Well I just decided to refresh my math skills so started to reread an old Calculus book I have had for a while.

I came across an end of chapter problem that inquired whether

0.999999... < 1.0000000

It asked to represent 0.9999999... as a fraction of integers to arrive at an answer

So I did this

x = 0.9999999999

100 x = 99.9999999999

- x = - 0.9999999999

99x = 99

x = 1

The second part of the question referred to finding a real number between 0.9999999999 and 1.000000000

So I did this 1/2(0.9999999999+1)

=1/2(1.9999999999)

=0.9999999999

Why would I get 1 when I tried to represent 0.9999999999... ( is it a rational number with repeating digits) as a fraction of integers