Operations of Integers Complete Guide to Rules and Concepts

We hope you are familiar with the number line given below. Traditionally, the number zero is placed in the centre of the number line. When we extend to the right of zero, we have positive numbers and the negative numbers extend to the left of zero. In order to add positive integers and negative integers, we will imagine that we move along a number line. We are going to discuss the addition and subtraction of integers on the number line, the addition and subtraction of integers on the number line.

What are Integers?

Integers refer to all numbers in Mathematics, which include positive numbers, negative numbers and zero but exclude all fractions. Operations on integers basically mean performing mathematical operations such as addition, subtraction, multiplication and division, i.e., basic arithmetic operations on these integers. 

What is a Number Line?

A number line in mathematics refers to a linear arrangement of numbers, where zero is placed at the centre. On the left side of zero lies all the negative integers, and on the right side of zero lies all the positive integers. 

1. Addition of Integers using Number Line

Let’s take a few examples for a better understanding of the addition of integers using number line -

For Example :

Example 1: 

If we are asked to add the numbers 4 and 3, we will start by moving to the number 4 present on the number line, exactly four units to the right of zero. Then we have to move three units to the right. Since we landed up seven units to the right of zero, we say that the sum of 3 and 4 is 7.

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Example 2: Suppose if we are asked to add the numbers 8 and -2, we would start by moving eight units to the right of zero and then we would move two units left from there as we know that negative numbers make us move to the left side of the number line. Since our last position is six units to the right of zero, we can say that the sum of 8 and -2 is 6.

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Here are the rules that you need to keep in mind in Addition of integers using number line-

Addition of Integers Using Number Line

2. Subtraction of Integers on Number Line

We will learn how to transform the subtraction problems into addition problems. Let’s discuss in detail the subtraction of Integers on a number line. Adding and subtracting integers using a number line are quite related.

The technique for changing subtraction problems into addition problems is quite simple. There are two steps you need to keep in mind:

Step 1) Change the subtraction sign in the given question into an addition sign.

Step 2) Take the opposite of the number that immediately follows the newly placed addition sign.

Now let's take a look at example number 1:

Example 1) 3 - 4

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According to the steps that we have discussed above, we have to change the subtraction sign to an addition sign in any question. We need to take the opposite of 4, which is -4. Therefore now the problem becomes:

3 + (-4) Now using the rules for addition, the answer we get is equal to -1.

 Here are a few other examples for better understanding:

Example 2) -2 – 7 =    -2 + (-7) = -9

Example 3) 6 - (-2) = 6 + 2 = 8

Example 4) -7 - (-2) = -7 + 2 = -5

Subtraction of Integers on Number Line

3. Multiplication of Integers on Number Line

It can be a little difficult to explain and show multiplication operations on a number line directly. This stands true for division as well. This is because multiplication usually leads to a value that is greater, while division yields a value that is usually lesser. Multiplication of integers also has certain rules associated with it. 

Now we have to understand the rules of multiplication. The first rule is very easy to remember because we have been learning it for a very long time. When we work with positive numbers under multiplication, it always yields positive answers. However, the last three rules are a bit more challenging to understand; here are the rules.

The second and third steps can be explained at the same time. This is because we can multiply numbers in any order. -7 x 3 has the same answer as 3 x -7, which is always true for all integers in Mathematics. (This property has a special name in mathematics which is known as the commutative property.) We can say thus that the second and third rules are equivalent.

4. Division of Integers on Number Line

The rules for division are exactly the same as those for the rules of multiplication. If you take the rules for multiplication and change the multiplication signs to division signs, then we would have an accurate set of rules for division.

Here are three examples given below:

  Example 1)   -27 ÷ 3 = -9

   Example 2)    24 ÷ (-4) = -6

   Example 3)   -21 ÷ (-3) = 7

Questions to be Solved

Question 1) Find the sum of the integers +25 and -5.

Solution) When we find the sum of integers +25 and -5 , we need to subtract the two numbers (25-5) = +20

Question 2) Divide the integers -27 and 3.

Solution) When we need to divide the two integers -27 and 3, on dividing we get the value as -8.

Rules of Integer Operations

From all the points noted above, it can be summarised that there are some rules when it comes to dealing with integers since it contains both positive and negative numbers. These rules are important and pivotal because all operations will fail if they are not applied properly. Thus, the rules can be summarised as: 

• The addition of two same-sign integers will yield a value in the same sign

• The addition of two different sign integers will yield the sign of the greater number

• Subtraction follows the same rules. 

• Both multiplication and division of two same-sign integers will yield a positive value.

• Both multiplication and division of two different sign integers will yield a negative value.