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The operation inverse to multiplication: To find an such that or for given and . The result of the division is known as the quotient or the ratio between and ; is the divided, while is the divisor. The operation of division is denoted by a colon , a horizontal stroke or an oblique stroke .

In the field of rational numbers, division (except for division by zero) is always possible, and the result of a division is unique. In the ring of integers division is not always possible. Thus, 10 is divisible by 5, but is not divisible by 3. If the division of an integer by an integer in the field of rational numbers yields a quotient which is also an integer, one says that is totally divisible (divisible without remainder) by ; this is noted as . Division of complex numbers is defined by the formula

while division of the complex numbers in their trigonometric form is given by the formula

Division with remainder is actually a separate operation, which is different from division as defined above. If and are integers, then division with remainder of by consists of finding integers and such that

Here is the divided, is the divisor, is the quotient, and is the remainder. This operation is always possible and is unique. If , one says that divides without remainder. The quotient will then be the same as in ordinary division.

Division with remainder of polynomials with coefficients in a given field is defined in a similar manner. It consists in finding, for two given polynomials and , polynomials and satisfying the conditions

where the degree of is less than that of . This operation is also always possible and is unique. If , one says that is divisible by without remainder.


Division (with remainder) is related to the Euclidean algorithm.

Division of a complex number by a complex number amounts to multiplying by and dividing by , i.e.

Here, is the complex conjugate of and is the norm of (cf. Complex number).

How to Cite This Entry:
Division. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article