Addition and Subtraction of Integers

In Mathematics, every concept involves numbers. In this article, you will be learning about how to add integers and how to subtract two integers. Addition and Subtraction operations come under the Arithmetics part of Mathematics. Arithmetic operations other than addition and subtraction are multiplication and division. When you have to find the sum of two variables, you apply the addition operation between the two numbers or variables. When the difference between the two numbers has to be found, then the subtraction operation is applied between the two numbers or variables. These arithmetic operations are applicable to not just integers, but also real numbers, whole numbers, rational numbers, etc. The table below shows what property is applicable to addition and subtraction. 


Property

Addition 

Subtraction

Closure

j + k ∈ Z

j - k ∈ Z

Commutative

j + k = k + j

j - k ≠ j - k

Associative

( j + k ) + l = j + ( k + l )

( j - k ) - l ≠  j - ( k - l )

Distributive

j x ( k + l ) = j x k + j x l

j x ( k - l ) = j x k - j x l 


Rules for Addition and Subtraction of Integers

Integers are a variety of numbers. They range from positive numbers to negative numbers and also includes the number zero. Integers are anything other than fractions.  The rules applicable for addition and subtraction are same for both natural numbers as well as integers because natural numbers are nothing but integers itself. Before we move forwards and learn the rules for addition and subtraction, let’s understand the positive and negative rules that are applied during addition and subtraction operations. 


Positive and Negative Rules

When two integers have the same sign or different signs, the product will have either a negative or positive sign. To understand it better, follow the table shown below.


Signs

Result

( + ) x ( + ) = ( + ) 

Positive x Positive = Positive

( + ) x ( - ) = ( - )

Positive x Negative = Negative

( - ) x ( + ) = ( - )

Negative x Positive = Negative

( - ) x ( - ) = ( + )

Negative x Negative = Positive


Rules

  • When a positive integer is multiplied by a positive integer the result is going to be positive.

  • When a negative integer is multiplied by a positive integer the result is going to be negative.

  • When a positive integer is multiplied by a negative integer the result is going to be positive.

  • When a negative integer is multiplied by a negative integer the result is going to be positive.

  • When there is no symbol present before the integer, it is considered to be positive. 


Addition of Integers

There are three possibilities when the addition operation is applied between two integers:

  • The addition operation is applied between two positive integers.

  • The addition operation is applied between negative positive integers.

  • The addition operation is applied between one positive integer and one negative integer.


Addition Rules for Integers

Types of Numbers

Operation Applied

Final Result

Example

( + ) + ( + )

Add

Positive ( + )

1449 + 1854 = 3303

( - ) + ( - ) 

Add 

Negative ( - ) 

(- 1449 ) + ( - 1854 ) = - 3303

( - ) + ( + )

Subtract

Positive ( + )

( - 1449 ) + ( 1854 ) = 405

( + ) + ( - )

Subtract

Negative ( - ) 

( 1449 ) + ( -  1854 ) = -405


Subtraction of Integers

There are three possibilities when the subtraction operation is applied between two integers:

  • The subtraction operation is applied between two positive integers.

  • The subtraction operation is applied between negative positive integers.

  • The subtraction operation is applied between one positive integer and one negative integer.


To Easily Solve Difference Problems, Follow the Steps Given Below:

Step 1: Convert the given negative sign to a positive sign. 

Step 2: After converting the negative sign, find the inverse of the upcoming number.


For example: find the difference between ( - 7654 ) and ( - 9342 )


Step 1: Convert the given negative sign to a positive sign ( - 7654 ) - ( - 9342 )


⇒ ( - 7654 ) + ( 9342 )


Step 2: After converting the negative sign, find the inverse of the upcoming number.


⇒ ( - 7654 ) + ( - 9342 ) ( opposite of 9 is - 9 )


⇒ ( - 7654 ) + ( - 9342 ) = -16,996 [ Perform addition operation ] 


Solved Problems

1. Evaluate the following: a ) 10 + 14, b) ( - 6 ) - ( 12 ), c) ( + 31 ) + ( - 21 )

Solution:

a ) 10 + 14 = 24 ( apply addition operation )


b ) ( - 6 ) - ( 12 ) = - 18 ( addition is performed two negatives numbers )


c) ( + 3 ) + ( - 2 ) = 10 


Step 1: Convert the given negative sign to a positive sign ( + 31 ) + (  - 21 )


⇒ ( + 31 ) - ( + 21 )


Step 2: After converting the negative sign, find the inverse of the upcoming number.


⇒ ( + 31 ) + ( - 21 ) ( opposite of 9 is - 9 )


⇒ ( + 31 ) + ( - 21 ) = 10 [ Perform addition operation ]

FAQ (Frequently Asked Questions)

1) What are the Positive and Negative Rules for Integers?

  • When a positive integer is multiplied by a positive integer the result is going to be positive.

  • When a negative integer is multiplied by a positive integer the result is going to be negative.

  • When a positive integer is multiplied by a negative integer the result is going to be positive.

  • When a negative integer is multiplied by a negative integer the result is going to be positive.

  • When there is no symbol present before the integer, it is considered to be positive.

2) What are the Rules for Addition and Subtraction Operations?

Types of Numbers

Operation Applied

Final Result

( + ) + ( + )

Add

Positive ( + )

( - ) + ( - ) 

Add 

Negative ( - ) 

( - ) + ( + )

Subtract

Positive ( + )

( + ) + ( - )

Subtract

Negative ( - )