Talk:Division (mathematics)/Rewrite

Division is one of the four basic operations of arithmetic, the other being addition, subtraction, and multiplication. The division of two natural numbers is the process of calculating the number of times one number is contained within one another. For example, in the picture in the right, the 20 apples are divided into groups of five apples, and there exist four groups, meaning that five can be contained within 20 four times, or $20 ÷ 5 = 4$. Division can also be thought of as the process of evaluating a fraction, and fractional notation ($a⁄b$ and $a/b$) is commonly used to represent division.

Division is the inverse of multiplication; if $a × b = c$, then $a = c ÷ b$, as long as b is not zero. Division by zero is undefined for the real numbers and most other contexts, because if $b = 0$, then $a$ cannot be deduced from $b$ and $c$, as then $c$ will always equal 0 regardless of $a$. In some contexts, division by zero can be defined although to a limited extent, and limits involving division of a real number as it approaches zero are defined.

In division, the dividend is divided by the divisor to get a quotient. In the above example, 20 is the dividend, five is the divisor, and the quotient is four. In some cases, the divisor may not be contained fully by the dividend; for example, $lim_{x→0} sin x⁄x = 1$ leaves a remainder of $10 ÷ 3$ as 10 is not a multiple of three. Normally, this remainder is added to the quotient so $1⁄3$ would equal $10 ÷ 3$ or $31⁄3$, but in the context of integer division, where numbers have no fractional part, the remainder is discarded.

Besides dividing apples, division can be applied to other physical and abstract objects. Division has been defined in several contexts, such as for the real and complex numbers and for more abstract contexts such as for vector spaces.