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In mathematics, especially in elementary arithmetic, division (denoted ÷ or / or —) is an arithmetic operation. Specifically, if ''b'' times ''c'' equals ''a'', written: :''a'' = ''b'' × ''c'' where ''b'' is not zero, then ''a'' divided by ''b'' equals ''c'', written: : ÷ For instance, :6 ÷ 3 = 2 since :3 × 2 = 6. In the above expressions, ''a'' is called the dividend, ''b'' is called the divisor, and ''c'' is called the quotient; in the expression or ''a'' is also called the numerator and ''b'' is also called the denominator. Conceptually, division of integers can be viewed in either of two distinct but related ways quotition and partition: * Partitioning involves taking a set of size ''a'' and forming ''b'' groups that are equal in size. The size of each group formed, ''c'', is the quotient of ''a'' and ''b''. * Quotition, or quotative division (also sometimes spelled quotitive) involves taking a set of size ''a'' and forming groups of size ''b''. The number of groups of this size that can be formed, ''c'', is the quotient of ''a'' and ''b''.〔Fosnot and Dolk 2001. ''Young Mathematicians at Work: Constructing Multiplication and Division''. Portsmouth, NH: Heinemann.〕 (Both divisions give the same result because multiplication is commutative.) Teaching division usually leads to the concept of fractions being introduced to school pupils. Unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder. To complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called. == Notation == Division is often shown in algebra and science by placing the ''dividend'' over the ''divisor'' with a horizontal line, also called a fraction bar, between them. For example, ''a'' divided by ''b'' is written : This can be read out loud as "''a'' divided by ''b''", "''a'' by ''b''" or "''a'' over ''b''". A way to express division all on one line is to write the ''dividend'' (or numerator), then a slash, then the ''divisor'' (or denominator), like this: : This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of ASCII characters. Some mathematical software, such as GNU Octave, allows the operands to be written in the reverse order by using the backslash as the division operator: : A typographical variation halfway between these two forms uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor: : Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (typically called the ''numerator'' and ''denominator''), and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus (or division sign), common in arithmetic, in this manner: : This form is infrequent except in elementary arithmetic. ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator. In some non-English-speaking cultures, "a divided by b" is written ''a'' : ''b''. This notation was introduced in 1631 by William Oughtred in his ''Clavis Mathematicae'' and later popularized by Gottfried Wilhelm Leibniz.〔(Earliest Uses of Symbols of Operation ), Jeff MIller〕 However, in English usage the colon is restricted to expressing the related concept of ratios (then "''a'' is to ''b''"). In elementary classes of some countries, the notation or is used to denote ''a'' divided by ''b'', especially when discussing long division; similarly, but less commonly, for short division (as shown in an example on that page). This notation was first introduced by Michael Stifel in ''Arithmetica integra'', published in 1544.〔 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「Division (mathematics)」の詳細全文を読む スポンサード リンク
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