Division

Division is one of the four basic operations of arithmetic, alongside addition, subtraction, and multiplication. It answers the question: if a number is divided into equal parts, how much is each part, or how many parts result? For example, 12 divided by 4 equals 3.

Division is written in different ways: 12 ÷ 4, 12 / 4, or as a fraction with 12 over 4. The number being divided is called the dividend, the number it is divided by is the divisor, and the result is the quotient. If the division does not complete evenly, the amount left over is called the remainder.

Relationship to multiplication

Division is the inverse operation of multiplication. If a is divided by b and the result is c, then b multiplied by c gives a. For this reason, division is often verified by multiplication. Similarly, division can be defined as multiplication by the reciprocal of the divisor: a ÷ b is the same as a multiplied by 1/b.

Division by zero

Division by zero is undefined in standard number systems. If 5 is divided by 0, there is no number that, when multiplied by 0, gives 5, so no result exists. Even 0 ÷ 0 is indeterminate and is called an undefined form. However, 0 divided by any non-zero number is zero.

Methods of division

Schools teach long division, in which a number is divided one digit at a time starting from the left. Other methods include repeated subtraction and mental arithmetic for simple numbers. Computers and calculators use specialized algorithms that work in binary systems.

Properties of the operation

Division is not commutative: 12 ÷ 4 and 4 ÷ 12 do not give the same result. It is also not associative. When all numbers are divided, the result is not always a whole number; this is one reason the systems of fractions and rational numbers were developed.