Division (mathematics)

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,

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.

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.

Say that 26 cannot be divided by 10.
Give the answer as a decimal fraction or a mixed number, so 26 ÷ 10 = 2.6 or $26/10=2{\frac {1}{5}}$ . This is the approach usually taken in mathematics.
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 a/b and c/d by

{a/b \over c/d}=(a\times d)/(b\times c)

All four quantities are integers, and only a may be 0.

Division in abstract algebra

${a \over b}$ is typically defined as $a\cdot {1 \over b}$ or $a\cdot b^{-1}$ in abstract algebra like matrix algebra and quaternion algebra.

External links

Printable Worksheets for Practicing Division