In its simplest form, division can be viewed either as a quotition or a partition. In terms of quotition, 20 ÷ 5 means the number of 5s that must be added to get 20. In terms of partition, 20 ÷ 5 means the size of each of 5 parts into which a set of size 20 is divided. For example, 20 apples divide into four groups of five apples, meaning that twenty divided by five is equal to four. This is denoted as 20 / 5 = 4, 20 ÷ 5 = 4, or 20/5 = 4. Notationally, the dividend is divided by the divisor to get a quotient. In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient.
Unlike the other basic operations, when dividing natural numbers there is sometimes a remainder that will not go evenly into the dividend; for example, 10 ÷ 3 leaves a remainder of 1, as 10 is not a multiple of 3. Sometimes this remainder is added to the quotient as a fractional part, so 10 ÷ 3 is equal to 3 1/3 or 3.33..., but in the context of integer division, where numbers have no fractional part, the remainder is kept separately or discarded. When the remainder is kept as a fraction, it leads to a rational number. The set of all rational numbers is created by every possible division using integers. In modern mathematical terms, this is known as extending the system.
Unlike multiplication and addition, Division is not commutative, meaning that a ÷ b is not always equal to b ÷ a. Division is also not, in general, associative, meaning that when dividing multiple times, the order of division can change the result. For example, (20 ÷ 5) ÷ 2 = 2, but 20 ÷ (5 ÷ 2) = 8 (where the use of parentheses indicates that the operations inside parentheses are performed before the operations outside parentheses).
Division is, however, distributive, in the sense that (a+b) ÷ c = (a ÷ c) + (b ÷ c) for every number. Specifically, division has the right-distributive property over addition and subtraction. That means:
This is the same as multiplication: . However, division is not left-distributive:
which is unlike the case in multiplication.
If there are multiple divisions in a row,the order of calculation traditionally goes from left to right, which is called left-associative: