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Multiplication (often denoted by the cross symbol "×", by a point "·" or by the absence of symbol) is one of the four elementary, mathematical operations of arithmetic; with the others being addition, subtraction and division. The multiplication of two whole numbers, when thinking of multiplication as repeated addition, is equivalent to adding as many copies of one of them (multiplicand) as the value of the other one (multiplier) : For example, 4 multiplied by 3 (often written as and said as "3 times 4") can be calculated by adding 3 copies of 4 together: : Here 3 and 4 are the "factors" and 12 is the "product". One of the main properties of multiplication is the commutative property, adding 3 copies of 4 gives the same result as adding 4 copies of 3: : 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. 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. A listing of the many different kinds of products that are used in mathematics is given in the product (mathematics) page. ==Notation and terminology== In arithmetics, multiplication is often written using the sign "×" between the terms; that is, in infix notation. For example, : (verbally, "two times three equals six") : : : The sign is encoded in Unicode at . There are other mathematical notations for multiplication: *Multiplication is also denoted by dot signs, usually a middle-position dot (rarely period): : :The middle dot notation, encoded in Unicode as , 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 mark, either the period or a middle dot is used for multiplication. *In algebra, multiplication involving variables is often written as a juxtaposition (e.g., ''xy'' for ''x'' times ''y'' or 5''x'' for five times ''x''). The notation can also be used for quantities that are surrounded by parentheses (e.g., 5(2) or (5)(2) 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 matrix 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 the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar. In computer programming, the asterisk (as in 5) is still the most common notation. This is due to the fact that 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.The numbers to be multiplied are generally called the "factors". The number to be multiplied is called the "multiplicand", while the number of times the multiplicand is to be multiplied comes from the "multiplier". Some persons interpret a multiplication expression such that the first of the two factors is the multiplicand, and the second is the multiplier, whereas other believe the ''first'' factor is the multiplier. There is no definitive authority speaking to this matter, however, and the uncertainty on which comes first and second extends even to the handheld calculator industry. Additonally, there are some sources in which 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 3''xy''2) is called a coefficient. The result of a multiplication is called a product. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「multiplication」の詳細全文を読む スポンサード リンク
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