Representation of Real Numbers on Number Line

The set of rational numbers Q and the set of irrational numbers Q’ together make up the set of real numbers. It is denoted as R. Subsets of whole numbers, natural numbers, integers, and rational and irrational numbers are all considered real numbers. Therefore, 
     R = Q ∪ Q’
 

On a number line, each number has a unique point known as a coordinate and has a distinct position. For instance, the real number 3 is positioned between 2 and 4. On a number line, two numbers cannot share the same position. The origin of a number line is at 0. On the right side of the origin are the positive numbers, while on the left side are the negative numbers.
 
The visual representation of a real number line is: 
 

Following the given steps will help us to easily indicate real numbers on a number line using graphs and coordinates. 

Step 1: Draw a straight line, mark the origin at 0, and draw arrows on both sides of the origin point.  

Step 2: Use a fixed scale for marking real numbers. Place real numbers on both sides of the origin at equal intervals. 

Step 3: Mark positive numbers to the right of the origin, and negative numbers to the left. 

Step 4: By identifying the correct positions of natural numbers, whole numbers, and integers, we can easily place them on a number line. If we have a large number, such as 100, then we can use a larger scale, for example, marking each unit 1 as unit 20.

Through this, we can reach 100 after taking 5 steps. 

Step 5: First, convert the rational or irrational numbers to a decimal form, to mark them on a number line.

Now we can mark real numbers such as -7/2, -2, 0, 1/2, and 3 on a number line. 
 

We use the number line to compare and arrange real numbers. Symbols such as greater than (>), less than (<), and equal to (=) are used to compare numbers.

Step 1: On a number line, the larger numbers are placed to the right and the smaller numbers are positioned to the left. 

Step 2: On the number line, negative numbers are always on the left side of the zero. The negative numbers closer to zero are considered greater. For example, -3 is greater than -13 because -3 is closer to the origin point than -13. 

Take a look at the given image to understand the comparison of real numbers on a number line.

 Here -3 > -13. 
 

The absolute value is the distance between a real number and the origin of the number line. It is denoted |x|, where x is the real number. Since distance is always positive, the absolute value will also be positive. For instance, 3 is a real number, then the absolute value will be |3| = 3. Take a look at the given image:


Here, the real number (3) is 3 units away from the origin. The distance between a negative number and the origin point is the same as its equivalent positive number. For example, |-3| = 3, here also -3 is 3 units far from the origin.   

The real number line plays an important role in various real-life situations by helping to compare and arrange numbers in order. The real-life applications of the real number line are as follows: 
 

• Weather forecasters and geologists use the real number line to represent the temperatures and their variations. For example, the temperature in Los Angeles is 17℃, and in New York it is -7℃. By using a number line, they can easily compare temperatures and find the temperature differences between these two cities. 
   
• In sports, scores are labeled on a number line, the positive scores of a team are marked on the right side, and the negative scores are on the left. For example, in a football match, team A scored + 3, which means they achieved 3 goals, whereas team B lost 2 points, bringing their score to -2.
   
• The savings and debts of an account holder can be represented using a number line. It helps the bank to understand the person’s balance and loans accurately. For example, if a person has $300 in his account and withdraws $500, now he has a balance of -$200. 
   
• When cooking, we can use a real number line to mark the quantity of items used for a recipe. For instance, to bake a cake we need 2/5 cups of flour and 3/4 cups of milk. These fractions can be easily marked on a number line which helps visualize and measure ingredients easily.