Then yes, you need to do the division by hand. Floating point (scientific notation) isn't going to be helpful. I expect you can find libraries for extended-precision computation, but if you're serious about pi, you're aiming for thousands of digits, not just 20 digits instead of 10.

My understanding is that nobody implements infinite-precision real numbers (analogous to infinite-precision integers) because even simple computations such as 1÷3 or 1÷10 require infinitely many bits to get a precise answer.

extract the right part of the final number (after the dot)

extract the left part of the final number (before the dot) and add the remember

assemble the final digit

An example:
1.1 + 22.22 + 333.999

Normalization
what is normalization (i invent this...):
1.1 + 22.22 + 333.999
become 001.100 + 022.220 + 333.999
(same amount of digits before and after for all numbers

number of digits after the dot: 3

add the digits after the dot of all numbers
100 + 220 + 999 = 1319

find the remember
at step 2 : number of digits = 3
i discard last 3 digits of 1319: remmber = 1

add the digits before the dot os all numbers
001 + 022 + 333 = 356

extract the right part of the final number (after the dot)
at step 2 : number of digits = 3
at step 3 : 1319 (last 3 digits) : 319

extract the left part of the final number (before the dot) and add the remember
at step 4 : remember = 1
at step 5 : sum = 356
356 + 1 = 357