Reciprocal of 89
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Theorem
The decimal expansion of the reciprocal of $89$ is as follows:
- $\dfrac 1 {89} = 0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$
This sequence is A021093 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
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
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $89$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $89$