In this article, you will learn about the Properties of Subtracting Integers. Properties of Subtracting Integers for Students present will ensure regular practice with various problems on the concept and even enhance your mathematics fundamentals. Subtraction means removing things from a group, the sign of subtraction is ‘-‘(minus sign).

The other names of subtraction are minus, difference, deduct, less, decrease, and exclude. For subtracting larger whole numbers, you’ll use subtraction with regrouping (borrowing) or quick subtraction or subtraction or addition methods. Interested students can read further sections to know the five different properties of subtraction of integers along with solved examples.

### Integers – Definition

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 are defined as:

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

Subtracting integers means the Subtraction is an inverse process to that of addition.

### Properties of Subtracting Integers

The below are the five properties of Subtracting integers. The properties are as follows:

1. Commutative Property

2. Associative Property

3. Additive Identity Property

4. Distributive Property

5. Closure Property

**1. Commutative Property:** In Commutative law when any two numbers say a and b, in subtraction, gives the result as c, then if the position of these two numbers is interchanged it will affect the difference value. So, subtraction is not commutative for integers and whole numbers. The Commutative Property is, a – b ≠ b-c.

Example: 6 – 5 = 1

5 – 6 = – 1 both the value will be different.

**2. Associative Property:** Associative Property of subtracting integers is not there for the subtraction of whole numbers. We can’t group any two numbers and subtract them first. The order of subtraction plays an important role. The Associative Property is, a-(b-c) ≠ (a-b)-c.

Example: 3-(6-4) = 3 – (2) = 1

(3-6)-4 = (-3) – 4 = -7

Both the difference value will not equal.

**3. Additive Identity Property:** In this property, if we subtract the integers from zero, the result will be the same as the given whole number. When zero is subtracted from an entire number, the result’s undefined. Additive Identity Property is also known as Zero Property. So, the Zero Property is

a- 0 =a ≠ 0-a, 0 – a is undefined.

Example: 13 – 0 = 13

0- 13 = – 13 is undefined.

**4. Closure Property:** In subtraction, the closure property states that the difference between any two integers will always be an integer. If a and b are any two integers, a+b and a-b will also be an integer.

Example:15 – 5 = 10

## Properties of Subtracting Integers Examples

**Example 1:**

Subtract the value 17-19 using the Commutative Property?

**Solution:**

Given the values are 17, and 19

Now, using the commutative property we will find the final value.

In Commutative Property, a-b ≠ b- a

So, the values are,

17 -19 = -2, and

19- 17 = 2

Both the difference values are not the same.

Therefore, we conclude that subtraction is not Commutative for integers the final values are -2 and 2.

**Example 2:**

Find the value using the Associative Property. The value is a =8, b =2, c =5.

**Solution:**

As given in the question, the values are 8, 2, and 5.

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

We know that Associate property is a-(b-c) ≠ (a-b)-c.

Substitute the given values within the above equation, we get

8-(2-5) = 8-(-3) = 8 +3 = 11

(8-2)-5 = 6-5 = 1

Both the difference values are not equal.

Hence, the subtraction of integers is not Associative.

**Example 3:**

Write a Positive Integer and Negative Integer whose difference is 18.

**Solution:**

The given difference value is 18.

Let the positive integer and negative integer be 9, -9.

We subtract the smallest number from the biggest one.

So, the values are

9-(-9) = 9+9 = 18.

So, the positive integer is 9, and the negative integer is -9.

**Example 4:**

Suresh has taken 8 steps forward and 3 steps backward. How many steps have you taken?

**Solution:**

Given the data,

Suresh has taken forward steps is 8.

He has taken backward steps is -3

Now, find how many steps he has taken.

So, the steps are 8 – 3 = 5.

Therefore, Suresh has taken 5 steps.

**Example 5: **

What is the value of 25-0 using the additive identity property?

**Solution:**

Given the values are 25, and 0.

Now, we need to find the value using the additive identity property.

The additive identity property states that a-0=a ≠ 0-a

So, the values are,

25 – 0 = 25, and

0-25 = -25

The difference values are not same.

Hence, the values are 25, and – 25.

**Example 6:**

An elevator is on the Sixteenth floor. It goes down 7 floors and then up 3 floors. What floor is the elevator on now?

**Solution:**

As given in the question, the data

An elevator is on the floor is 16

It goes down floors is 7

It goes up floors is 3

So, the elevator is on the floor is, 16 – 7 + 3 = 12

Therefore, the elevator is on the 12th floor.

## FAQ’s on Subtracting Integers Properties

**1. What are the Properties of Integers?
**The integers have 5 main properties which are as follows:

- Closure Property
- Associative Property
- Commutative Property
- Distributive Property
- Identity Property

**2. List the four integer operations?
**The four integer operations are:

- Addition
- Subtraction
- Multiplication
- Division

**3. What are the different types of Integers?
**There are 3 types of Integers, namely:

1. Zero

2. Positive Integers

3. Negative Integers

**4. What’s the key point that you simply should remember** **while solving problems on the subtraction of integers?**

While solving problems on the subtraction of integers always we have to change the subtraction problem into an addition problem.

**5. What are the possibilities, that we come across while solving problems on the subtraction of integers?**

We have three different scenarios that we come across while subtracting the integers and that they are listed as under

- Subtracting two positive integers
- Subtracting two negative integers.
- Subtraction in between positive integers and negative integers.