is a multiple of each factor if the other factor is an integer. For example, 15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5.

Computation

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.

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 algorithms

Methods of multiplication were documented in the Egyptian, Greek, Indian and Chinese civilizations.

The Ishango bone, dated to about 18,000 to 20,000 BC, hints at a knowledge of multiplication in the Upper Paleolithic era in Central Africa.

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 1 × 21 = 21, 4 × 21 = 84, 8 × 21 = 168. The full product could then be found by adding the appropriate terms found in the doubling sequence:

13 × 21 = (1 + 4 + 8) × 21 = (1 × 21) + (4 × 21) + (8 × 21) = 21 + 84 + 168 = 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.

Chinese

In the mathematical text Zhou Bi Suan Jing, 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. These place value decimal arithmetic algorithms were introduced by Al Khwarizmi to Arab countries in the early 9th century.

Modern method

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:

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.^[1]

Computer algorithms

The standard method of multiplying two n-digit numbers requires n² simple multiplications. Multiplication algorithms have been designed that reduce the computation time considerably when multiplying large numbers. In particular for very large numbers methods based on the Discrete Fourier Transform can reduce the number of simple multiplications to the order of n log₂(n) log₂ log₂(n).

Products of measurements

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 found applications in finance. One can only meaningfully add or subtract quantities of the same type but can multiply or divide quantities of different types.

A common example is multiplying speed by time gives distance, so

50 kilometers per hour × 3 hours = 150 kilometers.

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. 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=1}^{4}i=1\cdot 2\cdot 3\cdot 4,

that is

\prod _{i=1}^{4}i=24.

The subscript gives the symbol for a dummy 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. So, for example:

\prod _{i=1}^{6}i=1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6=720

More generally, the notation is defined as

\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},

where m and n are integers or expressions that evaluate to integers. 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.

Infinite products

One may also consider products of infinitely many terms; these are called infinite products. Notationally, we would replace n above by the lemniscate ∞. 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, 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.

Properties

For the real and complex numbers, which includes for example natural numbers, integers and fractions, multiplication has certain properties:

Commutative property

The order in which two numbers are multiplied does not matter:

x\cdot y=y\cdot x

.

Associative property

Expressions solely involving multiplication or addition are invariant with respect to order of operations:

(x\cdot y)\cdot z=x\cdot (y\cdot z)

Distributive property

Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions:

x\cdot (y+z)=x\cdot y+x\cdot z

Identity element

The multiplicative identity is 1; anything multiplied by one is itself. This is known as the identity property:

x\cdot 1=x

Zero element

Any number multiplied by zero is zero. This is known as the zero property of multiplication:

x\cdot 0=0

Zero is sometimes not included amongst the natural numbers.

There are a number of further properties of multiplication not satisfied by all types of numbers.

Negation

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

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

Negative one times negative one is positive one.

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

The natural numbers do not include negative numbers.

Inverse element: Every number x, except zero, has a multiplicative inverse, ${\frac {1}{x}}$ , such that $x\cdot \left({\frac {1}{x}}\right)=1$ .

Order preservation

Multiplication by a positive number preserves order:

if a > 0, then if b > c then ab > ac.

Multiplication by a negative number reverses order:

if a < 0 and b > c then ab < ac.

The complex numbers do not have an order predicate.

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.

Axioms

In the book Arithmetices principia, nova methodo exposita, Giuseppe Peano proposed axioms for arithmetic based on his axioms for natural numbers.^[2] Peano arithmetic has two axioms for multiplication:

x\times 0=0

x\times S(y)=(x\times y)+x

Here S(y) represents the successor of y, or the natural number that follows y. 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

x\times 1=x\times S(0)=(x\times 0)+x=0+x=x

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

(x_{p},\,x_{m})\times (y_{p},\,y_{m})=(x_{p}\times y_{p}+x_{m}\times y_{m},\;x_{p}\times y_{m}+x_{m}\times y_{p})

The rule that −1 × −1 = 1 can then be deduced from

(0,1)\times (0,1)=(0\times 0+1\times 1,\,0\times 1+1\times 0)=(1,0)

Multiplication is extended in a similar way to rational numbers and then to real 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.

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 n\cdot a=\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 $n\cdot a\,$ are exactly those of the Cartesian product $n\times a\,$ . The properties of the multiplicative operation as applying to natural numbers then follow trivially from the corresponding properties of the Cartesian product.

Multiplication in group theory

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 non-zero 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, look at the set of invertible square matrices of a given dimension, over a given field. Now it is straightforward to verify closure, associativity, and inclusion of identity (the identity matrix) and inverses. However, matrix multiplication is not commutative, therefore this group is nonabelian.

Another fact of note 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 a $\cdot$ b or ab. When referring to a group via the indication of the set and operation, the dot is used, e.g., our first example could be indicated by $\left(\mathbb {Q} \smallsetminus \{0\},\cdot \right)$

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 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).

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; $N\times (-M)=(-N)\times M=-(N\times M)$ and; $(-N)\times (-M)=N\times M$; The same sign rules apply to rational and real numbers.

Rational numbers: Generalization to fractions ${\frac {A}{B}}\times {\frac {C}{D}}$ is by multiplying the numerators and denominators respectively: ${\frac {A}{B}}\times {\frac {C}{D}}={\frac {(A\times C)}{(B\times D)}}$ . 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.

Real numbers: $(x)(y)$ 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. This gives the area of a rectangle x high and y wide. See Products of sequences, above.

Complex numbers: Considering complex numbers $z_{1}$ and $z_{2}$ as ordered pairs of real numbers $(a_{1},b_{1})$ and $(a_{2},b_{2})$ , the product $z_{1}\times z_{2}$ is $(a_{1}\times a_{2}-b_{1}\times b_{2},a_{1}\times b_{2}+a_{2}\times b_{1})$ . 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 Multiplication in group theory, above, and Multiplicative Group, which for example includes matrix multiplication. A very general, and abstract, concept of multiplication is as the "multiplicatively denoted" (second) binary operation in a ring. An example of a ring that is not any of the above number systems is a polynomial ring (you can add and multiply polynomials, but polynomials are not numbers in any usual sense.)

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 a division ring there are inverses, but ${\frac {x}{y}}$ may be ambiguous in non-commutative rings since $x\left({\frac {1}{y}}\right)$ need not the same as $\left({\frac {1}{y}}\right)x$ .

Exponentiation

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 2³, 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 to multiply base by itself, so that the expression

a^{n}=\underbrace {a\times a\times \cdots \times a} _{n}

indicates that the base a to be multiplied by itself n times.

Notes

^ Henry B. 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

References

Boyer, Carl B. (revised by Merzbach, Uta C.) (1991). History of Mathematics. John Wiley and Sons, Inc. ISBN 0-471-54397-7.{{cite book}}: CS1 maint: multiple names: authors list (link)

External links

[1] Henry B. 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

[2] PlanetMath: Peano arithmetic

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