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Compared to Liu Hui’s 3.1416, the answer 3.14159 doesn’t seem too bad. But you are right, now that we have computer technology there’s really no excuse for 率 deniers, or anyone else who thinks it’s cute to complain about irrationality.
Why didn`t you use sin [3]+3 and then a serie of sin[%]+% as ir will give Pi with 2709 digits after 5 repetitivos?