Multiplication
Multiplication 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. Multiplication is often denoted by the cross symbol,, by the mid-line dot operator,, by juxtaposition, or, in programming languages, by an asterisk,.
The multiplication of whole numbers may be thought of as 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. This is to be distinguished from terms, which are added.
Whether the first factor is the multiplier or the multiplicand may be ambiguous or depend upon context. For example, the expression can be phrased as "3 times 4" and evaluated as, where 3 is the multiplier, but also as "3 multiplied by 4", in which case 3 becomes the multiplicand. 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.
Systematic generalizations of this basic definition define the multiplication of integers, rational numbers, and real numbers.
Multiplication can also be visualized as counting objects arranged in a rectangle 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, usually with a derived unit of measurement. For example, multiplying the lengths of the two sides of a rectangle gives its area. Such a product 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.
Several mathematical concepts expand upon the fundamental idea of multiplication. The product of a sequence, vector multiplication, complex numbers, and matrices are all examples where this can be seen. These more advanced constructs tend to affect the basic properties in their own ways, such as becoming noncommutative in matrices and some forms of vector multiplication or changing the sign of complex numbers.
Notation
In arithmetic, multiplication is often written using the multiplication sign between the factors. For example,There are other mathematical notations for multiplication:
- To reduce confusion between the multiplication sign × and the common variable, multiplication is also denoted by dot signs, usually a middle-position dot :. The middle dot notation or dot operator is now standard in the United States and other countries. When the dot operator character is not accessible, the interpunct is used. In most European and other countries that use a comma as a decimal point, the multiplication sign or a middle dot is used to indicate multiplication. Historically, in the United Kingdom and Ireland, the middle dot was sometimes used for the decimal point to prevent it from disappearing in the ruled line, and the full stop was used for multiplication. However, since the Ministry of Technology ruled in 1968 that the period be used as the decimal point, and the International System of Units standard has since been widely adopted, this usage is now found only in the more traditional journals such as The Lancet.
- In algebra, multiplication involving variables is often written as a juxtaposition, also called implied multiplication. The notation can also be used for quantities that are surrounded by parentheses. This implicit usage of multiplication can cause ambiguity when the concatenated variables happen to match the name of another variable, when a variable name in front of a parenthesis can be confused with a function name, or in the correct determination of the order of operations.
- In vector multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a cross product of two vectors, yielding a vector as its result, while the dot denotes taking the dot product of two vectors, resulting in a scalar.
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; however, sometimes the first factor is considered the multiplicand and the second the multiplier.
Also, as the result of 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".
In algebra, a number that is the multiplier of a variable or expression is called a coefficient.
The result of a multiplication is called a product. When one factor is an integer, the product is a multiple of the other or of the product of the others. Thus, is a multiple of, as is. 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.
Definitions
The product of two numbers or the multiplication between two numbers can be defined for common special cases: natural numbers, integers, rational numbers, real numbers, complex numbers, and quaternions.Product of two natural numbers
The product of two natural numbers is defined as:Product of two integers
An integer can be either zero, a nonzero natural number, or minus a nonzero natural number. The product of zero and another integer is always zero. The product of two nonzero integers is determined by the product of their positive amounts, combined with the sign derived from the following rule:In words:
- A positive number multiplied by a positive number is positive,
- A positive number multiplied by a negative number is negative,
- A negative number multiplied by a positive number is negative,
- A negative number multiplied by a negative number is positive.
Product of two fractions
Product of two real numbers
There are several equivalent ways to define formally the real numbers; see Construction of the real numbers. The definition of multiplication is a part of all these definitions.A fundamental aspect of these definitions is that every real number can be approximated to any accuracy by rational numbers. A standard way for expressing this is that every real number is the least upper bound of a set of rational numbers. In particular, every positive real number is the least upper bound of the truncations of its infinite decimal representation; for example, is the least upper bound of
A fundamental property of real numbers is that rational approximations are compatible with arithmetic operations, and, in particular, with multiplication. This means that, if and are positive real numbers such that and then In particular, the product of two positive real numbers is the least upper bound of the term-by-term products of the sequences of their decimal representations.
As changing the signs transforms least upper bounds into greatest lower bounds, the simplest way to deal with a multiplication involving one or two negative numbers, is to use the rule of signs described above in. The construction of the real numbers through Cauchy sequences is often preferred in order to avoid consideration of the four possible sign configurations.
Product of two complex numbers
Two complex numbers can be multiplied by the distributive law and the fact that, as follows:The geometric meaning of complex multiplication can be understood by rewriting complex numbers in polar coordinates:
Furthermore,
from which one obtains
The geometric meaning is that the magnitudes are multiplied and the arguments are added.
Product of two quaternions
The product of two quaternions can be found in the article on quaternions. Note, in this case, that and are in general different.Computation
Many common methods for multiplying numbers using pencil and paper require a multiplication table of memorized or consulted products of small numbers. However, one method, the peasant multiplication algorithm, does not. The example below illustrates "long multiplication" :23958233
× 5830
———————————————
00000000
71874699
191665864
+ 119791165
———————————————
139676498390
In some countries such as Germany, the multiplication above is depicted similarly but with the original problem written on a single line and computation starting with the first digit of the multiplier:
23958233 · 5830
———————————————
119791165
191665864
71874699
00000000
———————————————
139676498390
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.