The Division reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

# Division

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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,
where b is nonzero, then
a = c ÷ b
(read as "c divided by b"). So for instance, 6 ÷ 3 = 2 since 2 × 3 = 6.

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.

The meaning of division by zero is not usually defined.

 Table of contents 1 Division of integers 2 Division of rational numbers 3 Division of real numbers 4 Division of complex numbers 5 Division in abstract algebra 6 External links

## 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 three possible approaches.

1. Say that 26 cannot be divided by 10.
2. Give the answer as a decimal fraction or a mixed number, so 26 ÷ 10 = 2.6 or . This is the approach usually taken in mathematics.
3. Give the answer as a quotient and a remainder, so 26 ÷ 10 = 2 remainder 6. This approach is often used in computer science. In some computer integer arithmetic, 26/10 (or 26i / 10i) is given as 2 while 26 modulo 10 (or 26i % 10i) is given as 6.

## 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

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:

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:

Again all four quantities are real numbers. r may not be 0.

## Division in abstract algebra

```is typically defined as  or  in abstract algebra like matrix algebra and quaternion algebra.
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