Addition and Subtraction of Integers

Addition and subtraction are two very common operations performed with integers. There are some rules followed while performing them.

Addition of Integers

There are some rules followed while adding two or more integers. A number line is often used to show the sum of two integers.

Adding Integers with Same Sign

Adding 2 positive integers gives an integer with a positive sign

For example, (+5) + (+4) = +9

The addition of 2 negative integers gives an integer with a negative sign

For example, (-6) + (-4) = -10

Adding Integers with Different Signs

While adding a positive and a negative integer, first, we need to find the absolute values of the number and then subtract the smaller absolute value from the larger one, and finally, we add the sign of the addend with the largest value.

 For example,

(-9) + (+3) = -6

(+9) + (-3) = +6

Properties

The properties of addition of whole numbers also hold true for integers.

1. Closure Property: It states that the sum of any 2 integers also gives an integer.

For example, (+15) + (+7) = 22 is also an integer

2. Commutative Property: It states that the addition result remains even if the position of the numbers is interchanged. 

For example, (-15) + (+7) = (+7) + (- 15) = – 8

3. Associative Property: It states that adding 3 or more integers gives the same result regardless of how it is grouped.

For example, (-11) + ((- 9) + 7) = (-11 + (- 9)) + 7 = – 13

4. Additive Identity: When 0 is added to any integer, the answer is an integer. The additive identity is the integer zero.

For example, 0 + 15 = 15

5. Additive Inverse: When an integer is added to its negative inverse, the result is always 0. The two converse integers are termed additive inverses of one another. 

For example, 15 + (-15) = 0

Subtraction of Integers

There are 2 general steps to be followed while performing subtraction. They are:

1. Converting the subtraction sign into an addition sign
2. Taking the inverse of the number which comes after the sign and then performing the addition

The above steps are how subtracting integers is related to adding integers.

For example, let us subtract (-8) – (3)

Changing the sign, we get

=> (-8) + (3)

Taking the inverse of the number which comes after the sign

⇒ (-8) + (-3) (inverse of 3 is -3)

Adding and putting the sign of the greater number

⇒ (-8) + (-3) = -11

Like addition, the subtraction of integers also has 3 possibilities.

Subtracting Integers with Same Sign

Subtraction between 2 positive integers is a normal subtraction.

For example, (+8) – (+1) => 8 – 1 = 7

When subtracting between 2 negative integers, the subtraction sign and the negative sign of the second digit become a positive sign.

For example, (-2) – (-8) => (-2) + 8 = 6

Subtracting Integers with Different Signs

When subtracting a positive from a negative number, we add the two numbers and put a negative sign before it.

For example, (-2) – (+8) => -2 – 8 = -10

Subtracting a negative from a positive number is just a simple addition.

For example, (+2) – (-3) => 2 + 3 = 5

Properties

1. Closure Property: It states that the difference between any 2 given integers also results in an integer.

For example, (+12) – (+7) = 5 is an integer

2. Commutative Property: It states that the difference between any 2 given integers changes if the order of the integers is interchanged. So, it is not true for subtraction.

For example, (+12) – (+7) = 5 but (+7) – (+12) = -5

So, (+12) – (+7) ≠ (+7) – (+12)

3. Associative Property: It states that subtraction between 3 or more integers changes the result if the grouping of the integers is changed. So, it is not true for subtraction.

For example, ((+60) – (+20)) – (+30) = – 10 but [(+60) – ((+20) – (+30)] = 70

So, ((+60) – (+20)) – (+30) ≠ [(+60) – ((+20) – (+30)]