Contents

**Division|Definition & Meaning**

**Definition**

The division is defined as the process of **distributing** a fixed **quantity of items** into **sub-quantities** in such a manner that **every sub-quantity** has an equal **number of items** in them. The term **quantity** could **be** anything like f**ruits, books, or arithmetic numbers**. Out of the **four essential arithmetic** operations of mathematics **Subtraction, Addition, and Multiplication**, **Division** is **one of them**.

**Concept Building**

Dividing simply means **making equal parts** or **dividing** into **equal amounts** among a group. An **equal-area triangle** consists of **two** equal** triangle**s **divided** by a **diagonal** in a **square**. **Depending** on the division operation, the **result** **may** be an** integer** **or not**. Decimal numbers may sometimes be used to represent the result.

Let us take an example of division. There are **50 students** and **5 classes**; we want to **divide** the **students among** the **5 classes.** What we will do is simply **split** the students into **5 batches** of **10 students** each, and send each batch to a **separate class**. Thus, we have assigned 50 students **equally** in 5 classes, with each class **containing 10 students**.

We can consider division as a **repetition process** of **successive subtraction** or the **opposite** of the **multiplication operation**.

**Symbol Representation**

There are generally **three different symbols** used for division shown above

**Divide ****($\boldsymbol\div$)**

Division of number 6 with 3 gives:

6 $\div$ 3 = 2

**Forward Slash ($\boldsymbol/$)**

Division of 4 books among 2 students:

4/2 = 2

**Bar (**$\boldsymbol{\frac{\phantom{a}}{\phantom{b}}}$**)**

Division of 8 apples among 4 people:

$\mathsf{\dfrac{8}{4}}$ = 2

**Mathematical Formula for Division**

$\mathsf{\dfrac{Dividend}{Divisor}}$ = Quotient + Remainder

**Dividend**

The dividend is the term or** item which we want to split** into different equal sub-parts. For example, if we **want** to **divide 10** oranges **among 5 people** then the 1**0 oranges** will be the **dividend**.

**Divisor**

The divisor is the **term** that **split** the **dividend** into **equal sub-parts** or in other words we can say the term with which we want to divide something. In the above example, **5 people** are the **divisor**.

**Quotient**

The** result** obtained by the **splitting** of the **two items** is known as the** quotient,** for example** dividing 10 oranges** among **5 people** gives a **quotient of 2**. In other words, the quotient is the term that decides the quantity of each subgroup.

**Remainder**

The remainder is the **residual result left** after the division operation for example division of **10 pencils** among **3 students** results in the **remainder** equal to **1**.

**Order of Division Operation**

In mathematics, Division is **highly prioritized** among other arithmetic operations. Division operation has the **first-order** meaning that whenever there is an arithmetic problem we will **first perform division operation**.

**Properties of Division**

- The
**division**of any number**by integer 1****gives**the**number itself**or in other words division of any number gives the dividend as the result. - The
**division**of any number**by integer 0****gives**infinity or more precisely**not defined**. - The
**division**of a number**by**the**number itself**gives the result as**1**or in other words, if the dividend and divisor are the same then the result will be equal to 1. - If the
**dividend**is**0**then the**division**operation**with**any**other number**gives**0**. - If we
**divide**any number**by 100**the**remainder**will be**equal**to the**number at**the**unit and tens position**and the**rest**of the number will be**considered quotient**.

**Dividing Fractions**

It is possible to divide fractions as well. **Division operators** need to be **transformed** into **multiplication operators** when dividing fractions. The following example shows the concept.

\[ \mathsf{ \frac{4}{3} \div \frac{4}{2} = \frac{4}{3} \times \frac{2}{4} = \frac{2}{3} } \]

In fraction, the** division sign** is **changed** into a **multiply sign** with the fraction being **reversed** as shown.

**Dividing Float Numbers**

Mathematics consists of a variety of concepts that divide decimal numbers. In many ways, the **decimal division** is **similar** to the **fractional division**. To understand how decimal division works, let’s look at the example below.

4.5 $\div$ 2.5 = $\mathsf{ \dfrac{45}{25} }$ = $\mathsf{\dfrac{9}{5}}$ = 1.8

**Dividing Polynomials**

It is possible to **divide polynomials** by other polynomials, as opposed to numbers and fractions. **Two methods** are available for **polynomial division,** however. As opposed to dividing numbers, the polynomial **long division** uses **polynomial expressions** to divide polynomial expressions. The **synthetic division method** is **another way** to divide polynomials.

**Steps for Division**

- First, figure out the
**data and define terms**such as divisor and dividend. - Write the
**dividend**inside the**division symbol hat**as shown below example. - Write the
**divisor outside**the**hat of the division**symbol as shown below example. - Now
**start dividing**the**dividend**by the**divisor**and**write**the**result**on the**top of the hat**. - This
**result**written**on the top**will be**considered**as the**quotient**. - If the
**dividend**is**not completely****divided**then**bring**the n**ext digit down**and**repeat**the process. **On repeating**the process if we got any**residual result left**at the end and further**dividend****cant be divided**then we will consider this**residual result**as the**remainder**.

**Examples of Division**

**Example 1**

**Consider** a **scenario** there are **51 players,** and we want to **make n teams** in such a way that **each team consists** of players **not more or less than 3. Describe** the **division process graphically** also. How many numbers of teams can be formed?

**Solution**

As from the problem statement, we are provided with the dividend as **51 players** and **divisor as 3,** so in order to find the n teams that can be formed, we will simply **divide 51 by 3,** which gives us **17**. So we **can make 17 teams** by s**plitting or dividing 51 players** into teams of **3 players each**. The calculation is shown below.

**Example 2**

**Harry bought** **16** Oranges, **4** Bananas, and **20** Apples for his family. The **number of** count of his **family member** is **4,** including him. How will you **divide all the fruits** **among family** members of Harry?

**Solution**

We illustrate the division process in figure 5 below.

**Division of Oranges**

Total oranges = 16

Number of family members = 4

Dividend = 16

Divisor = 4

Solving (leftmost in figure 5), we get:

**Quotient = 4**

**Division of Bananas**

Total bananas = 4

Number of family members = 4

Dividend = 4

Divisor = 4

Solving (middle in figure 5), we get:

**Quotient = 1**

**Division of Apples**

Total apples = 20

Number of family members = 4

Dividend = 20

Divisor = 4

Solving (rightmost in figure 5), we get:

**Quotient = 5**

So each family member got **4 oranges, 1 banana, **and** 5 apples**.

*All mathematical drawings and images were created with GeoGebra.*