Division

Division (also written uqeybin) is one of the four basic operations of arithmetic, following addition, subtraction, and multiplication. It shows how many times one number called the divisor fits into another number called the dividend. The result is called the quotient, while what remains is called the remainder.

The common symbols for division are ÷, /, and the fraction bar (a horizontal line with numbers above and below). Example: 12 ÷ 4 = 3, meaning that 4 added together three times produces 12. In this way, division is the operation opposite to multiplication.

Terminology and notation

When written as a ÷ b = c, a is the number being divided, b is the divisor, and c is the quotient. This relationship can also be written as a = b × c. The fraction a/b is likewise a way to express division, with the top number called the numerator and the bottom number called the denominator.

Remainder and division of whole numbers

When whole numbers are divided, the result often cannot be a whole number. In such cases, division with remainder is used: 17 ÷ 5 = 3, remainder 2, because 17 = 5 × 3 + 2. The remainder is always less than the divisor. When the remainder is zero, the divisor is said to divide the number evenly, a concept fundamental to ideas like prime numbers and greatest common divisor.

Division by zero

Division by zero is not defined in ordinary arithmetic. If a ÷ 0 had a result c, then it would necessarily follow that a = 0 × c = 0, which is impossible when a is not zero; if a is zero, any result would work, making it impossible to determine a single answer. For this reason, mathematics treats it as an undefined operation.

Calculation methods and extension

The long division method taught in schools divides large numbers step by step by looking at one digit at a time. The concept of division has also been extended to fractions, decimals, real numbers, and complex numbers, and to algebraic structures such as fields, where every non-zero number has a multiplicative inverse.