Multiplication

Related subjects: Mathematics

Background Information

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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. 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.

Multiplication is one of four main operations in elementary arithmetic, and most people learn basic multiplication algorithms in elementary school. The inverse of multiplication is division.

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 result is expressed with an equals sign. 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. In some countries that use a comma as a decimal point, the period is used for multiplication instead.
The asterisk (e.g. 5 * 2) is often used with computers because it appears on every keyboard. This usage originated in the FORTRAN programming language.
In algebra, multiplication involving variables is often written as a juxtaposition (e.g. xy for x times y or 5x for five times x). This notation can also be used for numbers that are surrounded by parentheses (e.g. 5(2) or (5)(2) for five times two).

The numbers to be multiplied are generally called the "factors" or "multiplicands". When thinking of multiplication as repeated addition, the number to be repeated is called the "multiplicand", while the number of repetitions is called the "multiplier". In algebra, a number that is multiplied by a variable or expression (i.e. the 3 in 3xy²) is called a coefficient.

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

Computation

The standard methods for multiplying numbers using pencil and paper require a multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9), however one method, the peasant multiplication algorithm, does not. Many mathematics curricula developed according to the 1989 standards of the NCTM do not teach standard arithmetic methods, instead guiding students to invent their own methods of computation. Though widely adopted by many school districts in nations such as the United States, they have encountered resistance from some parents and mathematicians, and some districts have since abandoned such curricula in favour of traditional mathematics.

Multiplying numbers to more than a couple of decimal places by hand is tedious and error prone. Common logarithms were invented to simplify such calculations. The slide rule allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early twentieth century, mechanical calculators, such as the Marchant, automated multiplication of up to 10 digit numbers. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand.

Historical algorithms

Methods of multiplication were documented in the Egyptian, Greece, Babylonian, Indus valley, 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 1 × 21 = 21, 2 × 21 = 42, 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 books, Chou Pei Suan Ching dated prior to 300 B.C., and the Nine Chapters on the Mathematical Art, multiplication calculations were written out in words, although the early Chinese mathematicians employed an abacus in hand calculations involving addition and multiplication.

Indus 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). Here a table was drawn up with the rows and columns labelled by the multiplicands. Each box of the table was divided diagonally into two, as a triangular lattice. The entries of the table held the partial products, written as decimal numbers. The product could then be formed by summing down the diagonals of the lattice.

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.

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 (∏) is defined a n-ary product for this purpose, distinct from U+03A0 (Π), the letter. This is defined as:

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

The subscript gives the symbol for a dummy variable ( $i$ in our case) and its lower value ( $m$ ); the superscript gives its upper value. 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.

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 (infinity symbol) ∞. 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. 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. 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 $a \times n\,$ . 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 · y = y · x.

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

Distributive property: Holds with respect to addition over multiplication. This identity is of prime importance in simplifying algebraic expressions.; x·(y + z) = x·y + x·z.

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

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

Inverse property: Every number x, except zero, has a multiplicative inverse, 1/x, such that x·(1/x) = 1.

Order preservation: Multiplication by a positive number preserves order: if a > 0, then if b > c then a·b > a·c. Multiplication by a negative number reverses order: if a < 0, then if b > c then a·b < a·c.

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

(−1) · x = (−x)

Negative one times negative one is positive one.

(−1) · (−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.

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 proved by means of the distributive property. We assume all the usual properties of addition and subtraction, and −x means the same as

x · 0

= (x · 0) + x − x

= (x · 0) + (x · 1) − x

= x · (0 + 1) − x

= (x · 1) − x

= x − x

= 0.

So we have proved:

x · 0 = 0.

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

(−1) · x

= (−1) · x + x − x

= (−1) · x + 1 · x − x

= (−1 + 1) · x − x

= 0 · x − x

= 0 − x

= −x

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

(−1) · (−1)

= −(−1)

= 1.

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.

a×1=a
a×b'=(a×b)+a

Here, b' represents the successor of b, or the natural number which follows b. With his other nine axioms, it is possible to prove common rules of multiplication, such as the distributive or associative properties.

Multiplication with set 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 inverse.

Multiplication also is an abelian group, 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 matrices) or don't look much like numbers (such as quaternions).

Integers N × 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 × -M) = - (N × M).

Rationals Generalization to fractions A/B × C/D is by multiplying the numerators and denominators respectively: A/B × C/D = (A × B) / (C × D). This gives the area of a rectangle A/B high and 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. This gives the area of a rectangle x high and y wide. See above.

Complex Considering complex numbers z1 and z2 as ordered pairs or real numbers (a1, b1) and (a2, b2), the product z1 × z2 is (a1 × a2 - b1 × b2, a1 × b2 + a2 × b1). This is the same as for reals, a1 × a2, when the imaginary parts b1 and b2 are zero.

Further generalizations See 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 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.)

Division Often division x / y is the same as multiplication by an inverse, x × (1/y). Multiplication for some types of "numbers" may have corresponding division, without inverses; in an Integral domain x may have no inverse "1/x" but x/y may be defined. In a Division ring there are inverses but they are not commutative (since 1/x X 1/y is not the same as 1/y X 1/x, x/y may be ambiguous).

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