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# Division – Explanation & Examples

Division is one of the four basic operations which distributes a number into equal parts. It is denoted by several symbols: the slash, the horizontal line, and the division sign. The horizontal line was introduced by Arabs and used by European mathematicians in the 13^{th} century. It was officially first used by a Swedish mathematician, Johann Rahn, in 1659.

## What is Division?

**The division is a mathematical technique where a number is shared into smaller groups or a technique of distributing quantities into equal parts.** It is normally one of the basic operations in arithmetic, which results in fair sharing.

The division is an inverse operation of multiplication. For example, the multiplication of 5 by 2 gives 10. Either of the factors 2 and 5 can be obtained by dividing 10 by any of the numbers.

## Parts of Division

**The Dividend**

In the division sentence, the dividend is the number that is to be divided. For example, in an expression: 12 ÷ 3 = 4 ^{1}/_{3}, the dividend is the number 12.

**The Divisor**

The divisor in the division sentence is the number that divides the dividend. For instance, in an equation: 12 ÷ 3 = 4 ^{1}/_{3,} the number 3 is the divisor.

**The Quotient**

The quotient is the number of times the divisor divides the dividend. In this 12 ÷ 3 = 4 ^{1}/_{3, }4 is the quotient.

**The Remainder**

The number which is left over after the operation of division is known as the remainder. For example, in 12 ÷ 3 = 4 ^{1}/_{3, }the number 1 is the remainder. It can be noted that the divisor is the denominator of the answer.

## Properties of Division

**Closure Property**

In division, the closure property states that the division of two whole numbers does not give a quotient a whole number. For example, in 10 ÷ 5, the quotient is a whole number, but for 5 ÷ 10, the quotient is not a whole number.

**Commutative Property**

The commutative property does not apply to the division of numbers. For instance, a ÷ b ≠ b ÷ a.

**Associative Property**

The Associative property does not apply to the division of numbers. In general, a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c

## How to Divide Numbers?

- When a number is divided by 1, the quotient is the number itself.

Example: 45 ÷ 1= 45.

- The quotient is 1 if a number is divided by itself.

Example: 5 ÷ 5 = 1

- In the division of any negative or positive number by zero, the result is always undefined. It is therefore meaningless to divide any number by 0.

Example: 2 ÷ 0 = Undefined

- The division of zero by any positive or negative number gives zero as the quotient.

Example: 0 ÷ 2 = 0

- The decimal point is moved to the left in the division of any number by another number in multiples of 10, 100, 1000, etc.

Example: 5 ÷ 10 = 0.5 and 5 ÷ 1000 = 0.005

- Positive number / Positive number = Positive quotient

Negative number / Negative number = Positive quotient

Negative number / Positive number = Negative quotient

Positive number / Negative number = Negative quotient