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 groups, how much does each group contain? Or: how many times does a smaller number fit into a larger one?

The standard symbols used are ÷ (obelus), / (slash, commonly used in computers), and the fraction form with the numerator written above the denominator. Examples: 12 ÷ 3 = 4, which can also be written as 12/3 = 4.

Parts of the operation

The division operation has three named parts. The number being divided is called the dividend. The number it is divided by is the divisor. The result is the quotient. When division does not finish evenly, the remaining number is called the remainder. For example, 13 ÷ 4 gives a quotient of 3 and a remainder of 1.

Relationship to multiplication

Division is the inverse operation of multiplication. If a ÷ b = c, then c × b = a. This relationship is used to verify the correctness of an answer. Therefore, dividing a non-zero number always yields a single answer that can be verified by multiplication.

Division by zero

Division by zero is undefined in ordinary arithmetic. The reason is that no number c can satisfy c × 0 equalling a non-zero number. If the dividend is also zero (0 ÷ 0), the answer is indeterminate and is left undefined. However, dividing zero by a non-zero number is possible: 0 ÷ 5 = 0.

Properties

Division is not commutative: 8 ÷ 2 does not equal 2 ÷ 8. It is also not associative. When whole numbers are divided, the result is not always a whole number; this led to the use of fractions and rational numbers. When negative numbers are divided, the sign follows a rule: two numbers with the same sign give a positive result, while different signs give a negative result.

Calculation methods

Schools teach a method called long division, in which large numbers are divided step by step, starting from the leftmost digits. There are also faster mental methods, such as breaking down the number. Mathematical tools and computers use internal algorithms that produce the same result.

Applications

Division is used in calculating ratios, percentages, rates (such as distance divided by time), unit prices, and averages. It is fundamental to fractions, proportion, and many areas of higher mathematics.