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

By BYJU'S Exam Prep

Updated on: September 25th, 2023

To convert: 0.99999…. in the form of p/q

Let us assume, 0.99999…. = x (equation 1)

In the equation 1, 9 is the only repeating digit. Hence multiply 10 on both the sides that are LHS and RHS.

x = 0.99999….

10x = 10 x 0.99999….

10x = 9.9999… (equation 2)

Now, on subtracting equation 1 from equation 2 we will get

10x – x = (9.9999…) – (0.99999….)

9x = 9

x = 9/9

x = 1

As 0.999 goes on forever and there is no space between 1 and 0.9999, we can therefore state that 0.99999…. = 1.

Table of content

1. 0.99999…. in the p/q form
2. Express 0.99999…. in the form p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

0.99999…. in the p/q form

As discussed above 0.99999…. = 1. Now, let us understand by taking two numbers that are 0.9 and 1. We are aware that the numbers that will fall within this range are 0.9, 0.91, 0.92,…0.99, 1. 0.99 is closer to 1 in this case. Take the range from 0.99 to 1 now. In this range, the numbers 0.99, 0.991,…,0.999, and 1 will fall.

As 0.999 is closer to 1 than 0.999. Similar to this, the intervals are enlarged further until we reach the interval of 0.9999 to 1.
0.99991, 0.99992,…, 0..99999, and 1 are the numbers that will fall within this range.
Hence, we can see that 0.99999… always lies closer to 1 regardless of the number of intervals we take.
We can therefore state that 0.99999 = 1.

What is a Rational Number?

Any number with the form p/q, where p and q are integers and q is not equal to 0, is a rational number. Rational numbers include, among others, 1/3, 2/4, 1/5, 9/3, and so on. The number “0” is also a rational number because there are many ways to represent it, including 0/1, 0/2, 0/3, etc.

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