In this article, you will learn about the Properties of Adding Integers, along with few examples. In mathematics, the whole numbers and negative numbers together are called integers. Don’t bother about the different Addition properties available in mathematics, because this platform makes it quite simple for you to find the sum.

Students can check out the various properties of addition like identity, commutative, associative, etc. in this article. Find the sum value with the help of these additional properties and learn the complete concept in no time.

Also, See: Properties of Subtracting Integers

### What are Integers?

Integers are defined as the set of all whole numbers they also include negative numbers. So, integers can be negative that is -6, -5, -4, -3, -2, -1 and positive integers are 1, 2, 3, 4, 5, 6 and even include 0. An integer will not be a fraction, a decimal, or a percent. The integer set is denoted by the symbol is ‘Z’. The set of integers will be defined as:

So, the integer Z is { -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

### Properties of Adding Integers

The following are the five properties of Adding Integers. The properties are:

1. Commutative Property

2. Associative Property

3. Additive Identity Property

4. Distributive Property

5. Closure Property

**1. Commutative Property:** Once we add two or more whole numbers, their sum is that the same no matter for the order of the addends. The Commutative Property is, a + b = b + c.

Example: 5 + 6 = 6 + 5 both the value will be same.

**2. Associative Property:** It refers to grouping. This rule can be applied for addition and multiplication. If the Associative property for addition and multiplication operation is carried out despite the order of how they are grouped, the result remains constant. The Associative Property is,

x+(y+z) = (x +y)+z.

Example: 2+( 6+(-9)) = (2 +6)+(-9)

**3. Additive Identity Property:** Identity Property states that when any zero is added to any number it will give the sum as the same given number. Zero is called Additive Identity. Adding 0 to a number doesn’t change the value of the number. This property is a + 0 = a.

Example : 12 + 0 = 12

**4. Distributive Property:** This Property means to divide the given operations on the numbers so as to order that the equation becomes easier to resolve. It states that ‘the Multiplication is distributed over the addition’. The distributive property is x *(y+z) = (x*y) + (x*z)

Example: 3(1+9) = 3 x 1 + 3 x 9

**5. Closure Property: **The Closure property in addition and subtraction states that the sum or difference of any two integers will always be an integer. If a and b are any two integers, a+b and a-b both will also be an integer.

Example: 3+1 = 4 and 4 +(-3) = 4 -3 = 1

### Properties of Addition of Integers with Examples

**Example 1: **

Find the value using Distributive Property of Adding 4, 2, 3 numbers?

**Solution:**

As given in the question, the values are 4, 2, 3

Now, we have to find the value using Distributive Property.

The Adding Distributive Property is A x (B + C) = A x B + A x C.

So, the values are A = 4, B = 2, C = 3

Substitute the values in above equation, then

A × (B + C) = 4 × (2 + 3) = 4 × (5) = 20.

A × B + A × C = (4 × 2) + (4× 3) = 8 + 12 = 20.

20 = 20.

Therefore, the value using distributive property is 21.

**Problem 2: **

Using Additive Identity Addition Property, find the value 12 + 0?

**Solution:**

Given the values, 12 and 0.

Now, find the value using Additive Property.

Additive Identity Property means adding 0 to a number does not change the value of a number.

Thus, the value is

12 + 0 = 12, and

0 + 12 = 12.

Both values are the same.

Hence, after performing addition the value is 12.

**Problem 3:**

Add 9 and 0 using Additive Identity Property of Addition.

**Solution: **

Given,

Using the Additive Identity Property, the finding value is,

9 + 0 = 9, and 0 + 9 = 9.

Thus, the final value is 9.

**Problem 4: **

What is the sum 12, 3, and 1 using Associative Property of Addition?

**Solution:**

The given numbers are 12, 3, and 1.

We know that, Associative Property of Addition is A + (B + C) = (A + B) + C.

Using the given values in above equation,

A = 12, B = 3, and C = 1.

A + (B + C) = 12 + (3 + 1) = 12 + (4) = 16.

(A + B) + C = (12 + 3) + 1 = (15) + 1 = 16.

16 = 16.

Hence, the sum value is 16.

**Problem 5: **

Add 32 and 21 using Commutative Property of Addition?

**Solution:**

Given the numbers are 32 and 21.

Now add the numbers using the Commutative Property.

So, the numbers are,

32 with 21 is, 32 + 21 = 53.

Add number 21 with 32 is 53.

The result is 53 for both 32+ 21 and 21+ 32.

Therefore, 32 + 21 = 21 + 32.

**Problem 6:**

Find the value 8 with -6 using Closure Property?

**Solution:**

As given in the question, the values are 8 and -6.

Using the Closure Property to find the value.

So, the value is,

8 +(-6) = 8 – 6 = 2

Hence, the value is 2.

**Problem 7:**

What is the sum value of 19 + (-23)?

**Solution:**

Given the values,

Now, we have to find the value using the closure property.

In this property, the set of integers is closed under addition.

So, the sum value is,

19 + (-23) = 19 – 23 = – 4.

Therefore, the sum value is -4. -4 is an integer.

### FAQ’s on Properties of Adding Integers

**1. What are the important properties of addition?**

There are 4 important properties of addition are

(i) Commutative property

(ii) Identity Property

(iii) Associative property

(iv) Distributive property

**2. What is the additive identity of 46?**

As per the Additive Identity Property of Addition, any number that is added to a unique real number (zero) gives the number itself as an output. Therefore, the additive identity of 46 + 0 is 46.

**3. What is the commutative property of addition?**

The commutative property of addition states that Even the order of addends is changed the sum will never change, the value remains the same.

**4. Write the difference between Commutative and Associative Properties of Adding Integers?**

In commutative property, the integers are often rearranged in any way, and therefore the result will still be similar. In the case of associative property, integers can be grouped in any way using parenthesis and the result will still be the same.

- Commutative Property of Adding integer can be a + b = b + a
- Associative Property of Adding Integer will be (a + b) + c = a + (b + c)

**5. Write the property using both addition and ****multiplication operations?**

The Distributive Property of Addition will be used in both addition and multiplication operations.