Reciprocal of 59
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Theorem
The decimal expansion of the reciprocal of $59$ is as follows:
- $\dfrac 1 {59} = 0 \cdotp \dot 01694 \, 91525 \, 42372 \, 88135 \, 59322 \, 03389 \, 83050 \, 84745 \, 76271 \, 18644 \, 06779 \, 66 \dot 1$
This sequence is A021063 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
Performing the calculation using long division:
0.016949152542372881355932203389830508474576271186440677966101... ------------------------------------------------------------------ 59)1.000000000000000000000000000000000000000000000000000000000000... 59 ---- 410 150 170 350 230 500 370 380 570 354 118 118 295 177 472 354 354 531 --- --- --- --- --- --- --- --- --- 560 320 520 550 530 280 160 260 390 531 295 472 531 472 236 118 236 354 --- --- --- --- --- --- --- --- --- 290 250 480 190 580 440 420 240 360 236 236 472 177 531 413 413 236 354 --- --- --- --- --- --- --- --- --- 540 140 80 130 490 270 70 400 60 531 118 59 118 472 236 59 354 59 --- --- -- --- --- --- -- --- -- 90 220 210 120 180 340 110 460 100 59 177 177 118 177 295 59 413 59 -- --- --- --- --- --- --- --- --- 310 430 330 200 300 450 510 470 ... 295 413 295 177 295 413 472 413 --- --- --- --- --- --- --- --- 150 170 350 230 500 370 380 570
$\blacksquare$