20 ÷ 5 = 4, because 20 apples contain five apples four times. That is an example of division as quotition rather than as partition. One can also say that 20 ÷ 5 = 4 because when 20 apples are divided into 5 equal sets of apples, then there are 4 in each such set. That is division as partition rather than as quotition.
Division is one of the four basic operations of arithmetic, the others 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 another one.:7 For example, in the picture on the right, the 20 apples are divided into four groups of five apples, meaning that twenty divided by five gives four, or four is the result of division of twenty by five. This is denoted as 20 / 5 = 4, 20 ÷ 5 = 4, or 20/5 = 4.
Division can be viewed either as quotition or as partition. In quotition, 20 ÷ 5 means the number of 5s that must be added to get 20. In partition, 20 ÷ 5 means the size of each of 5 parts into which a set of size 20 is divided.
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,:246 because if b = 0, then a cannot be deduced from b and c, as then c will always equal zero 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.[a]
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 four is the quotient. In some cases, the divisor may not be contained fully by the dividend; for example, 10 ÷ 3 leaves a remainder of one, as 10 is not a multiple of three. Sometimes this remainder is added to the quotient as a fractional part, so 10 ÷ 3 is equal to 31/3 or 3.33 . . ., but in the context of integer division, where numbers have no fractional part, the remainder is kept separately or 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 and fields.
Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a fraction bar, between them. For example, a divided by b is written
This can be read out loud as "a divided by b", "a by b" or "a over b". A way to express division all on one line is to write the dividend (or numerator), then a slash, then the divisor (or denominator), like this:
A typographical variation halfway between these two forms uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor:
Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (typically called the numerator and denominator), and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus (or division sign), common in arithmetic, in this manner:
This form is infrequent except in elementary arithmetic. ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator. The obelus was introduced by Swiss mathematician Johann Rahn in 1659 in Teutsche Algebra.:211
In some non-English-speaking countries, "a divided by b" is written a : b. This notation was introduced by Gottfried Wilhelm Leibniz in his 1684 Acta eruditorum.:295 Leibniz disliked having separate symbols for ratio and division. However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b").
Since the 19th century US textbooks have used or to denote a divided by b, especially when discussing long division. The history of this notation is not entirely clear because it evolved over time.