Session 1: What are Integers? – Definition, Symbol, Number line, Absolute Value and Examples

Can you imagine how our life would be if we didn’t have any way to represent time, bank accounts, age and scores? When we first learnt to count, we relied on the ten mathematical digits (0 to 9) used to define all quantities. Later on, we learnt that there are numbers that are used to represent parts or portions of a whole. 

In the lesson “What are Whole Numbers?”, we learned whole numbers’ properties and operations. We have seen that if a smaller whole number is subtracted from a larger whole number, we get a whole number—for example, 10−6=4,42−16=26 and so on.

But, if a larger whole number is subtracted from a smaller whole number, we have no whole number to represent the difference—for example, 6−10,16−42 and so on.

Hence, we need to expand our knowledge to negative numbers to represent such differences.  

Positive and negative numbers show up everywhere in the world around us. We come across many situations where we use integers in the opposite sign. Let us see some cases where we use integers as directed numbers.

• Geographically, the altitude of a location is its distance above or below sea level. The altitude above sea level is represented with positive integers and the altitude below sea level with negative integers. Consider, Mount Everest which is 8849 m above sea level, so we say the altitude is +8849m. On the other hand, the Mariana Trench, the deepest part of the Earth’s oceans, lies 10,994 m below sea level, so we say the altitude is -10,994 m, a negative number.

[Figure 1]

• If a positive number represents the temperature above 0° C, then the temperature below 0° C is represented by a negative number. 
• If a positive number represents profit made by selling an article, then the loss incurred is represented by a negative number. 

What are Integers?

All natural and whole numbers belong to the set of numbers known as integers.

What are Natural Numbers?

Natural numbers are the numbers we use for counting, or enumerating items. The set of natural numbers is denoted by .Thus ={1,2,3,4,…}. The set of natural numbers is infinite.  

What are Whole Numbers?

The set of whole numbers is the set of natural numbers plus zero. The set of whole numbers is denoted by . Thus ={0,1,2,3,4,…}. 

Set of Integers

The set of integers adds the opposites of the natural numbers to the set of whole numbers: {⋯−3,−2,−1,0,1,2,3,…}. Integers are made of three distinct subsets: negative integers, zero and positive integers. The set of integers is denoted by  or . 

• Positive Integers: The numbers 1, 2, 3, 4, … i.e. natural numbers are called positive integers.
• Negative Integers: The numbers -1, -2, -3, -4, … are called negative integers.
• Zero: It is neither positive nor negative.

DID YOU KNOW? In 1742, Swedish astronomer Anders Celsius created a temperature scale that we use today for temperature measurement. He used 0° for the boiling point and 100° for the freezing point of water to avoid negative numbers. This was later inverted to put 0° on the cold end and 100° on the hot end, and it is in this form, it continues to be used today. Since 1948, it has been most commonly referred to as the Celsius scale in honour of its originator.

Representation of Integers on a Number Line

To represent whole numbers on the number line, we draw a line and choose an arbitrary point as 0. Any whole number corresponds to a unique position on the number line. 

Steps to Represent Integers on a Number Line

To represent integers on a number line, we follow the following steps:

Step 1: Draw a line and choose an arbitrary point as 0. 

Step 2: The length between two successive integers is called a unit length. Mark points at unit lengths on both sides of 0. 

Step 3: Label the points with negative numbers to the left of 0 and positive numbers to the right of 0. 

The arrows on each side of the number line show that the line continues indefinitely in both directions. 

Thus, every integer can be represented by some point on the number line. 

We observe the following about the number line. 

• There is no largest integer and no smallest integer. 
• An integer is greater than all those integers that lie to its left on the number line. 
• An integer is lesser than all those integers that lie to its right on the number line. 

Thus, a number line can help us see the relationship between integers. 

Note: If a number has no sign, it usually means a positive number.

The Pairing of Opposites on a Number Line

The figure below shows how each positive number matches a negative number, the same distance from zero. These pairs are called opposites. 

Note: The opposite of 0 is 0. 

Thus, two numbers at the same distance from zero but on opposite sides are called opposites. 

What is Absolute Value?

The absolute value of a number is the distance of that number from zero. There are always two numbers on the number line that are the same distance from zero. For instance, the numbers 4 and -4 are each a distance of 4 units away from zero.

The absolute value will never be negative because it represents the distance of a number from zero with no reference to direction. The absolute value symbol is written using a straight vertical line on either side of the number or expression.

|4| represents the distance from 4 to zero, which equals 4.

|−4| represents the distance from -4 to zero, which equals 4. 

Absolute value does not affect a positive number but changes a negative number into its positive inverse.

Note: While comparing two negative integers, the one with the smaller absolute value will be greater.

−7>−10⇒|−7|<|−10|⇒7<10

What are Integers? – Examples

Example 1

Write out each of the following situations as an integer.

i. The head of the West Mata volcano is located 4000 feet below sea level. 

ii. Kartik spent ₹1056 on Pokémon cards. 

iii. New Delhi experienced a heatwave of 40°C on July 1st.

iv. There was no gain or loss in the stock market for Tina. 

i. First, determine whether the location is positive or negative. 

The sea level is zero in this example. This means that “below sea level” is negative. 

Then, write it as a negative integer.

The answer is -4000. 

ii. First, determine whether Kartik gained or lost money. 

He “spent” it. That means he no longer has it. So it is a loss. 

Then, write it as a negative integer. 

The answer is -1056. 

iii. First, determine if the temperature is above or below zero. 

A “heatwave” means it’s very hot. If it were below zero, it would be very cold. That means it is positive. 

Then, write it as a positive integer. 

The answer is +40. 

iv. There is no loss or gain. Therefore, the integer used to describe it is zero, neither negative nor positive. 

Example 2

Write the opposite of each of the following situations. 

i. A loss of ₹378

ii. 890 feet above sea level

i. A loss of ₹378.

First, determine whether the given integer is positive or negative. 

“Loss” means negative. 

Then, write the original integer. 

It is -378. 

Finally, write the opposite. 

The answer is +378, i.e., a gain of ₹378.

ii. 890 feet above sea level

“Above sea level” means positive. 

Then, write the original integer. 

It is +890. 

The opposite of +890 is -890, i.e. 890 feet below sea level. 

Example 3

Using the number line, write the integer, which is:

i. 3 more than 1

ii. 4 more than -5

iii. 2 less than 1

iv. 3 less than -3

i. We want to obtain an integer 3 more than 1. So, we start from 1 and proceed with 3 units to the right to obtain 4. 

ii. Here, we want to know the integer 4 more than -5. So, we start from -5 on the number line and move through 4 units to its right to obtain -.1. 

iii. Here, we want to know the integer 2 less than 1. So, we start from 1 and proceed 2 units to its left to obtain -1. 

iv. Here, we want to know the integer 3 less than -3. So, we start from -3 on the number line and move through 3 units to its left to obtain -6. 

Example 4

Find the value of each of the following.

i. |−10|

First, determine what the distance is. 

In this case, it is 10. 

Then, remember that the distance is always a positive number. 

∴ |−10|=10.

ii. |8|

First, determine what the distance is. 

In this case, it is 8.

∴ |8|=8.

Remember this!

An integer is greater than all those integers that lie to its left on the number line.An integer is lesser than all those integers that lie to its right on the number line.0 is less than every positive integer and greater than every negative integer. Every positive integer is greater than every negative integer. Two integers that are at the same distance from 0, but on opposite sides of it are called opposite numbers.