Specifically, if c times b equals a, written:
In the above expression, a is called the dividend, b the divisor and c the quotient.
Conceptually, division describes two distinct but related settings. 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. Quotative division 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.
When division is taught to students in elementary school, which occurs following the teaching of multiplication, this usually results in the concept of non-integers being introduced to students. Unlike addition, subtraction, and multiplication, which always produce integers when the numbers previously involved were integers, division problems involving two or more integers do not always lead to one.
Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a vinculum or fraction bar, between them. For example, a divided by b is written
A typographical variation, which is 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 (although typically called the numerator and denominator), and there is no implication that the division needs to be evaluated further.
A second way to show division is to use the obelus (or division sign), common in arithmetic, in this manner:
Modern computers compute division by methods that are faster than long division: see Division (digital).
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.
In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus. We can calculate division by multiplication in such a case. This approach is useful in computers that do not have a fast division instruction.
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.
One has to be careful when performing division of integers in a computer program. Some programming languages, such as C, 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.
Names and symbols used for integer division include div, /, , and %. Definitions vary regarding integer division when the quotient is negative: rounding may be toward zero or toward minus infinity.
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. We may define division of two rational numbers p/q and r/s by
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 not 0. It is defined such a/b = c if and only if a = cb and b ≠ 0.
Division of any number by zero (where the divisor is zero) is not defined. This is because zero added to zero, no matter how many times the equation is repeated, will always result in a sum of zero. Entry of such an equation into most calculators will result in an error message being issued.
Dividing two complex numbers results in another complex number when the divisor is not 0, defined thus:
All four quantities are real numbers. r and s may not both be 0.
Division for complex numbers expressed in polar form is simpler than the definition above:
Again all four quantities are real numbers. r may not be 0.
Note that with left and right division defined this way, A/(BC) is in general not the same as (A/B)/C and nor is (AB)C the same as A(BC), but A/(BC) = (A/C)/B and (AB)C = B(AC).
In abstract algebras such as matrix algebras and quaternion algebras, fractions such as are typically defined as or where is presumed to be an invertible element (i.e. there exists a multiplicative inverse such that where is the multiplicative identity). In an integral domain where such elements may not exist, division can still be performed on equations of the form or by left or right cancellation, respectively. More generally "division" in the sense of "cancellation" can be done in any ring with the aforementioned cancellation properties. If such a ring is finite, then by an application of the pigeonhole principle, every nonzero element of the 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 normed division algebra must be isomorphic to either the real numbers R, the complex numbers C, the quaternions H, or the octonions O.
There is no general method to integrate the quotient of two functions.
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