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, .https://files.eric.ed.gov/fulltext/EJ1246116.pdf Basics of multiplication
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", or "Three groups of four", notating the multiplication symbol with the word of, 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 (including negative numbers), rational numbers (fractions), and real numbers.
Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers) or as calculating the area of a rectangle with sides of given lengths. The area of a rectangle remains the same regardless of which side is measured first—a result of the commutative property.
The product of two measurements (or physical quantities) is a new type of measurement (or new quantity), usually with a derived unit of measurement. For example, multiplying the lengths (in meters or feet) of the two sides of a rectangle gives its area (in square meters or square feet). Such a product is the subject of dimensional analysis.
The inverse operation of multiplication is considered 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 (either or ) between the factors (that is, in infix notation). For example, ("two times three equals six")
The cross sign was first used by Edward Wright in 1618. There are other mathematical notations for multiplication: Multiplication is also denoted by a middle-position dot: . Middle dot was first used by G. W. Leibniz in 1698, because the multiplication sign × is easily confused with the common mathematical variable . 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 (and a period as a thousands separator), 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 (period) 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 (SI) 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 (e.g., for times or for five times ), also called implied multiplication. The notation can also be used for quantities that are surrounded by parentheses (e.g., , or for five times two). 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 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.
In computer programming, the asterisk (as in 5*2) is still the most common notation. This is because 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. (Even modern compilers do not recognize × or ⋅ as multiplication operators.)
The numbers to be multiplied are generally called the factors (as in factorization). 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 (e.g., the 3 in ) 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:
(This rule is a consequence of the distributivity of multiplication over addition, and is not an additional rule.)
In words:
A positive number multiplied by a positive number is positive (product of natural numbers), 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 Two fractions can be multiplied by multiplying their numerators and denominators:
which is defined when .
Product of two real numbers
There are several equivalent ways to define the real numbers formally; 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 (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", "grade-school 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 ) 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.
Historical algorithms Methods of multiplication were documented in the writings of ancient Egyptian, 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.
Egyptians
The Egyptian method of multiplication of integers and fractions, which is documented in the Rhind Mathematical 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 , , . 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.