nyuuzyou/wikis huggingface dataset:
Reciprocal of 131

== Theorem ==
The decimal expansion of the reciprocal of $131$ is as follows:
:$\dfrac 1 {131} = 0 \cdotp \dot 00763 \, 35877 \, 86259 \, 54198 \, 47328 \, 24427 \, 48091 \, 60305 \, 34351 \, 14503 \, 81679 \, 38931 \, 29770 \, 99236 \, 64122 \, 13740 \, 45801 \, 52671 \, 75572 \, 51908 \, 39694 \, 65648 \, 85496 \, 18320 \, 61068 \, 7022 \dot 9$
== Proof ==
Performing the calculation using long division (zoom out for best effect):
 0.0076335877862595419847328244274809160305343511450381679389312977099236641221374045801526717557251908396946564885496183206106870229
 --------------------------------------------------------------------------------------------------------------------------------------
131)1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
 917 655 262 131 393 131 786 1179 786
 --- --- --- --- --- --- --- ---- ---
 830 550 980 190 170 290 940 910 240
 786 524 917 131 131 262 917 786 131
 --- --- --- --- --- --- --- --- ---
 440 260 630 590 390 280 230 1240 1090
 393 131 524 524 262 262 131 1179 1048
 --- --- --- --- --- --- --- ---- ----
 470 1290 1060 660 1280 180 990 610 420
 393 1179 1048 655 1179 131 917 524 393
 --- ---- ---- --- ---- --- --- --- ---
 770 1110 1200 500 1010 490 730 860 270
 655 1048 1179 393 917 393 655 786 262
 --- ---- ---- --- --- --- --- --- ---
 1150 620 210 1070 930 970 750 740 800
 1048 524 131 1048 917 917 655 655 786
 ---- --- --- ---- --- --- --- --- ---
 1020 960 790 220 1300 530 950 850 140
 917 917 786 131 1179 524 917 786 131
 ---- --- --- --- ---- --- --- --- ---
 1030 430 400 890 1210 600 330 640 900
 917 393 393 786 1179 524 262 524 786
 ---- --- --- --- ---- --- --- --- ---
 1130 370 700 1040 310 760 680 1160 1140
 1048 262 655 917 262 655 655 1048 1048
 ---- --- --- ---- --- --- --- ---- ----
 820 1080 450 1230 480 1050 250 1120 920
 786 1048 393 1179 393 1048 131 1048 917
 --- ---- --- ---- --- ---- --- ---- ---
 340 320 570 510 870 200 1190 720 300
 262 262 524 393 786 131 1179 655 262
 --- --- --- --- --- --- ---- --- ---
 780 580 460 1170 840 690 1100 650 380
 655 524 393 1048 786 655 1048 524 262
 --- --- --- ---- --- --- ---- --- ---
 1250 560 670 1220 540 350 520 1260 1180
 1179 524 655 1179 524 262 393 1179 1179
 ---- --- --- ---- --- --- --- ---- ----
 710 360 150 410 160 880 1270 810 1000
 655 262 131 393 131 786 1179 786 ....
 --- --- --- --- --- --- ---- ---
 550 980 190 170 290 940 910 240
Category:131
Category:Examples of Reciprocals
Reciprocal of 142,857

== Theorem ==
The decimal expansion of the reciprocal of $142 \, 857$ is as follows:
:$\dfrac 1 {142 \, 857} = 0 \cdotp \dot 00000 \, \dot 7$
== Proof ==
Performing the calculation using long division:
 0.000007000007...
 ------------------
142857)1.000000000000...
 1 000000
 999999
 --------
 1000000
 999999
 -------
== Sources ==
* : $142,857$
* : $142,857$
Category:142,857
Category:Examples of Reciprocals
Reciprocal of 142,857/Mistake

== Source Work ==
:
:The Dictionary
::$142,857$
:
:The Dictionary
::$142,857$
== Mistake ==
:Because $142,857 \times 7 = 999,999$, the decimal period of $1 / 7$ is $142857$ and the decimal period of $1 / 142,857$ is $7$. In fact $1 / 142,857 = 0.000007 \, 000007 \, 000007 \, \ldots$
== Correction ==
Careful counting reveals that the period is in fact $6$, not $7$.
== Sources ==
* : $142,857$
* : $142,857$
Category:Mistakes/Curious and Interesting Numbers
Reciprocal of 17

== Theorem ==
The decimal expansion of the reciprocal of $17$ is as follows:
:$\dfrac 1 {17} = 0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$
== Proof ==
Performing the calculation using long division:
 0.058823529411764705...
 ------------------------
17)1.000000000000000000000
 85 68
 -- -- 
 150 20 
 136 17
 --- --
 140 30
 136 17
 --- --
 40 130
 34 119
 -- ---
 60 110
 51 102
 -- ---
 90 80
 85 68
 -- --
 50 120
 34 119
 -- ---
 160 100
 153 85
 --- ---
 70 ...
 68
Category:17
Category:Examples of Reciprocals
Reciprocal of 19

== Theorem ==
The decimal expansion of the reciprocal of $19$ is as follows:
:$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$
== Proof ==
Performing the calculation using long division:
 0.05263157894736842105...
 ------------------------
19)1.00000000000000000000000
 95 152 76 
 -- --- --
 50 180 40
 38 171 38
 -- --- --
 120 90 20
 114 76 19
 --- -- --
 60 140 100
 57 133 95
 -- --- ---
 30 70 ...
 19 57
 -- --
 110 130
 95 114
 -- ---
 150 160
 133 152
 -- ---
 170 80
 152 76
Category:19
Category:Examples of Reciprocals
Reciprocal of 19 from Sum of Powers of 2 Backwards

== Theorem ==
The decimal expansion of the reciprocal of $19$ can be constructed by summing the powers of $2$, offset progressively backwards by $1$ digit:
 1
 2
 4
 8
 16
 32
 64
 128
 256
 512
 1024
 2048
 4096
 8192
 16384
 32768
 65536
 131072
+ 262144
 ......
--------------------------
.......1052631578947368421
while:
:$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$
== Proof ==
We confirm that from Reciprocal of $19$:
:$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$
The construction above can be expressed as the sum:
:$\ds \sum_{k \mathop \ge 0} \paren {2^k \times 10^k} = \sum_{k \mathop \ge 0} 20^k$
Obviously this sum does not converge.
However we can still analyse the last $N$ digits from this construction for every positive integer $N$.
From $k \ge N$ onwards, the last $N$ digits will not be affected by further additions of $20^k$ because:
:$\forall k \ge N: 20^k \equiv 20^{k - N} \times 2^N \times 10^N \equiv 0 \pmod {10^N}$
By Number times Recurring Part of Reciprocal gives 9-Repdigit/Generalization, the last $N$ digits of this number coincides with the decimal expansion of the reciprocal of $19$ :
:$\ds \sum_{k \mathop = 0}^{N - 1} 20^k \times 19 \equiv -1 \pmod {10^N}$
We have:
Hence the result.
== Sources ==
* : $052,631,578,947,368,421$
* : $052,631,578,947,368,421$
Category:19
Category:Examples of Reciprocals
Reciprocal of 21

== Theorem ==
The decimal expansion of the reciprocal of $21$ is as follows:
:$\dfrac 1 {21} = 0 \cdotp \dot 04761 \, \dot 9$
== Proof ==
Performing the calculation using long division:
 0.04761901...
 --------------
21)1.00000000000
 84
 --
 160
 147
 ---
 130
 126
 ---
 40
 21
 ---
 190
 189
 ---
 100
 84
 ---
 ...
== Sources ==
* : $47,619$
* : $47,619$
Category:21
Category:Examples of Reciprocals
Reciprocal of 23

== Theorem ==
The decimal expansion of the reciprocal of $23$ is as follows:
:$\dfrac 1 {23} = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$
== Proof ==
Performing the calculation using long division:
 0.043478260869565217391304...
 -----------------------------
23)1.00000000000000000000000000
 92 184 23
 -- --- --
 80 160 170
 69 138 161
 -- --- ---
 110 220 90
 92 207 69
 --- --- --
 180 130 210
 161 115 207
 --- --- ---
 190 150 30
 184 138 23
 --- --- --
 60 120 70
 46 115 69
 -- --- --
 140 50 100
 138 46 92
 --- -- ---
 200 40 ...
 184 23
Category:23
Category:Examples of Reciprocals
Reciprocal of 239

== Theorem ==
The decimal expansion of the reciprocal of $239$ is as follows:
:$\dfrac 1 {239} = 0 \cdotp \dot 00418 \, 4 \dot 1$
== Proof ==
Performing the calculation using long division:
 0.0041841004...
 -------------
239)1.0000000000...
 956 
 ---- 
 440
 239
 ---
 2010
 1912
 ----
 980
 956
 ---
 240
 239
 ---
 1000
 956
 ----
 ...
Category:239
Category:Examples of Reciprocals
Reciprocal of 27

== Theorem ==
The decimal expansion of the reciprocal of $27$ is as follows:
:$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$
== Proof ==
Performing the calculation using long division:
 0.037...
 --------
27)1.00000
 81
 --
 190
 189
 ---
 100
 81
 ---
 ...
It is to be noted that this is because $999 = 27 \times 37$.
Category:27
Category:Examples of Reciprocals
Reciprocal of 29

== Theorem ==
The decimal expansion of the reciprocal of $29$ is as follows:
:$\dfrac 1 {29} = 0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$
== Proof ==
Performing the calculation using long division:
 0.034482758620689655172413793103...
 ------------------------------------
29)1.000000000000000000000000000000...
 87
 -- 
 130 180 150 230
 116 174 145 203
 --- --- --- ---
 140 60 50 270
 116 58 29 261
 --- -- -- ---
 240 200 210 90
 232 174 203 87
 --- --- --- --
 80 260 70 30
 58 232 58 29
 -- --- -- --
 220 280 120 100
 203 261 116 87
 -- --- --- ---
 170 190 40 ...
 145 174 29
 --- --- --
 250 160 110
 232 145 87
 --- --- ---
 180 150 230
Category:29
Category:Examples of Reciprocals
Reciprocal of 3

== Theorem ==
The decimal expansion of the reciprocal of $3$ is as follows:
:$\dfrac 1 3 = 0 \cdotp \dot 3$
== Proof ==
Performing the calculation using long division:
 0.33...
 --------
 3)1.00...
 9
 ---
 10
 9
 --
 ..
Category:3
Category:Examples of Reciprocals
Reciprocal of 31

== Theorem ==
The decimal expansion of the reciprocal of $31$ is as follows:
:$\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$
== Proof ==
Performing the calculation using long division:
 0.03225806451612903...
 -----------------------
31)1.00000000000000000000
 93 155
 -- ---
 70 50
 62 31
 -- --
 80 190
 62 186
 -- ---
 180 40
 155 31
 --- --
 250 90
 248 62
 --- --
 200 280
 186 279
 --- ---
 140 100
 124 93
 --- ---
 160 ...
 155
 ---
Category:31
Category:Examples of Reciprocals
Reciprocal of 37

== Theorem ==
The decimal expansion of the reciprocal of $37$ is as follows:
:$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$
== Proof ==
Performing the calculation using long division:
 0.027...
 --------
37)1.00000
 74
 --
 260
 259
 ---
 100
 74
 ---
 ...
It is to be noted that this is because $999 = 27 \times 37$.
Category:37
Category:Examples of Reciprocals
Reciprocal of 41

== Theorem ==
The decimal expansion of the reciprocal of $41$ is as follows:
:$\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$
== Proof ==
Performing the calculation using long division:
 0.0243902...
 -------------
41)1.0000000...
 82 
 ---- 
 180
 164
 ---
 160
 123
 ---
 370
 369
 ---
 100
 82
 ---
 ...
Category:41
Category:Examples of Reciprocals
Reciprocal of 43

== Theorem ==
The decimal expansion of the reciprocal of $43$ is as follows:
:$\dfrac 1 {43} = 0 \cdotp \dot 02325 \, 58139 \, 53488 \, 37209 \, \dot 3$
== Proof ==
Performing the calculation using long division:
 0.02325581395348837209302...
 -----------------------------
43)1.00000000000000000000000...
 86 
 ----
 140 410 310
 129 387 301
 --- --- ---
 110 230 90
 86 215 86
 --- --- --
 240 150 400
 215 129 387
 --- --- ---
 250 210 130 
 215 172 129
 --- --- ---
 350 380 100
 344 344 86
 --- --- ---
 60 360 ...
 43 344
 -- ---
 170 160
 129 129
 --- ---
 410 310
Category:43
Category:Examples of Reciprocals
Reciprocal of 451

== Theorem ==
The decimal expansion of the reciprocal of $451$ is as follows:
:$\dfrac 1 {451} = 0 \cdotp \dot 00221 \, 7294 \dot 9$
== Proof ==
Performing the calculation using long division:
 0.0022172949002...
 -------------------
451)1.0000000000000
 902
 ---
 980
 902
 ---
 780
 451
 ---
 3290
 3157
 ----
 1330
 902
 ----
 4280
 4059
 ----
 2210
 1804
 ----
 4060
 4059
 ----
 1000
 902
 ----
 ....
== Also see ==
* Sequence of Smallest Numbers whose Reciprocal has Period n
Category:451
Category:Examples of Reciprocals
Reciprocal of 47

== Theorem ==
The decimal expansion of the reciprocal of $47$ is as follows:
:$\dfrac 1 {47} = 0 \cdotp \dot 02127 \, 65957 \, 44680 \, 85106 \, 38297 \, 87234 \, 04255 \, 31914 \, 89361 \, \dot 7$
== Proof ==
Performing the calculation using long division:
 0.021276595744680851063829787234042553191489361702...
 ------------------------------------------------------
47)1.000000000000000000000000000000000000000000000000...
 94 
 ---- 
 60 270 400 140 160 150 440
 47 235 376 94 141 141 423
 -- --- --- --- --- --- ---
 130 350 240 460 190 90 170
 94 329 235 423 188 47 141
 --- --- --- --- --- -- ---
 360 210 50 370 200 430 290
 329 188 47 329 188 423 282
 --- --- -- --- --- --- ---
 310 220 300 410 120 70 80
 282 188 282 376 94 47 47
 --- --- --- --- --- -- --
 280 320 180 340 260 230 330
 235 282 141 329 235 188 329
 --- --- --- --- --- --- ---
 450 380 390 110 250 420 100
 423 376 376 94 235 376 94
 --- --- --- --- --- --- ---
 270 40 140 160 150 440 ... 
Category:47
Category:Examples of Reciprocals
Reciprocal of 49

== Theorem ==
The decimal expansion of the reciprocal of $49$ is as follows:
:$\dfrac 1 {49} = 0 \cdotp \dot 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 5 \dot 1$
== Proof ==
Performing the calculation using long division:
 0.02040816326530612244897959183673469387755102...
 ---------------------------------------------
49)1.00000000000000000000000000000000000000000000000
 98 294 196 441 196 245 
 -- --- --- --- --- ---
 200 260 240 90 340 250
 196 245 196 49 294 245
 --- --- --- -- --- ---
 400 150 440 410 460 50
 392 147 392 392 441 49
 --- --- --- --- --- --
 80 300 480 180 190 100
 49 294 441 147 147 98
 -- --- --- --- --- ---
 310 60 390 330 430 ...
 294 49 343 294 392
 --- -- --- --- ---
 160 110 470 360 380
 147 98 441 343 343
 --- --- --- --- ---
 130 120 290 170 360
 98 98 245 147 343
 --- --- --- --- ---
 320 220 450 230 270
 294 196 441 196 245
 --- --- --- --- ---
Category:49
Category:Examples of Reciprocals
Reciprocal of 49 shows Powers of 2 in Decimal Expansion

== Theorem ==
The decimal expansion of the reciprocal of $49$ contains the powers of $2$:
:$\dfrac 1 {49} = 0 \cdotp \dot 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 5 \dot 1$
== Proof ==
From Reciprocal of $49$:
Adding up powers of $2$, shifted appropriately to the right:
02
 04
 08
 16
 32
 64
 128
 256
 512
 1024
 2048
 4096
 8192
 16384
 32768
 65536
 131072
 262144
 524288
 1048576
 2097152
 4194304
 8388608
 16777216
 33554432
 67108864
 134217728
 268435456
 536870912
 1073741824
 2147483648
 4294967296
 8589934592
 ...........
--------------------------------------------------------------------
020408163265306122448979591836734693877551020408163265305947144192
--------------------------------------------------------------------
 1 1 1111 11 11212122222112222222322222111
As can be seen, the decimal expansion of $\dfrac 1 {49}$ matches the sum of the shifted powers of $2$ (to the limits of calculation) and the pattern is apparent.
This is what is to be expected, because:
:$0 \cdotp 02 + 0 \cdotp 0004 + 0 \cdotp 000008 + 0 \cdotp 00000016 + \cdots$
is nothing else but:
:$\dfrac 1 {50} + \paren {\dfrac 1 {50} }^2 + \paren {\dfrac 1 {50} }^3 + \paren {\dfrac 1 {50} }^4 + \cdots = \ds \sum_{k \mathop \ge 1} \paren {\dfrac 1 {50} }^k$
Hence:
== Sources ==
* : $49$
* : $49$
Category:49
Category:Examples of Reciprocals
Reciprocal of 53

== Theorem ==
The decimal expansion of the reciprocal of $53$ is as follows:
:$\dfrac 1 {53} = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3$
== Proof ==
Performing the calculation using long division:
 0.018867924528301...
 ---------------------
53)1.000000000000000000
 53 212 
 -- ---
 470 280
 424 265
 --- ---
 460 150
 424 106
 --- ---
 360 440
 318 424
 --- ---
 420 160
 371 159
 --- ---
 490 100
 477 53
 --- ---
 130 ...
 106
 ---
 240
 212
 ---
Category:53
Category:Examples of Reciprocals
Reciprocal of 59

== Theorem ==
The decimal expansion of the reciprocal of $59$ is as follows:
:$\dfrac 1 {59} = 0 \cdotp \dot 01694 \, 91525 \, 42372 \, 88135 \, 59322 \, 03389 \, 83050 \, 84745 \, 76271 \, 18644 \, 06779 \, 66 \dot 1$
== 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
Category:59
Category:Examples of Reciprocals
Reciprocal of 61

== Theorem ==
The decimal expansion of the reciprocal of $61$ is as follows:
:$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$
== Proof ==
Performing the calculation using long division:
 0.01639344262295081967213114754098360655737704918032786885245901...
 --------------------------------------------------------------------
61)1.00000000000000000000000000000000000000000000000000000000000000000
 61 122 61 61 488 183 183 305 
 -- --- --- -- --- --- --- ---
 390 380 590 90 220 470 170 150
 366 366 549 61 183 427 122 122
 --- --- --- -- --- --- --- ---
 240 140 410 290 370 430 480 280
 183 122 366 244 366 427 427 244
 --- --- --- --- --- --- --- ---
 570 180 440 460 400 300 530 360
 549 122 427 427 366 244 488 305
 --- --- --- --- --- --- --- ---
 210 580 130 330 340 560 420 550
 183 549 122 305 305 549 366 549
 --- --- --- --- --- --- --- ---
 270 310 80 250 350 110 540 100
 244 305 61 244 305 61 488 61
 --- --- -- --- --- --- --- ---
 260 500 190 600 450 490 520 ...
 244 488 183 549 427 488 488
 -- --- --- --- --- --- ---
 160 120 70 510 230 200 320
 122 61 61 488 183 183 305
 --- --- -- --- --- --- ---
Category:61
Category:Examples of Reciprocals
Reciprocal of 67

== Theorem ==
The decimal expansion of the reciprocal of $67$ is as follows:
:$\dfrac 1 {67} = 0 \cdotp \dot 01492 \, 53731 \, 34328 \, 35820 \, 89552 \, 23880 \, 59 \dot 7$
== Proof ==
Performing the calculation using long division:
 0.01492537313432835820895522388059701...
 -----------------------------------------
67)1.00000000000000000000000000000000000...
 67
 ----
 330 210 560 640 590
 268 201 536 603 536
 --- --- --- --- ---
 620 90 240 370 540
 603 67 210 335 536
 --- -- --- --- ---
 170 230 390 350 400
 134 201 335 335 335
 --- --- --- --- ---
 360 290 550 150 650
 335 268 536 134 603
 --- --- --- --- ---
 250 220 140 160 470
 201 201 134 134 469
 --- --- --- --- ---
 490 190 600 260 100
 469 134 536 201 67
 --- --- --- --- ---
 210 560 640 590 ...
Category:67
Category:Examples of Reciprocals
Reciprocal of 7

== Theorem ==
The decimal expansion of the reciprocal of $7$ is as follows:
:$\dfrac 1 7 = 0 \cdotp \dot 14285 \, \dot 7$
== Proof ==
Performing the calculation using long division:
 0.142857...
 ------------
 7)1.000000000
 7
 --
 30
 28
 --
 20
 14
 --
 60
 56
 --
 40
 35
 --
 50
 49
 --
 10
 7
 --
 ..
Category:7
Category:Examples of Reciprocals
Reciprocal of 71

== Theorem ==
The decimal expansion of the reciprocal of $71$ is as follows:
:$\dfrac 1 {71} = 0 \cdotp \dot 01408 \, 45070 \, 42253 \, 52112 \, 67605 \, 63380 \, 2816 \dot 9$
== Proof ==
Performing the calculation using long division:
 0.0140845070422535211267605633802816901...
 -------------------------------------------
71)1.0000000000000000000000000000000000000...
 71
 ----
 290 160 80 400 580
 284 142 71 355 568
 --- --- -- --- ---
 600 180 90 450 120
 568 142 71 426 71
 --- --- -- --- ---
 320 380 190 240 490
 284 355 142 213 426
 --- --- --- --- ---
 360 250 480 270 640
 355 213 426 213 639
 --- --- --- --- ---
 500 370 540 570 100
 497 355 497 568 71
 --- --- --- --- ---
 300 150 430 200 ...
 284 142 426 142
 --- --- --- ---
 160 80 400 580
Category:71
Category:Examples of Reciprocals
Reciprocal of 73

== Theorem ==
The decimal expansion of the reciprocal of $73$ is as follows:
:$\dfrac 1 {73} = 0 \cdotp \dot 01369 \, 86 \dot 3$
== Proof ==
Performing the calculation using long division:
 0.0136986301...
 -------------
73)1.0000000000...
 73
 ----
 270
 219
 ---
 510
 438
 ---
 720
 657
 ---
 630
 584
 ---
 460
 438
 ---
 220
 219
 ---
 100
 73
 ---
 ...
Category:73
Category:Examples of Reciprocals
Reciprocal of 79

== Theorem ==
The decimal expansion of the reciprocal of $79$ is as follows:
:$\dfrac 1 {79} = 0 \cdotp \dot 01265 \, 82278 \, 48 \dot 1$
== Proof ==
Performing the calculation using long division:
 0.012658227848101...
 ---------------------
79)1.000000000000000...
 79
 ----
 210 670
 158 632
 --- ---
 520 380
 474 316
 --- ---
 460 640
 395 632
 --- ---
 650 80
 632 79
 --- --
 180 100
 158 79
 --- ---
 220 ...
 158
 ---
 620
 553
 ---
 670
Category:79
Category:Examples of Reciprocals
Reciprocal of 8

== Theorem ==
The reciprocal of $8$ can be expressed as the summation of the powers of $2$ multiplied by the reciprocal powers of $10$:
:$\dfrac 1 8 = \ds \sum_{k \mathop \ge 0} \dfrac {2^k} {10^{k + 1} }$
.1
 2
 4
 8
 16
 32
 64
 128
 256
 512
 1024
 ....
-------------
.12499999744
-------------
== Proof ==
We have that:
:$\dfrac {2^k} {10^{k + 1} } = \dfrac 1 {10} \paren {\dfrac 1 5}^k$
and so by Sum of Infinite Geometric Sequence:
== Sources ==
* : $125$
* : $125$
Category:8
Reciprocal of 81

== Theorem ==
The decimal expansion of the reciprocal of $81$ has a particularly interesting pattern:
:$\dfrac 1 {81} = 0 \cdotp \dot 01234 \, 567 \dot 9$
== Proof ==
Performing the calculation using long division:
 0.0123456790...
 ----------------
81)1.0000000000000
 81
 --
 190
 162
 ---
 280
 243
 ---
 370
 324
 ---
 460
 405
 ---
 550
 486
 ---
 640
 567
 --
 730
 729
 ---
 100
 81
 ---
 ...
== Also see ==
* Reciprocal of Square of 1 Less than Number Base
== Sources ==
* : $81$
* : $81$
Category:81
Category:Examples of Reciprocals
Reciprocal of 83

== Theorem ==
The decimal expansion of the reciprocal of $83$ is as follows:
:$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$
== Proof ==
Performing the calculation using long division:
 0.0120481927710843373493975903614457831325301...
 -------------------------------------------------
83)1.0000000000000000000000000000000000000000000...
 83
 ----
 170 640 310 810 120 260 100
 166 581 249 747 83 249 83
 --- --- --- --- --- --- ---
 400 590 610 630 370 110 
 332 581 581 581 332 83
 --- --- --- --- --- ---
 680 90 290 490 380 270
 664 83 249 415 332 249
 --- -- --- --- --- ---
 160 700 410 750 480 210
 83 664 332 747 415 166
 --- --- --- --- --- ---
 770 360 780 300 650 440
 747 332 747 249 581 415
 --- --- --- --- --- ---
 230 280 330 510 690 250
 166 249 249 498 664 249
 --- --- --- --- --- ---
 640 310 810 120 260 100
Category:83
Category:Examples of Reciprocals
Reciprocal of 83 has Prime Period

== Theorem ==
The decimal expansion of the reciprocal of $83$ has a prime period, that is $41$:
:$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$
== Proof ==
From Reciprocal of $83$:
:$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$
Counting the digits, it is seen that this has a period of recurrence of $41$.
Indeed, $41$ is the $13$th prime number.
Category:83
Category:Examples of Reciprocals
Reciprocal of 89

== Theorem ==
The decimal expansion of the reciprocal of $89$ is as follows:
:$\dfrac 1 {89} = 0 \cdotp \dot 01123 \, 59550 \, 56179 \, 77528 \, 08988 \, 76404 \, 49438 \, 20224 \, 719 \dot 1$
== 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
== Sources ==
* : $89$
* : $89$
Category:89
Category:Examples of Reciprocals