There is a carry from the 5+8=13. Add these up and you get closer and closer to 1 / 89: 0.0 0.01 0.001 0.0002 0.00003 0.000005 0.0

There is a carry from the 5+8=13. Add these up and you get closer and closer to 1 / 89: 0.0 0.01 0.001 0.0002 0.00003 0.000005 0.0000008 0.00000013 0.000000021 0.0000000034 0.00000000055 0.000000000089 0.0000000000144 0.00000000000233 0.000000000000377 0.0000000000000610 0.00000000000000987 0.000000000000001597 0.0000000000000002584 0.00000000000000004181 0.000000000000000006765 0.0000000000000000010946 0.00000000000000000017711 0.000000000000000000028657 0.0000000000000000000046368 0.00000000000000000000075025 0.000000000000000000000121393 0.0000000000000000000000196418 0.00000000000000000000000317811 0.000000000000000000000000514229 0.0000000000000000000000000832040 0.00000000000000000000000001346269 0.000000000000000000000000002178309 0.0000000000000000000000000003524578 0.00000000000000000000000000005702887 0.000000000000000000000000000009227465 0.0000000000000000000000000000014930352 0.00000000000000000000000000000024157817 0.000000000000000000000000000000039088169 0.0000000000000000000000000000000063245986 0.00000000000000000000000000000000102334155 0.000000000000000000000000000000000165580141 0.0000000000000000000000000000000000267914296 0.00000000000000000000000000000000000433494437 0.000000000000000000000000000000000000701408733 0.0000000000000000000000000000000000001134903170 0.00000000000000000000000000000000000001836311903 0.000000000000000000000000000000000000002971215073 0.0000000000000000000000000000000000000004807526976 0.00000000000000000000000000000000000000007778742049 0.000000000000000000000000000000000000000012586269025 0.0000000000000000000000000000000000000000020365011074 0.00000000000000000000000000000000000000000032951280099 0.000000000000000000000000000000000000000000053316291173 0.0000000000000000000000000000000000000000000086267571272 0.00000000000000000000000000000000000000000000139583862445 0.000000000000000000000000000000000000000000000225851433717 0.0000000000000000000000000000000000000000000000365435296162 0.00000000000000000000000000000000000000000000000591286729879 + 0.000000000000000000000000000000000000000000000000956722026041 -------------------------------------------------------------- 0.011235955056179775280898876404494382022471910112174867308031 ≈ 1 / 89 ============================================================== For those interested in running this with more iterations, here is a Python script: from decimal import * n = 60 getcontext().prec = n t = Decimal(0) p, q = 0, 1 for i in range(1, n+1): s = f'0.{p:0{i}}' print(s) t += Decimal(s) p, q = q, p+q print('-' * (n + 2)) print(t, '≈ 1 / 89') print('=' * (n + 2)) print() print(1 / t)