Division

Division in mathematics is the process of splitting a number into equal parts or groups. It is one of the four basic arithmetic operations and is the inverse of multiplication, helping us break a total into smaller, equal parts. In a division problem, the number being divided is called the dividend, the number you divide by is the divisor, and the result is the quotient, with any leftover amount known as the remainder. The division symbol (÷) is used to represent this operation.

The concept of division has existed for thousands of years, dating back to ancient civilizations such as the Egyptians and Babylonians around 2000 BCE, who used simple methods to divide quantities equally. Later, mathematicians such as Euclid studied division in terms of geometry and ratios, enriching its mathematical foundation. Division became more systematic with the invention of modern arithmetic symbols, including the division sign (÷) introduced by Johann Heinrich Rahn in 1659. Today, division is a fundamental skill taught worldwide and applied in countless real-life situations.

Division remains a core mathematical skill used in daily life and advanced studies. From solving basic problems to performing long division polynomial calculations in algebra, division continues to play a vital role in developing logical and analytical thinking in students.

The properties of division help us understand how division works and how it interacts with other mathematical operations. These include rules of division by 1, division by itself, and how division differs from multiplication in terms of commutativity and associativity.

 

 

Property 1: Traditionally, division has been seen as left-associative. In other words, the computation order is left to right if there are several divisions in a row.

a ÷ b ÷ c = (a ÷ b) ÷ c 

 

 

Property 2:  Any number divided by itself will always result in 1.

a ÷ a = 1 

 

 

Property 3: Any number divided by 1 will always result in the number itself. 

a ÷ 1 = a

 

 

Property 4: Any number divided by 0 will always result in a null value because there is no defined answer for a calculation such as this. For example, 
a ÷ 0 = undefined

 

 

 

There are many types of division applied in mathematics. Some include exact division and division with remainders. It also involves methods like long division and mental division techniques. Let’s understand them further.

 

 

Single-digit Division

 

Single-digit division is when you divide one number by another, and both numbers are single digits. That is numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9. Here is a step-by-step guide for solving single-digit division problems.

 

Step 1: Think of the multiplication table for 2

        2 × 1 = 2

        2 × 2 = 4

        2 × 3 = 6

        2 × 4 = 8

 

Step 2: Look for the number in the multiplication table that equals 8 (or comes close without going over it).

2 × 4 = 8

 

Step 3: Since 2 × 4 = 8, the answer is 4

Therefore, 8 ÷ 2 = 4

 

Step 4: To check, multiply your answer (4) by the divisor (2).

4 × 2 = 8

Which matches the original number.

 

 

Multi-digit Division 

 

Multi-digit division can feel a bit tricky at first, but you can do it step-by-step.

 

Step 1: Write the given multi-digit number. For example, let’s take 96 (the dividend) and 4 (the divisor)

 

Step 2: Look at the first digit of 96, which is 9
    
Ask yourself, how many times does 4 fit into 9 without going over?

The answer is 2, because 4 × 2 = 8 (and 4 × 3 = 12, which is too big)

Write 2 above the division bar, over the 9.

 

Step 3: Multiply the divisor (4) by the number you just wrote (2)

4 × 2 = 8

Subtract 8 from 9

9 – 8 = 1

Write the 1 below the 9

 

Step 4: Now, bring down the next digit of the dividend, which is 6.

This makes the number 16

 

Step 5: Ask yourself, how many times does 4 fit into 16?

The answer is 4, because 4 × 4 = 16.

Write 4 above the division bar, next to the 2.

 

Step 6: Multiply the divisor (4) by the number you just wrote (4)

4 × 4 = 16

Subtract 16 – 16 = 0

There is no remainder.

 

Step 7: The quotient (answer) is 24.

So, 96 ÷ 4 = 24

 

 

Dividing Decimals

 

Dividing decimals involves using the same steps as a regular division, but with an added focus on correctly placing the decimal point in the answer. To make it easier, you can shift the decimal point in both the divisor and dividend to work with whole numbers before dividing.

 

Step 1: Write the given decimal. For example, let’s take 3.6 as the dividend (inside the division bar) and 1.2 as the divisor (outside the bar). 

 

Step 2: Division is easier when the divisor is a whole number. To do this, move the decimal point in the divisor to the right.
        

1.2 becomes 12

Whatever you do to the divisor, you must also do to the dividend. So move the decimal point in 3.6 one place to the right, making it 36.

 

Step 3: Now, treat it like a normal long division. 

Ask, how many times do 12 fit into 36?

The answer is 3, because 12 × 3 = 36. Write 3 above the division bar.

 

Step 4: Multiply the divisor (12) by the number above the bar (3)

12 × 3 = 36

Subtract 36 – 36 = 0.

 

Step 5: Since you moved the decimal point in Step 2, the numbers are already whole, so the decimal in your answer is placed directly above its position in the dividend. 

In this case, the quotient is 3.0 or simply 3.

Final answer 3.6 ÷ 1.2 = 3

 

 

Dividing Fractions

 

Dividing fractions might sound tricky, but it’s simple when you know the steps. To divide fractions, you just flip the second fraction (called the divisor) and multiply it by the first one. 

 

Step 1: When dividing fractions, you flip the second fraction (called the divisor) and multiply. This is called “multiplying the reciprocal”.

 

Step 2: The first fraction stays the same ⅔

Flip the second fraction  (4/5) it becomes 5/4 

So, the problem now looks like this.

2/3 × 5/4

 

Step 3: Multiply the numerators (top numbers)

2 × 5 = 10

Multiply the denominators (bottom numbers)

3 × 4 = 12

Now you have 10/12

 

Step 4: Find the greatest common factor (GCF) of 10 and 12, which is 2.

Divide the numerator and denominator by 2

10 ÷ 2 = 5

12 ÷ 2 = 6

The simplified answer is ⅚

Final answer 2/3 ÷ 4/5 = 5/6

 

 

Dividing Negative Numbers

 

Dividing negative numbers follows a simple rule: if both numbers have the same sign, the result is positive, and if they have different signs, the result is negative. This rule helps make sense of how negative values interact when divided.

 

Step 1: When both numbers have the same sign (both positive and negative), the result is positive.

(-6) ÷ (-3) = +2

 

Step 2: When the numbers have different signs (one positive and one negative), the result is negative.

(-6) ÷ 3 = – 2

 

 

Dividing by Zero

 

Dividing by zero is a special math problem where you try to split something into zero parts, doesn’t make sense. In math, division by zero is not allowed because it leads to results that don’t work or make sense.

 

Step 1: Division is the process of splitting something into equal groups.

For example, 6 ÷ 3 = 2 means you can split 6 into 3 groups, and each group has 2. 

 

Step 2: Let’s imagine you want to divide 6 by 0 (6 ÷ 0)

This means, how many groups of 0 can you make out of 6?

But there is a problem.

You cannot create any groups with a size of 0 because a group of size 0 does not exist.

No matter how many times you try to divide something into groups of zero, you will never get a meaningful answer.

  

 Step 3: Division and multiplication are inverse operations. For example, 

If 6 ÷ 3 = 2, then 2 × 3 = 6

For 6 ÷ 0 = X, this means X ÷ 0 = 6

But no number multiplied by 0 equals 6 because anything multiplied by 0 is always 0.

 

Step 4: Since no number makes sense when dividing by 0, we say division by zero is undefined. 

 

 

Long Division

 

Long division is a method used to large numbers by breaking the problem into smaller, easier steps. It helps us find the quotient and remainder when dividing one number by another.

 

Step 1: Write the division as a long division problem. Put the dividend (752) inside the division bar and the divisor (4) outside the division bar.

 

Step 2: Look at the first digit of the dividend (7 in 752)

Ask yourself, how many times does 4 go into 7 without going over?

The answer is 1, because 4 × 1 = 4 (and 4 × 2 = 8, which is too big).

 

Step 3: Now, multiply the divisor (4) by the number you just wrote (1)

4 × 1 = 4
    

Write 4 under the 7 and subtract it from 7

7 – 4 = 3

Now you have a remainder of 3

 

Step 4: Now, bring down the next digit from the dividend, which is 5. This makes the number 35 (you now have 35 below the remainder 3).

 

Step 5: Ask yourself, how many times does 4 go into 35 without going over?

The answer is 8, because 4 × 8 = 32 (and 4 × 9 = 36, which is too big)

 

Step 6: Now, multiply the divisor (4) by the number you just wrote (8)

4 × 8 = 32

Write 32 under the 35 and subtract it from 35

35 – 32 = 3

 

Step 7: Now, bring down the last digit from the dividend, which is 2.

This makes the number 32 (you now have 32 below the remainder of 3)

 

Step 8: Ask yourself, how many times does 4 go into 32 without going over?

The answer is 8, because 4 × 8 = 32
    

Write 8 above the division bar, aligning it with the last digit of the dividend. In this case, next to the 18.

 

Step 9: Now, multiply the divisor (4) by the number you just wrote (8)

4 × 8 = 32

Write 32 under the 32 and subtract it from 32

32 – 32 = 0

Now, there is no remainder.

 

Step 10: The quotient (final answer) is 188.

52 ÷ 4 = 188

 

 

 

Division methods and techniques involve various ways to split numbers into equal parts, such as long division, short division, and using multiplication to check answers. These methods help break down complex division problems into simpler steps for easier understanding. Let’s look at the three methods of division.

 

 

Standard Long Division

 

Standard long division is the classic method of dividing a larger number (dividend) by a smaller number (divisor). The process involves dividing step by step, using multiplication and subtraction. 

 

Step 1: Write the dividend (153) inside the division symbol and the divisor (6) outside it.

 

Step 2: Start by dividing the first digit of the dividend (1) by the divisor (6).
 

Ask yourself: How many times does 6 fit into 1?

The answer is 0 because 6 is larger than 1. So, we don’t write any number yet. 

Since 6 doesn’t fit into 1, we move on to the next digits of the dividend. Combine the first two digits of 153 (which is 15). 

 

Step 3: Now, divide 15 by 6

Ask yourself, how many times does 6 fit into 15?

The answer is 2, because \(6 × 2 = 12\) (and \(6 × 3 = 18\), which is too large)

Write 2 above the division bar over the second digit of the dividend (which is 5)

 

Step 4: Multiply 6 (divisor) by 2 (the quotient you just found)

\(6 × 2 = 12\)

Write 12 under the 15

Subtract \(15 – 12 = 3\). Write the remainder 3 below the 12

 

Step 5: Now, bring down the next digit from the dividend, which is 3. This makes the new number 33.

 

Step 6: Divide 33 by 6

Ask yourself, how many times does 6 fit into 33?

The answer is 5, because \(6 × 5 = 30\) (and \(6 × 6 = 36\), which is too large)

Write 5 above the division bar next to the 2

 

Step 7: Multiply 6 by 5 (the quotient you just found)
\(6 × 5 = 30\)

Write 30 under the 33

Subtract \(33 – 30 = 3\). Write the remainder 3 below.

 

Step 8: Now, that you’ve finished the division process and there are no more digits to bring down, we have a remainder of 3.

 

Step 9: The quotient is 25 and the remainder is 3

So, \(153 ÷ 6 = 25\), remainder 3

 

Short Division
 

The short division is a concise and efficient method for dividing numbers, particularly when the divisor is a single-digit number. It simplifies the division process by performing calculations directly above the dividend, making it ideal for quick mental or written calculations.

Step 1: Write the dividend (the number you are dividing) under the division bar. Write the divisor (the number you are dividing by) outside the division bar. Let’s take 175 by 5 for example.

 

Step 2: Look at the first digit of the dividend (1)

Determine how many times 5 can fit into 1. It fits 0 times.

Write 0 above the division bar.

 

Step 3: Multiply 0 by 5 to get 0.

Subtract 0 from 1 to get 1
 

 

Step 4: Bring down the next digit (7) to make it 17

 

Step 5: Determine how many times 5 can fit into 17. It fits 3 times

Write 3 above the division bar

 

Step 6: Multiply 3 by 5 to get 15

Subtract 15 from 17 to get 2.

 

Step 7: Bring down the next digit (5) to make it 25

 

Step 8: Determine how many times 5 can fit into 25. It fits 5 times

Write 5 above the division bar

 

Step 9: Multiply 5 by 5 to get 25

Subtract 25 from 25 to get 0

 

Step 10: The quotient is 35 and there is no remainder.
 

 

 

Chunking Method

 

The chunking method is an alternative division technique that involves repeatedly subtracting “chunks” of the divisor from the dividend. This method is particularly useful for visual learners and helps build a deeper understanding of division by breaking the problem into manageable parts.

 

Step 1: Write the dividend (the number you are dividing) and the divisor (the number you are dividing by). Let’s consider 175 by 5.

 

Step 2: Estimate how many times the divisor can fit into the dividend. Choose a manageable number (a “chunk”) that is easy to work with, such as a multiple of the divisor.

\( 30 × 5 = 150 \)

 

Step 3: Multiply the chosen chunk by the divisor 

Subtract 150 from 175

\(175 – 150 = 25\)

 

Step 4: Estimate another chunk. Let’s use 5 chunks of 5. 

\(5 × 5 = 25\)

Subtract 25 from 25

\(25 – 25 = 0\)

Add up all the chunks \(30 + 5 = 35\)

 

Step 5: The quotient is 35

The remainder is 0

So, 175 divided by 5 is 35 with no remainder.

 

 

Division with Remainders

 

Division with remainders is a fundamental arithmetic operation where a number (dividend) is divided by another number (divisor), resulting in a quotient and a remainder. The remainder is the part of the dividend that is left over after the division process, indicating that the divisor does not fit evenly into the dividend. 

 

Step 1: Write the dividend inside the division bracket (or long division symbol). Write the divisor outside the division bracket. 

 \(   23 ÷ 5 \)

 

Step 2: Find how many times the divisor fits into the dividend without exceeding it.

    \(5 × 4 = 20\), which is the closest without exceeding 23.

 

Step 3: Multiply \(4 × 5 = 20\)

 

Step 4: Subtract \(23 – 20 = 3\)

 

Step 5: The remainder is 3 because 3 is smaller than the divisor

 

Final answer \(23 ÷ 5 = 4\) remainder 3

Division is an essential math skill that helps students understand how to break down larger numbers into smaller, more manageable parts, making it easier to solve problems and understand real-life situations like sharing or grouping. Mastering division also builds a strong foundation for more advanced math concepts. 
 

There are many tips and tricks we use to master division, here are a few tips and tricks to make division easier.

 

Master Multiplication Facts: Students can solve division problems if they have a solid understanding of the multiplication table. For example, if the students know that \(7 × 8 = 56\), then they can know that \(56 ÷ 8 = 7.\)

Break Down Large Numbers: Breaking down the number into two or three parts and then dividing them separately might be easier. This is useful, especially for larger numbers. For example, dividing 156 by 3, students can break down 156 into 150 and 6 and then divide both by 3: \(150 ÷ 3 = 50\), and \(6 ÷ 3 = 2\). So, if the students group the answers, it would be 52. Hence, \(156 ÷ 3 = 52\).

Practice Long Division: If the students want to learn division faster and more efficiently, they can practice the long division method. If the students have regular practice in solving long-division method problems, it will improve their speed and accuracy.  

 

Estimate First: Before dividing, round numbers to the nearest tens or hundreds to estimate the quotient. This helps students know if their final answer is reasonable.

 

Check with Multiplication: After dividing, multiply the quotient by the divisor. If the result equals the dividend, the answer is correct.

Teach Division as Repeated Subtraction
Explain that division is repeated subtraction. This helps the students to grasp the concept intuitively before moving on to formal methods such as the long division steps.

Break Down Long Division Steps Clearly
Guide children through each part of long division, divide, multiply, subtract, and bring down. Repeating these long division steps with small numbers first builds confidence and accuracy.

Introduce Synthetic Division for Advanced Learners
Once students are comfortable with numerical division, introduce synthetic division as a shortcut for dividing polynomials. This approach simplifies the calculations and is especially useful in higher grades.

Connect to Polynomial Division
Teachers can show how polynomial division extends the same principles of long division to algebraic expressions. Using examples with variables helps students see the link between arithmetic and algebra.

 

There are a lot of real-world applications of division. Some real-world applications are given below:

 

Everyday Life: We use division in our daily tasks like dividing a pizza among friends, we use it in cooking to divide the ingredients, we use it in shopping to find the discounts on particular prices and items.

 

Business and Finance: We use division in business and finance, as well as to calculate profit margins, to determine the average stock levels, to calculate the sales per employee, and to calculate the return on investments.

 

Science and Engineering: We use division in Science and Engineering like, in physics we use it to calculate the speed, acceleration, and density; it is also used in chemistry to determine the concentration of solutions; in engineering, it is used to design structures, calculate the load and stresses; in computer science, it is used to divide the data into packets for transmission, allocating memory and developing algorithms.

 

Healthcare: We use division in healthcare as well, like to calculate the amounts of medications based on the patient's weight, it is also used to analyze the data of the patients to identify patterns.

 

Education: In education, we use division to calculate the average scores, dividing the marks earned by the number of assignments submitted; it is also used to divide the classroom among the students; and is used to solve problems related to division.