Question

explain brief on topic representing irrational numbers on number line?​

Answer 1

Answer:

HEY MATE ,

Step-by-step explanation:

. Here is the answer:

. Draw a number line and mark a point o , representing zero , on it . suppose point A represents 1 as shown in fig . 1.6 Then , OA = 1 .Now ,draw a right triangle OAB such that AB = OA = 1

. ( refer the attachment )

→ By Pythagoras theorem , we have

→      OB² = OA² + AB²

→     OB²  = 1² + 1²

→     OB² = $\sqrt2$

→ Now , draw a circle with centre O and radius OB . We find that the circle cuts the number line at A₁.

→ clearly , OA 1= OB = Radius of the circle = $\sqrt2$

→ Thus , A1 represents $\sqrt2$ on the number line .

→ But ,we have seen that  $\sqrt2$ is not an irrational number . Thus ,we find that there is a point on the Number which is not a rational number .

→ Now , draw a right triangle OA1 B1 such that A₁B1 = AB = 1

→ Again ,by pythagoras theorem , we have

→  OB₁² = OA ₁ ² + A₁B₁²

→ OB₁² = ( √2 ) ² + 1²

→ OB₁ = √ 3

→ Now , draw a circle with centre o  and radius OB₁ =  √3. This circle cuts → the number line at A₂ as shown in the fig 1.6.

clearly ,  OA₂ = OB₁ = √3

→ Thus ,A₂ represents √ 3 on the number line .

→ Also , we know that √ 3 is not an irrational number .

→ Thus ,A₂is apoint on the number line not representing a rational number .

→ Continuing in this manner we can show that there are many other points on the number line representing √ 5 , √ 6 , √7 , √ 8 etc, which are not rational numbers .In fact ,such numbers are irrational numbers .

→In the same manner , we can represent √n for any positive integer n , after  √ n - 1 has been irrational number.

→ it follows from the above discussion that there are points to represent irrational number on the number line .

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Thank you

Answer 2

Answer:

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