Representing Real Numbers on the Number Line | Advance Learner Course: Mathematics (Maths) Class 8 PDF Download

• Any real number has a decimal expansion. There are many decimal numbers or real numbers present between two integers. 
• Every real number is represented by a unique point on the number line. 
• Also every point on a number line represents one & only one real number

“The process of visualization of representation of decimal number on number line through a magnifying glass, is known as successive magnification”

Example 1
Visualise the representation of 4.36% on the number line upto 4 decimal places.

Here, we can understand representation of 4.36% on the number line upto 4 decimal places with the help of following steps:

Step 1

Here, we know that, the number 4.36% lies between 4 and 5. Hence, first draw the number line and look at the portion between 4  and 5 by a magnifying glass.     

Step 2

Divide the above part into ten equal parts and mark first point to the right of 4 as 4.1, the second as 4.2 and so on.    

Step 3

Now 4.36 lies between 4.3 and 4.4. So, divide this portion again into ten equal parts and mark first point to the right of 4.3 as 4.31, second 4.32 and so on.     

Step 4 

Now, 4.366 lies between 4.36 and 4.37. So, divide this portion again into ten equal parts and mark first point to the right of 4.36 as 4.361, second 4.362 and so on.     

Step 5

To visualize 4.36 more accurately, again divide the portion between 4.366 and 4.367 into 10 equal parts and visualize the representation of 4.36% as in the figure given below
We can proceed endlessly in this manner.
Thus, 4.3666 is the 6th mark in this subdivision.         Example 2 

1. Visualise 2.565 on the number line, using successive magnification.      
   We know that, 2.565 lies between 2 and 3.
2. So, divide the part of the number line between 2 and 3 into 10 equal parts and look at the portion between 2.5 and 2.6 through a magnifying glass.    
3. Now, 2.565 lies 2.5 and 2.6 hence first draw the number line and look at the portion between 2.5 and 2.6 by a magnifying glass.          
4. Now, we imagine and divide this again into 10 equal parts. The first mark will represent 2.51, the next 2.52 and so on. To see this clearly we magnify this as shown in the following figure,         
5. Again 2.565 lies between 2.56 and 2.57 so, now focus on this portion of the number line and imagine to divide it again into 10 equal parts as shown in the following figure       

This process is called visualization of representation of number on the number line through a magnifying glass.
Thus, we can visualise that 2.561 is the first mark and 2.565 is the fifth mark in these subdivision.

Practice Questions: 

Represent √(9.3) on the number line. 

Step 1: Draw a line and mark a point A on it.
Step 2: Mark a point B on the line drawn such that AB = 9.3cm.
Step 3: Mark a point C on AB produced such that BC = 1 cm.
Step 4: Find mid-point of AC (9.3+1). Let the mid-point be O.
Step 5: Taking O as center and OC = OA [10.3/2] as radius draw a semi-circle. Also draw a line passing through B perpendicular to OB. Let it cut the semi-circle at D
Step 6: Taking B as the center and BD as radius draw an arc cutting OC produced at E.Point E so obtained represents √(9.3). 

 
Visualize 3.765 on the number line using successive magnification. 

Step 1: Since the given number lies between 3 and 4, look at the portion of the number line between 3 and 4.
Step 2: Divide the portion between 3 and 4 into 10 equal parts and mark each point of the division as shown in Fig. 1.
Step 3: The 7th mark and 8th mark of this sub-division corresponds to 3.7 and 3.8 respectively and 3.765 lies between them (3.7 < 3.765 < 3.8)
Step 4: Again divide the portion between 3.7 and 3.8 into 10 equal parts. Now 3.765 lies between its 6th and 7th mark (3.76 < 3.765 < 3.77).
Step 5: Divide the portion between 3.76 and 3.77 again into ten equal parts. Therefore, the 5th mark of this sub-division mark represents 3.765 as shown in fig.3