division

an army unit large enough to sustain combat

"two infantry divisions were held in reserve" one of the portions into which something is regarded as divided and which together constitute a whole

"the written part of the exam"

"the finance section of the company"

"the BBC's engineering division" the act or process of dividing an administrative unit in government or business discord that splits a group a league ranked by quality

"he played baseball in class D for two years"

"Princeton is in the NCAA Division 1-AA" (biology) a group of organisms forming a subdivision of a larger category (botany) taxonomic unit of plants corresponding to a phylum a unit of the United States Air Force usually comprising two or more wings a group of ships of similar type an arithmetic operation that is the inverse of multiplication

the quotient of two numbers is computed the act of dividing or partitioning

separation by the creation of a boundary that divides or keeps apart

This article is about the arithmetic operation. For other uses, see Division (disambiguation).''In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication, and sometimes it can be interpreted as repeated subtraction. Specifically, if :$c\; \backslash times\; b\; =\; a$where ''b'' is not 0 (number)zero, then:$\backslash frac\; ab\; =\; c$that is, ''a'' divided by ''b'' equals ''c''. For instance, $\backslash frac\; 63\; =\; 2$ since $2\; \backslash times\; 3\; =\; 6\backslash ,$.In the above expression, ''a'' is called the dividend, ''b'' the divisor and ''c'' the quotient.Division by zero (i.e. where the divisor is zero) is usually not defined.

Notation - Division is most often shown by placing the ''dividend'' over the ''divisor'' with a horizontal line between them. For example, ''a'' divided by ''b'' is written $\backslash frac\; ab$. This can be read out loud as either "a over b" or "a divided by b".A way to express division all on one line is to write the ''dividend'', then a Slash (punctuation)slash, then the ''divisor'', like this: !$a/b\backslash ,$ .This? is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of characters.A typographical variation which is halfway between these two forms uses a slash but elevates the dividend, and lowers the divisor: !^a⁄ ;_bAny? of these forms can be used to display a fraction (mathematics)fraction. A fraction is a division expression where both dividend and divisor are integers (although typically called the ''numerator'' and ''denominator''), and there is no implication that the division needs to be evaluated further.A less common way to show division is to use the obelus (or division sign) in this manner: $a\; \backslash div\; b$. This form is infrequent except in elementary arithmetic. 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 languageEnglish-speaking cultures, "a divided by b" has sometimes been written ''a'' : ''b''. However, in English usage the colon (punctuation)colon is restricted to expressing the related concept of ratios.

Computing division - With a knowledge of multiplication tables, two integers can be divided on paper using the method of long division. If the dividend has a fraction (mathematics)fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, one can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction.Division can be calculated with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset.In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus. In such a case, division can be calculated by multiplication. This approach is useful in computers that do not have a fast division instruction.

Division of integers - Division of integers is not closed. Apart from division by zero being undefined, the quotient will not be an integer unless the dividend is an integer multiple of the divisor; for example 26 cannot be divided by 10 to give an integer. In such a case there are four possible approaches.# Say that 26 cannot be divided by 10.# Give the answer as a decimal fraction or a !Vulgar_fraction#Other_ways_of_ writing_fractionsmixed number, so $\backslash frac\; =\; 2.6$ or $26/10\; =\; 2\; \backslash frac\; 35$. This is the approach usually taken in mathematics.# Give the answer as a ''quotient'' and a ''remainder'', so $\backslash frac\; =\; 2$ remainder 6.# Give the quotient as the answer, so $\backslash frac\; =\; 2$. This is sometimes called ''integer division''. One has to be careful when performing division of integers in a computer program. Some programming languages, such as C programming languageC, will treat division of integers as in case 4 above, so the answer will be an integer. Other languages, such as MATLAB, will first convert the integers to real numbers, and then give a real number as the answer, as in case 2 above.

Division of rational numbers - The result of dividing two rational numbers is another rational number when the divisor is not 0. We may define division of two rational numbers ''p''/''q'' and ''r''/''s'' by:$=\; (p\; \backslash times\; s)/(q\; \backslash times\; r).$All four quantities are integers, and only ''p'' may be 0. This definition ensures that division is the inverse operation of multiplication.

Division of real numbers - Division of two real numbers results in another real number when the divisor is not 0. It is defined such ''a''/''b'' = ''c'' if and only if ''a'' = ''cb'' and ''b'' ≠ 0.

Division of complex numbers - Dividing two complex numbers results in another complex number when the divisor is not 0, defined thus::$=\; +\; i\; .$All four quantities are real numbers. ''r'' and ''s'' may not both be 0.Division for complex numbers expressed in polar form is simpler and easier to remember than the definition above::$\backslash over\; re^\; \}\; =\; e^\; .$Again all four quantities are real numbers. ''r'' may not be 0.

Division of polynomialpolynomials - One can define the division operation for polynomials. Then, as in the case of integers, one has a remainder. See polynomial long division.

Division in abstract algebra - In abstract algebras such as Matrix (mathematics)matrix algebras and quaternion algebras, fractions such as $$ are typically defined as $a\; \backslash cdot$ or $a\; \backslash cdot\; b^$ where $b$ is presumed to be an invertible element (i.e. there exists a multiplicative inverse $b^$ such that $bb^\; =\; b^\; b\; =\; 1$ where $1$ is the multiplicative identity). In an integral domain where such elements may not exist, ''division'' can still be performed on equations of the form $ab\; =\; ac$ or $ba\; =\; ca$ by left or right cancellation, respectively. More generally "division" in the sense of "cancellation" can be done in any ring (mathematics)ring with the aforementioned cancellation properties. By a Wedderburn's theoremtheorem of Wedderburn, all finite division rings are fields, hence every nonzero element of such a ring is invertible, so ''division'' by any nonzero element is possible in such a ring. To learn about when ''algebras'' (in the technical sense) have a division operation, refer to the page on division algebras. In particular Bott periodicity can be used to show that any real numberreal normed division algebra must be isomorphic to either the real numbers R, the complex numbers C, the quaternions H, or the octonions O.

Division and calculus - The derivative of the quotient of two functions is given by the quotient rule::$\text{'}\; =\; \backslash frac$There is no general method to integralintegrate the quotient of two functions.

See also -

Rational number

Vulgar fraction

Reciprocal

Inverse element

Division by two

Division by zero

Quasigroup

Group (mathematics)Group

Field (algebra)

Division algebra

Division ring

Long division

Vinculum

External links -

mathsisfun.com - Method for Dividing Decimals

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webhome.idirect.com - Division on a Japanese abacus selected from webhome.idirect.com - Abacus: Mystery of the Bead

webhome.idirect.com - Chinese Short Division Techniques on a Suan PanCategory:Elementary !arithmeticCategory:Arithmeticb g:Делениеda:Division? (matematik)de:Division (Mathematik)es:División !(matemáticas)fr:Divisionis:De ilingit:Divisione? !(matematica)nl:Delenja:除法p l:Dzieleniept:Divisãosimple:D ivisionsv:Division? !(matematik)th:การหา� ��zh:除法

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