5.36363636... = 5.36•• = 5 + 0.36••

(Sorry, I have no idea how to put two dots on top)

For two dots, we use 10 x 10 = 100 as a multiplier, as follows.

Let u = 0.36•• = 0.36363636...
Then 100u = 36.363636... (shifting the decimal place 2 units to the right) = 36.36••

So 100u - u = 36.36•• - 0.36•• = 36
99u = 36
u = 36/99 = 4/11
So 5.36•• = 5 + 4/11 = 59/11.

Note that although 36.3636... is lagging behind 0.363636... by 2 digits, we treat them to be the same for continuous, non-stopping digits. Hence, 36.3636... - 0.363636... = 36.

Thank you.

You can put 59/11 (sorry, typo previously) in calculator to verify this.