Division Guide, Meaning , Facts, Information and Description
This article is about the arithmetic operation. For other uses, see Division (disambiguation).In mathematics, especially elementary arithmetic, division is an arithmetic operation which is the reverse operation of multiplication and sometimes can be interpreted as repeated subtraction.
Specifically, if
- a × b = c,
- a = c ÷ b
In the above expression, a is called the quotient, b the divisor and c the dividend.
The expression c ÷ b is also written "c/b" (read "c over b"), especially in higher mathematics (including applications to science and engineering) and in computer programming languages. This form is also often used as the final form of a fraction, without any implication that it needs to be evaluated further.
In most non-English languages, c ÷ b is written c : b. In English usage the colon is restricted to the related concept of ratios.
The meaning of division by zero is not usually defined.
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 three possible approaches.
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
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.
Dividing two complex numbers results in another complex number when the divisor is not 0, defined thus:
Division for complex numbers expressed in polar form is simpler and easier to remember than the definition above:
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. By a theorem 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 normed division algebra must be isomorphic to either the real numbers R, the complex numbers C, the quaternions H, or the octonions O.
The derivative of the quotient of two functions is given by the quotient rule:
Printable Worksheets for Practicing Division
This is an Article on Division. Page Contains Information, Facts Details or Explanation Guide About Division Division of integers
Division of rational numbers
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 complex numbers
All four quantities are real numbers. r and s may not both be 0.
Again all four quantities are real numbers. r may not be 0.Division in abstract algebra
Division and calculus
There is no general method to integrate the quotient of two functions.External links
See also: Rational number, Vulgar fraction, Reciprocal, Inverse element, Divisor, Division by two, Division by zero, Quasigroup, Group, Field (algebra), Division algebra, Division ring, Long division, Vinculum