Q.4. Express 0.99999…in the form $\frac{\mathrm{p}}{\mathrm{q}}$. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Solution:

Let x = 0.9999… (1)

10x = 9.9999… (2)

Subtracting (1) from (2), we get,

9x = 9

x = 1

As the number of digits increases in the decimal representation, the difference between 1 and the actual number decreases. For example difference between 1 and 0.999999 is 0.000001 which is negligible. Thus, 0.999… is very near 1.