Question

Find the value of 1.999.... in the form of p
q
where p and q are integers.

Answer 1

1.999… is exactly the number 2.

Let x = 1.999… . Then 1000x = 1999.999… . Subtract x from this to eliminate the fractional part, and you get

1000x - x = 1999.999… - 1.999…

1000x - x = 1999 - 1

999x = 1998

x = 1998/999 = 2/1 = 2

So p = 2 and q = 1.