# Mastering top 5 properties in Math

- Author: Noreen Niazi
- Last Updated on: January 25, 2024

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ToggleDo you ever use any properties and identities in Math? Mathematical identities and properties make the calculation simple. Whether we talk about the commutative property or deal with the distributive property, all properties have their impacts.

Let’s dive into details about the key properties in math and how to use them. Also, explore different mathematical identities and their use for making calculations simple.

## What are top 5 Properties in Math?

Number properties in math are some rules that make the calculation simple. So, they help us easily find the answer when dealing with mathematical operations.

There are five basic number properties.

**Commutative Property****Associative property****Distributive Property****Identity Property.****Inverse Property**

We use these properties to solve the problems of basic calculations. Algebraic operations use these properties, but not all algebraic operations satisfy them. For example, subtraction and division do not satisfy some properties of numbers.

Lets dive in details and discuss one by one about the properties of number.

## 1. Commutative Property

Commutative property means order does not matter. It means that when we talk about adding and multiplying a number, it does not matter that you place which number before and other number after the operations .

### Commutative Property for Addition :

Commutative property hold for the addition.

Let $$m$$ and $$n$$ be a positive number, then the commutative property for addition is:

$$m+n=n+m$$

**For example :**

**Prove that the numbers $$2$$ and $$7$$ satisfy the commutative property for addition.**

Now add $$2$$ and $$7$$ we get nine i.e. $$2+7=9.$$

Similarly, by adding 7 and 2 we get 9 i.e. $$7+2=9$$

Hence $$2+7=7+2$$ . Therefore 2 and 7 satisfy the commutative property of addition.

### Commutative Property for subtraction :

The commutative property does not hold for the subtraction.

**For example :**

**Lets the numbers 3 and 5.**

Now subtract $$3$$ and $$5$$ we get $$-2$$ i.e. $$3- 5=-2.$$

Similarly, by subtraction $$5$$ and $$3$$, we get $$2$$ i.e. $$5-3=2$$

Therefore $$2$$ and $$-2$$ are not equal, so subtraction does not satisfy the commutative property of subtraction.

### Commutative Property for Multiplication :

The commutative property also holds for the multiplication.

Let $$m$$ and $$n$$ be a number, then the commutative property for multiplication is:

$$m \times n=n \times m$$

**For example :**

**Prove that the numbers $$2$$ and $$4$$ satisfy the commutative property for multiplication.**

Now multiply $$2$$ and $$4$$ we get eight i.e. $$2\times 4=8.$$

Similarly, by multiplying $$4$$ and $$2$$, we get $$8$$ i.e. $$4\times 2=8$$

Hence $$2\times 4=4\times 2$$. Therefore $$2$$ and $$4$$ satisfy the commutative property of multiplication.

### Commutative Property for division:

The commutative property does not hold for the division.

**For example :**

**Let the numbers 4 and 8.**

Now divide $$8$$ by $$4$$ we get $$2$$ i.e. $$\frac{8}{4}=2.$$

Similarly, by dividing $$4$$ by $$8$$ i.e. $$\frac{4}{8}$$, we get $$\frac{1}{2}$$

Therefore $$2$$ and $$\frac{1}{2}$$ are unequal, so division is not commutative. .

## 2. Associative Property:

Associative property tells us that bracket does not matter. If three numbers satisfy the associative property, then it does not matter in which type of group they are made. Let’s discuss them in detail with example.

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### Associative property for addition

Lets $$a$$ , $$b$$ and $$c$$ are three numbers then associative property of addition is defined as

$$a+(b+c)=(a+b)+c$$.

For instance consider the number $$2,3$$ and $$4$$ then there sum can be written as

$$2+(3+4)=2+7=9$$

We can also write it as

$$(2+3)+4=5+4=9$$

Since the results of both the expressions are the same, the given number satisfies the associative property.

### Associative property for multiplication:

Lets $$a$$ , $$b$$ and $$c$$ are three numbers then associative property of multiplication is defined as

$$a\times (b\times c)=(a\times b)\times c$$.

For instance consider the number $$2,3$$ and $$4$$ then there sum can be written as

$$2\times (3\times 4)=2\times 12=24$$

We can also write it as

$$(2\times 3)\times 4=6\times 4=24$$

Since the results of both expressions are the same, the given number satisfies the associative property.

### Associative property for subtraction and divisions

The associative property does not hold for subtraction and divisions.

For instance consider the number $$2,3$$ and $$4$$ then there difference can be written as

$$2- (3-4)=2-(-1)=2+1=3$$

We can also write it as

$$(2-3)-4=-1-4=-5$$

Since the results of both expressions differ, the given number does not satisfy the associative property.

The associative property is also not valid for divisions.

Let’s take an example. 16, 8 and 4.

$$16 \div (8 \div 4)=16 \div 2 =8$$

$$(16 \div 8) \div 4=2 \div 4 =\frac{1}{2}$$

Since the result of both the expressions is not the same. Hence, the associative property does not hold in divisions.

## 3. Distributive property

The distributive property is a very important part of mathematics that helps to simplify expressions and also solve equations. This property is true for many arithmetic operations such as addition, multiplication, subtraction or also division.

### Distributive Property for Addition:

The distributive property for addition is thet given real numbers $$a, b$$ and $$c$$ then the expression

$$a(b + c) = ad + ac.$$

When distributing a number outside the parentheses among terms within them, one adds multiplication of that figure and each term separately.

Example 1:$\mathrm{}$

$$3(2+5)=3×2+3×5$$

$\mathrm{\$\$3\; \backslash times\; 7=6+15\$\$}\mathrm{}$

$$21=21$$

### Distributive Property for Multiplication:

The distributive property for multiplication is similar but involves multiplying a factor outside the parentheses with each term inside.

Example 2

$$4(3+6)=4×3+4×6$$

$$4 \times 9= 12+24$$

$\mathrm{\$\$36=36\$\$}$

### Distributive Property for Subtraction:

The distributive property for subtraction is similar to addition properties, but instead of addition, here is the minus sign. i.e.

$$a(b – c) = ad – ac.$$

**Example 3**

$$2(8−3)=2×8−2×3$$

$$ 2 \times 5= 16-6$$

$$10=10$$

### Distributive Property for Division:

he distributive property for division can be expressed as a fraction. For real numbers a, b, and c (where c is not zero), the expression $\frac{a}{b}(c+d)$ equals $\frac{a}{b}\times c+\frac{a}{b}\times d$.

**Example 4**

$$ \frac{9}{3}(6+2)= \frac{9}{3} \times 6 + \frac{9}{3} \times 2$$

$$3 \times 8 = 18+6$$

$$24=24$$

## 4. Identity Property:

Mathematics is the world of many properties and laws, each intended to serve a particular purpose in solving problems and understanding relations between different numbers. An example of a fundamental property is the Identity Property, which serves an important purpose in many mathematical operations

### Addition Identity Property

The Identity Property of Addition states that for any number ‘a,’ the sum of ‘a’ and zero is equal to ‘a’:

$$a+0=a$$

**Example:**

$$7+0=7$$

### Subtraction Identity Property

Even though subtraction does not have a unique identity property, it is closely related to addition. The idea here is that subtracting zero from any number results in the original number:

$$a−0=a$$

**Example:** $\mathrm{}$

$\mathrm{}$$$12−0=12$$

### Multiplication Identity Property

The Identity Property of Multiplication asserts that for any number ‘a,’ the product of ‘a’ and one is equal to ‘a’:

$$ a \times 1=a=1 \times a$$

**Example:**

$$ 5 \times 1=5=1 \times 5$$

### Division Identity Property

Similar to subtraction, division doesn’t have a distinct identity property. However, dividing any number by one results in the original number:

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

**Example:**

$$ \frac {9} {1}=9=\frac{1}{9}$$

## 5. Inverse Property

Math is a world of structures and relationships in which properties are essential to define what the numbers do during different operations. An important principle is the “Inverse Property” that plays a very crucial role in addition, multiplication, as well subtraction and division.

### Addition Inverse Property:

The Addition Inverse Property states that for any real number $a$, there exists a unique real number $-a$ such that $a+(-a)=0$. In simpler terms, adding a number to its additive inverse results in zero.

**Example:** Consider $a=7$; the additive inverse of 7 is $-7$. Thus, $7+(-7)=0$.

### Multiplication Inverse Property:

The Multiplication Inverse Property states that for any real number $c\mathrm{\ne}0$, there exists a unique real number $\frac{1}{c}$ (reciprocal) such that $c\cdot \frac{1}{c}=1$. In essence, multiplying a number by its multiplicative inverse yields 1.

**Example:**

Let $c=5$, the multiplicative inverse of 5 is $\frac{1}{5}$. Thus, $5\cdot \frac{1}{5}=1$.

## Practice questions on Properties in Math

**Commutative Property of Addition:**

Evaluate $5+8$ and $8+5$. Are the results the same?

If $m+n=17$, find the value of $n+m$.

**Commutative Property of Multiplication:**

Calculate $3\times 7$and $7\times 3$. Do you get the same answer?

If $p\times q=24$, what is $q\times \mathrm{p?}$

**Associative Property of Addition:**

Evaluate $(4+6)+8$ and $4+(6+8)$. Are the results equal?

If $a+(b+c)=21$, find the value of $(a+b)+c$.

**Associative Property of Multiplication:**

Calculate $(2\times 3)\times 5$ and $2\times (3\times 5)$. Is the product the same?

If $x\times (y\times z)=48$, determine the value of $(x\times y)\times z$.

Distributive Property:

Simplify $4\times (9+2)$ using the distributive property.

If $m\times (n=40$ and $m\times p=16$, find the value of $m\times n$.

## Conclusion:

The study of number property helps to build a strong base for the undertaking more difficult mathematical notions. Regardless of whether the operators are manipulating equations, simplifying expressions or solving real-world problems, knowledge in these properties is important. With this practice, not only will your mathematical skills improve but also you will find the pleasure of beauty and harmony in mathematics.

## FAQs on properties in math

The Commutative Property specifies that the order in which the numbers are arranged to be added or multiplied does not influence the resultant value obtained.

**For addition, $a+b=b+a$, and for multiplication, $a×b=b×a$.**

The Associative Property states that the organization of numbers in an addition or multiplication operation does not alter the answer.

For addition $(a+b)+c=a+(b+c)$, and for multiplication, $(a×b)×c=a×(b×c)$

Distributive Property links multiplication with addition and also establishes that

a×(b+c)=a×b+a×c

The Identity Property for addition holds that, for any number

a+0=a

where 0 is the additive zero. In multiplication, the identity is 1 and also a×1=a

The inverse Property asserts the existence of additive and multiplicative inverses. For addition, $a+(-a)=0$, and for multiplication ($a\mathrm{\ne}0$, $a\times \frac{1}{a}=1$.

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