Properties of Division

Division

By Alex David

April 6, 2025

Division is the fundamental operator of mathematics. Teaching division and properties of division is an essential part of primary mathematics education. Understanding the properties of division helps the students to develop a strong foundation in the field of logical reasoning.

Division has unique properties apart from other basic operators of arithmetic. These unique properties of division make the it a fascinating arithmetic operator to explore. Mentally, these properties of division help you to solve the basic problems in a few seconds and complex problems in quick time. Here is a look at what makes division special:

Basic Division Properties:

Here is the discussion of those properties of division that are primary applicable in division.

Division by itself

If a number (except zero) is divided by itself, the quotient is always 1.

Example:

$\frac{25}{25} = 1$

$\frac{10.5}{10.5} = 1$

Division of zero

If zero is divided by any number, then the result will always be zero. This is one of the most essential property among all properties of division.

Example:

$\frac{0}{100} = 0$

$\frac{0}{0.01} = 0$

If any number is divided by zero, then the result will always be undefined.

Example:

$\frac{5}{0} = \text{undefined}$

One of the Property of division that is division by 1.

Division by one

If a number is divided by 1, then the answer is always itself again.

Example:

$\frac{12}{1} = 1$

$\frac{4.5}{1} = 4.5$

Division of 10 and multiple of 10

If any number is divided by 10, then always the remainder will be the digit at the ones place in the dividend.

Example:

$\frac{252}{10}$

the quotient is 25 and the remainder is 2

If any number is divided by 100, then always the remainder will be the digits at the ones and tens places in the dividend.

Example:

$\frac{2521}{100} = 21$

the quotient is 25 and the remainder is 21

Mentally understand it and follow the trends to get more results about it. As shown, divided by 10 has digits at ones place, and divided by 100 has digits at ones and tens places. Keep going to follow the trend and get more results.

Division of whole number by whole number:

If a whole number is divided by another whole number, the quotient may not necessarily be a whole number.

Example:

$\frac{15}{2} = 7.5$

Use the ability of mental math and trends of mathematics and get the results. This concept extends to other types of numbers as well:

Fractions

Dividing a fraction by another fraction often results in another fraction.

Example

$\frac{1}{2}$ ÷ $\frac{1}{4}$ = 2

This works because division of fractions involves multiplying by the reciprocal.

Negative Numbers

Dividing negative numbers follows sign rules, and the result can be positive or negative depending on the signs of the numbers involved.

For example,

$\frac{-10}{-2} = 5$

Similarly, this concept becomes the root for many other concepts, like whether a dividend or divisor is negative, then the answer will always be negative.

Some Other Properties of Division

After the basic properties of division, there are some other properties of division that help you to solve even the complex problems in very quick time.

Scaling Property

It is the special case of division. If the dividend and divisor both are divided or multiplied by the same non-zero number, then the answer will remain unchanged. This property is also known as the division property of equality.

Example:

If $\frac{12}{6} = 2$

then, $12 \times 100 \div 6 \times 100 = 2$

Unit Fractions

This property will elaborate on the concept of multiplicative inverse. It is stated as:

“Division by a number is equivalent to multiplying by its reciprocal.”

Example:

12 ÷ 2 = 12 ÷ $\frac{1}{2}$

Relationship with Multiplication

This is also a special property from properties of division. This property works in the case of exact division (with no remainder), the divisor multiplied by the quotient is the dividend. This property holds true only if all three numbers are non-zero whole numbers.

Example:

$\frac{30}{5} = 6$ , then 5 × 6 = 30

Properties That Don’t Apply to Division

After the discussion of properties of division, it is necessary to explain which properties are not applicable in division.

Non-Commutative Property

Unlike other basic operations of arithmetic, changing the order in division completely changes the result. It’s pretty simple; division doesn’t follow commutative property.

Mathematical Form:

a ÷ b ≠ b ÷ a

Non-Associative Property

Division does not support associative property. It means grouping is not allowed in division.

Mathematical Form:

a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c

Non-Distributive Property

Division does not distribute over addition or subtraction.

Mathematical Form:

(a ± b) ÷ c ≠ (a ÷ c) ± (b ÷ c)

Identity Property

Unlike other operators of arithmetic, division does not have any identity element for all numbers. Dividing by 1 leaves the number unchanged, but dividing by any other number alters the result.

Inverse

Division by a number does not always have an inverse unless the divisor is nonzero. There is no direct inverse operation like in multiplication.

Conclusion:

Properties of division help you a lot to solve complex problems in a very short time. These properties mentally prepare you to think, remember, and act instantly. Mental math will help you a lot in this regard. Keep going to remember properties of division and follow the steps to make the process pretty simple.

Frequently Asked Questions (FAQs)

Q # 01: What is the definition of division in mathematics?

Division is the mathematical operation of splitting a quantity into equal parts. It is the inverse of multiplication and helps in determining how many times one number is contained within another.

Q # 02: How does division relate to multiplication?

Division and multiplication are inverse operations. If a × b =c, then dividing c by b gives back a.
For example:

6 × 4 = 24
24 ÷ 4 = 6

Q # 03: Why do we use division in real-world scenarios?

Division is widely used in:

Finance (splitting bills, calculating tax percentages)
Cooking (halving or adjusting recipes)
Science (average speed, density calculations)
Business (profit distribution, employee salary calculations)

Q # 04: Why is division by zero undefined?

When dividing by zero, there is no possible number that, when multiplied by the divisor, gives the dividend. Mathematically:

6 ÷ 0 = ?

No number multiplied by 0 can result in 6, so the operation is undefined.

Q # 05: Does division have an identity property like multiplication?

Yes, but with a limitation. In multiplication, the identity is 1 because a × 1=a. In division, dividing by 1 keeps the number unchanged, but division lacks a universal identity element for all numbers.

Q # 06: Why is division not commutative like multiplication?

In multiplication, changing the order doesn’t affect the result:
3 × 4 = 4 × 3

But in division, order matters:
12 ÷ 3 = 4 ≠ 3 ÷ 12

Q # 07: Why does dividing a number by 10 shift the decimal place?

Dividing by 10 moves all digits one place to the right because our number system is base 10.
Example:

250 ÷ 10=25.0
25.0 ÷ 10=2.5

Q # 08: How does division apply to algebra?

In algebra, division helps solve for unknowns:
If 5x = 20, dividing by 5 gives x = 4

It is also essential in working with fractions, rational expressions, and equations.

Q # 09: What are common mistakes people make while dividing numbers?

Some common errors include:

Misplacing the decimal point when dividing decimals
Dividing by zero incorrectly
Forgetting remainder rules in integer division

Q # 10: Can negative numbers be divided?

Yes. Division follows sign rules:

Positive ÷ Positive = Positive
Negative ÷ Negative = Positive
Positive ÷ Negative = Negative

Example:
−12 ÷ 4 = −3

Q # 11: What happens when we divide an irrational number?

Dividing an irrational number may result in another irrational number.

Example:

π÷2=1.57…(still irrational)

But sometimes, it results in a rational number:
2 ÷ 2 = 1

Q # 12: What are the best mental math tricks for quick division?

Divisibility rules help identify factors quickly.
Splitting numbers: Instead of dividing 96 by 4 directly, split as

(80 ÷ 4) + (16 ÷ 4) = 20 + 4 =24.

Rounding method: Approximate 198 ÷ 6 by using 200 ÷ 6, then adjust.

Q # 13: How can estimation help in division problems?

Estimation simplifies calculations by rounding numbers.

Example:

To solve 412÷8, round 412 to 400.
400÷8=50, so the actual answer is close to 50.

Q # 14: How does Vedic math simplify division?

Vedic math techniques like Nikhilam and Paravartya make division faster by using complements and special cases. These methods avoid long division steps, making mental division quicker.