Integers Chapter Notes | Mathematics for Class 5 PDF Download

Introduction

Natural Numbers: Natural numbers are the counting numbers, such as 1, 2, 3, 4, and so on, with no end to their sequence. 

Whole Numbers: Whole numbers encompass all the natural numbers along with zero, so they include 0, 1, 2, 3, and so forth. 

• If you visualize a plane mirror positioned at point 0 on a number line, perpendicular to it, the reflection of each number creates what we term mirror numbers. 
• These mirror numbers are represented with a negative sign, such as -3 or -5. We associate positive signs with the original numbers, like +2 or +3. 
• This convention helps in comprehending the concept of negative numbers and their relationship with positive numbers on the number line.

On the number line, integers consist of positive numbers, negative numbers, and zero. 

• Positive integers are those with a '+' sign, while negative integers have a '-' sign. Zero is also considered an integer, but it's neither positive nor negative. 
• It's important to note that positive integers are equivalent to counting numbers, but negative integers are not. 

• The mirror number corresponding to +4 is -4, and conversely, the mirror number corresponding to -4 is +4. 
• This is why +4 and -4 are referred to as opposites or opposite numbers. 
• Opposite numbers are located on opposite sides of zero on the number line and are equidistant from it.

Comparison of Integers

When comparing counting numbers, we follow the rule that the number to the right on the number line is greater than the number to the left. Since counting numbers are a subset of integers, this rule also applies to all integers. Therefore, for integers as well, the integer positioned to the right on the number line is greater than the one on the left.

Now, see the number line above from which we can make the following observations :

• +5 > +4, since +5 is to the right of +4 on the number line.
• +1 > 0, since +1 is to the right of 0 on the number line.
• 0 > –1, since 0 is to the right of –1 on the number line.
• –1 > –3, since –1 is to the right of –3 on the number line.

From the number line, we can have the following facts :

(i) 0 is less than every positive integer.
(ii) 0 is greater than every negative integer.
(iii) Every negative integer is less than every positive integer.

Arranging Integers in Ascending/ Descending Order

Look at the number line with all integers marked on it; O being the point centrally located, represents the number 0.
The above marking of the number line should not be taken as to mean that –4 and 4 are the end points. Arrows on both sides indicate that there are endless points on both the sides.
In ascending order, we write ... –4 < –3 < –2 < –1 < 0 < 1 < 2 < 3 < 4 < ...
In descending order, we have ... 4 > 3 > 2 > 1 > 0 > –1 > –2 > –3 > –4 > ...

Absolute Value of an Integer

The absolute value of an integer is its numerical value regardless of its sign.

We denote it by | | . To write the absolute value of an integer, we omit its sign.

For example, 
(a) absolute value of –2 = | –2| = 2
(b) absolute value of –7 = | –7| = 7
(c) absolute value of +3 = | +3| = 3
(d) absolute value of 0 = | 0 | = 0.

Addition of Integers

In the case of counting numbers, we noted that addition means, moving to the right on the number line. The same holds good in case of positive integers.
Following rules should be kept in mind while moving on the number line.
Adding a positive integer means moving to the right and adding a negative integer means moving to the left on the number line.

Adding Integers on a Number Line

We need to remember the following rules for adding integers on a number line.

• While adding a negative number we move towards the left side of the number line.
• While adding a positive number we move towards the right side of the number line.

Now let us consider an example in which we need to add negative and positive numbers using a number line.

Example: The temperature of a city was -4° C. It increased by 5º C. What is the temperature now?
Sol: The temperature of the city increased by 5°, so it became (-4 + 5 = 1°). Observe the number line given below which shows how we added a negative and a positive number. When we add a positive number on a number line we always move to the right, here we moved 5 steps to the right of -4 and we reached 1. This means -4 + 5 = 1.

It should be noted that when we add a negative number, we move towards the left side of the number line. Let us recollect all the rules for adding integers using the following number lines.

Subtraction of Integers 

We know that subtraction is an inverse process of addition.

So, to subtract an integer from another integer, we add the opposite of first integer to the second

integer.

Subtracting Integers on a Number Line

We need to remember the following rules for subtracting integers on a number line.

• After we change the subtraction fact to an addition fact, the operation changes to addition so we can follow the same rules of addition of integers on a number line.
• We should remember that while adding a negative number we move towards the left side of the number line and while adding a positive number we move towards the right side of the number line.

Let us understand this with the help of an example using a number line.

Example: Subtract -7 - (-4)
Sol:

Observe the number line given below to understand the steps.

• Since every subtraction fact can be written as an addition fact, we change the subtraction sign to an addition sign and reverse the sign of the subtrahend. Here, -7 - (-4) = -7 + 4
• While adding a positive number we move towards the right side of the number line. In this case, we will move to the right and reach -3