Divisions

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For example, " a divided by b" can written as:. A way to express division all on one line is to write the dividend or numerator , then a slash , then the divisor or denominator , as follows:.

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This is the usual way of specifying division in most computer programming languages , since it can easily be typed as a simple sequence of ASCII characters. 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 The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator.

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In some non- English -speaking countries colon is used to denote division. However, in English usage the colon is restricted to expressing the related concept of ratios. The history of this notation is not entirely clear because it evolved over time. Division is often introduced through the notion of "sharing out" a set of objects, for example a pile of lollies, into a number of equal portions.

Distributing the objects several at a time in each round of sharing to each portion leads to the idea of " chunking " — a form of division where one repeatedly subtracts multiples of the divisor from the dividend itself.

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By allowing one to subtract more multiples than what the partial remainder allows at a given stage, more flexible methods, such as the bidirectional variant of chunking, can be developed as well. More systematic and more efficient but also more formalised, more rule-based, and more removed from an overall holistic picture of what division is achieving , a person who knows the multiplication tables can divide two integers with pencil and paper using the method of short division , if the divisor is small, or long division , if the divisor is larger.

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Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition. Division or divider may refer to: Contents. 1 Mathematics; 2 Science; 3 Technology; 4 Society; 5 Places; 6 Music; 7 Other uses; 8 See also. Mathematics[ edit].

If the dividend has a 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. A person can calculate division 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.

A person can use logarithm tables to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result. A person can calculate division with a slide rule by aligning the divisor on the C scale with the dividend on the D scale. The quotient can be found on the D scale where it is aligned with the left index on the C scale. The user is responsible, however, for mentally keeping track of the decimal point. Modern computers compute division by methods that are faster than long division, with the more efficient ones relying on approximation techniques from numerical analysis.

For division with remainder , see Division algorithm.

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Administrative geography. Other countries Atlas. Find more information on your section using the Volunteer Leadership Directory. Meet IRIS. Germany on Saturday marks 30 years since the fall of the Berlin Wall that ushered in the end of communism and national reunification, as the Western alliance that helped secure those achievements is riddled with divisions.

In modular arithmetic modulo a prime number and for real numbers , nonzero numbers have a multiplicative inverse. In these cases, a division by x may be computed as the product by the multiplicative inverse of x. This approach is often associated with the faster methods in computer arithmetic. The Euclidean division is the mathematical formulation of the outcome of the usual process of division of integers.

Integers are not closed under division. Apart from division by zero being undefined, the quotient is not an integer unless the dividend is an integer multiple of the divisor. For example, 26 cannot be divided by 11 to give an integer. Such a case uses one of five approaches:. Dividing integers in a computer program requires special care. Some programming languages , such as C , treat integer division as in case 5 above, so the answer is an integer. Other languages, such as MATLAB and every computer algebra system return a rational number as the answer, as in case 3 above.

These languages also provide functions to get the results of the other cases, either directly or from the result of case 3. Divisibility rules can sometimes be used to quickly determine whether one integer divides exactly into another. The result of dividing two rational numbers is another rational number when the divisor is not 0. All four quantities are integers, and only p may be 0.

This definition ensures that division is the inverse operation of multiplication. Division of two real numbers results in another real number when the divisor is nonzero. Dividing two complex numbers when the divisor is nonzero results in another complex number, which is found using the conjugate of the denominator:. All four quantities p , q , r , s are real numbers, and r and s may not both be 0. Again all four quantities p , q , r , s are real numbers, and r may not be 0.

One can define the division operation for polynomials in one variable over a field. Then, as in the case of integers, one has a remainder. See Euclidean division of polynomials , and, for hand-written computation, polynomial long division or synthetic division. One can define a division operation for matrices.

An elementwise division can also be defined in terms of the Hadamard product. Examples include matrix algebras and quaternion algebras. A quasigroup is a structure in which division is always possible, even without an identity element and hence inverses. In an integral domain , where not every element need have an inverse, division by a cancellative element a can still be performed on elements of the form ab or ca by left or right cancellation, respectively.

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