Express 0.99999 .... in the form of p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Solution:

Let x = 0.99999   ......(1)

Since one digit is repeated after the decimal, we will multiply both the sides of equation (1) by 10.

10x = 9.9999

10x = 9 + 0.9999

10x = 9 + x [From equation (1)]

10x - x = 9

9x = 9

x = 1

Hence, 0.99999 = 1

Let's think about it by visualisation.

Let's take two numbers 0.9 and 1. We know that the numbers falling in this interval will be 0.9, 0.91, 0.92,...0.99, 1. Here, 0.99 lies closer to 1.

Now, let's take the interval 0.99 to 1. The numbers falling in this interval will be 0.99, 0.991,...0.999, 1. We see that 0.999 lies closer to 1.

Similarly, this process of magnifying the intervals continues and we now reach the interval of 0.9999 to 1. The numbers falling in this interval will be 0.99991, 0.99992,...,0.99999, 1.

Thus, we see that no matter whatever the number of intervals we take, 0.99999... always lies closer to 1.

Hence, we can say that 0.99999 = 1 which is algebraically proven.

☛ Check: Class 9 NCERT Solutions Maths Chapter 1

Video Solution:

Express 0.99999 .... in the form of p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

NCERT Solutions Class 9 Maths Chapter 1 Exercise 1.3 Question 4

Summary:

0.99999... can be expressed as 1 in the p/q form. We have also used visualization to understand the same.

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