In mathematics, division is one of the four basic arithmetic operations (along with addition, subtraction, and multiplication). It is essentially the process of determining how many times one number, known as the divisor, is contained within another number, known as the dividend.

Formal Definition

Given two numbers, ( a ) (the dividend) and ( b ) (the divisor, where ( b \neq 0 )), the division of ( a ) by ( b ) is written as:

$$\frac{a}{b}$$

This expression represents the number ( x ) such that:

$$b \times x = a$$

In other words, division is the inverse operation of multiplication.

Alternative View

Division can also be understood in terms of multiplication by the multiplicative inverse (reciprocal). For any nonzero number ( b ), its reciprocal is ( \frac{1}{b} ). Therefore, dividing ( a ) by ( b ) is equivalent to multiplying ( a ) by ( \frac{1}{b} ):

$$\frac{a}{b} = a \times \frac{1}{b}$$

Key Points

• Dividend: The number being divided.
• Divisor: The number by which the dividend is divided.
• Quotient: The result of the division.
• Division by Zero: Division by zero is undefined because there is no number that can multiply by 0 to yield a nonzero dividend.

Example

Consider dividing 12 by 3:

$$\frac{12}{3} = 4$$

This is because:

$$3 \times 4 = 12$$

Division is a fundamental operation that allows us to partition or distribute quantities into equal parts and is essential for solving many types of mathematical problems.

Latex

\frac{a}{b} = a \times \frac{1}{b}