Question

Represent √6, √7, √8 on the number line.

Answer 1

Draw a number line and mark point O, representing zero, on it

Suppose point A represents 2 as shown in the figure Then OA = 2. Now, draw a right triangle OAB such that AB = 1. By Pythagoras theorem, we have 

 OB² = OA² + AB²

=> OB² = 2² + 1²

=> OB² = 4 + 1 = 5 => OB = √5

 Now, draw a circle with center O and radius OB. We find that the circle cuts the number line at A Clearly, OA₁ = OB = radius of circle = √5 Thus, A₁ represents √5 on the number line. But, we have seen that √5 is not a rational number. Thus we find that there is a point on the number which is not a rational number. Now, draw a right triangle  OA₁B₁ = AB = 1 ,Such that A₁B₁ = AB = 1  Again, by Pythagoras theorem, we have 

Draw a circle with center O and radius OB₁ = √6. This circle cuts the number line at A₂ as shown in figure Clearly  OA₂ = OB₁ = √6.Thus, A₂ represents √6on the number line. Also, we know that √6 is not a rational number. Thus, A₂ is a point on the number line not representing a rational number Continuing in this manner, we can represent √7 and √8 also on the number of lines as shown in the figure 

Thus,  OA₃ = OB₂  = √7 and  OA₄ OB₃ = √8