# Reciprocal of 89

## Theorem

The decimal expansion of the reciprocal of $89$ contains the Fibonacci sequence:

$\dfrac 1 {89} = 0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$

## Proof

Performing the calculation using long division:

   0.0112359550561797752808988764044943820224719101...
------------------------------------------------------
89)1.0000000000000000000000000000000000000000000000000
89       534       712       801        89
--       ---       ---       ---       ---
110       160       880       390       810
89        89       801       356       801
---       ---       ---       ---       ---
210       710       790       340        90
178       623       712       267        89
---       ---       ---       ---        --
320       870       780       730        100
267       801       712       712         89
---       ---       ---       ---        ---
530       690       680       180       ...
445       623       623       178
---       ---       ---       ---
850       670       570        200
801       623       534        178
---       ---       ---        ---
490       470       360        220
445       445       356        178
--       ---       ---        ---
450       250        400       420
445       178        356       356
---       ---        ---       ---
500      720        440       640
445      712        356       623
---      ---        ---       ---
550       800       840       170
534       712       801        89


$\blacksquare$