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. Whole numbers, natural numbers, integers, rational, and irrational numbers are all 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. For large numbers, such as 100, we can use a larger scale, e.g., marking every 20 units as one step.
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 \( -\frac{7}{2} \), -2, 0, \( \frac{1}{2} \), 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: Negative numbers are always to the left of zero on the number line. 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.   

Using a number line effectively helps visualize positive, negative, fractional, and decimal numbers, making comparisons and calculations easier.
 

• Always start by marking zero as the reference point.
   
• Place positive numbers to the right and negative numbers to the left of zero.
   
• Ensure equal spacing between consecutive numbers for accuracy.
   
• Divide intervals properly to represent fractions and decimals.
   
• Use the number line to compare values and check positions.

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: 
 

• Meteorologists use number lines to compare temperatures. For example, if Los Angeles is at 17℃ and New York is at -7℃, the number line shows the temperature difference clearly.
   
• In sports, number lines help represent scores, positive scores to the right and negative to the left. For instance, if team A scores +3 goals and team B loses 2 points, their score is -2.
   
•  Banks use number lines to track savings and debts. For example, if a person has $300 and withdraws $500, their balance becomes -$200.
   
• In cooking, number lines help measure ingredients accurately. Fractions on the line can represent portions of flour, milk, or other ingredients.
   
• Scientists use number lines to show positions above and below sea level. A ship sailing at +10 m and a diver at -20 m can be easily compared.