3 × 4 = 12, so twelve dots can be
arranged in three rows of four (or four columns of three).
Multiplication is the
mathematical operation of scaling
one number by another. It is one of the four basic operations in
elementary arithmetic (the
others being
addition,
subtraction and
division).
Multiplication is defined for
whole
numbers in terms of repeated addition; for example, 3
multiplied by 4 (often said as "3 times 4") can be calculated by
adding 4 copies of 3 together:
- 3 \times 4 = 3 + 3 + 3 + 3 = 12.\!\,
Multiplication of
rational numbers
(fractions) and
real numbers is defined
by systematic
generalization
of this basic idea.
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 (for numbers generally).
The inverse of multiplication is division: as 3 times 4 is equal to
12, so 12 divided by 3 is equal to 4.
Multiplication is generalized further to other types of numbers
(such as
complex numbers) and to more
abstract constructs such as
matrices.
Notation and terminology
The multiplication sign.
Multiplication is often 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 = 6\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 other
countries that use a comma as a
decimal point, either the period or a middle dot is used for
multiplication.
- 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.
- 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 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,
while the dot denotes a scalar multiplication.
- A like convention distinguishes between the cross product and the dot product of two vectors.
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
xy^{2}) is
called a
coefficient.
The result of a multiplication is called a
product, and is a
multiple of each factor that 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,
Babylonian,
Indus valley, 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, 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, 2
n, ...,
20
n; followed by the multiples of 10
n:
30
n 40
n, and 50
n. Then to compute any
sexagesimal product, say 53
n, one only needed to add
50
n and 3
n computed from the table.
Chinese
In the mathematical text
Zhou Bi
Suan Jing, 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
The early
Indian mathematicians
of the
Indus Valley
Civilization 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 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=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. 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.
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 ∞. 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 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.
Properties
For integers, fractions, and 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.
- Associative
property
- Expressions 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. 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
- 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.
- 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 <&NBSP;0 and=""
b > c then
ab <&NBSP;ac.
- Negative one times any number is equal to the opposite 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.
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.
\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
definition for multiplication based on his axioms for natural
numbers.
- 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. 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
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 which 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 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).
Exponentiation
When multiplication is repeated, the resulting operation is known
as
exponentiation. For instance, the product 2×2×2
of three factors of two 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 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.
See also
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
External links