DECIMAL DATA
The first two tables below give the decimal expansion of 1/b for 2 ≤ b ≤ 90. Some expansions do
not have their periodic part at the beginning (e.g., 1/44 = .0227 = .0227272727 . . . has an initial 02
which is not in the periodic part). Those 1/b having a purely periodic decimal expansion (that is,
it repeats right from the start) are printed in bold. A finite decimal expansion could be viewed as
an expansion with periodic part 0 (e.g., 1/16 = .0625 = .06250).
Fraction
Decimal Expansion
1/2
1/3
1/4
1/5
1/6
1/7
1/8
1/9
1/10
1/11
1/12
1/13
1/14
1/15
1/16
1/17
1/18
1/19
1/20
1/21
1/22
1/23
1/24
1/25
1/26
1/27
1/28
1/29
1/30
1/31
1/32
1/33
1/34
1/35
1/36
1/37
1/38
1/39
1/40
.5
.3
.25
.20
.16
.142857
.125
.1
.1
.09
.083
.076923
.0714528
.06
.0625
.0588235294117647
.05
.052631578947368421
.05
.047619
.045
.0434782608695652173913
.0416
.04
.0384615
.037
.03571428
.0344827586206896551724137931
.03
.032258064516129
.03125
.03
.02941176470588235
.0285714
.027
.027
.0263157894736842105
.025641
.025
2
DECIMAL DATA
Fraction
Decimal Expansion
1/41
1/42
1/43
1/44
1/45
1/46
1/47
1/48
1/49
1/50
1/51
1/52
1/53
1/54
1/55
1/56
1/57
1/58
1/59
1/60
1/61
1/62
1/63
1/64
1/65
1/66
1/67
1/68
1/69
1/70
1/71
1/72
1/73
1/74
1/75
1/76
1/77
1/78
1/79
1/80
1/81
1/82
1/83
1/84
1/85
1/86
1/87
1/88
1/89
1/90
.02439
.0238095
.023255813953488372093
.0227
.02
.02173913043478260869565
.0212765957446808510638297872340425531914893617
.02083
.020408163265306122448979591836734693877551
.02
.0196078431372549
.01923076
.0188679245283
.01851
.018
.017857142
.017543859649122807
.01724137931034482758620689655
.0169491525423728813559322033898305084745762711864406779661
.016
.016393442622950819672131147540983606557377049180327868852459
.0161290322580645
.015873015873
.015625
.0153846
.015
.014925373134328358208955223880597
.014705882352941176
.0144927536231884057971
.0142857
.01408450704225352112676056338028169
.0138
.01369863
.0135
.013
.01315789473684210526
.012987
.0128205
.0126582278481
.0125
.012345679
.012195
.01204819277108433734939759036144578313253
.01190476
.01176470588235294
.0116279069767441860465
.0114942528735632183908045977
.01136
.01123595505617977528089887640449438202247191
.01
DECIMAL DATA
3
The next table indicates the period length for the decimal expansion of 1/p when p is a prime up
to 113, other than 2 and 5.
p
3
length(1/p) 1
p
59
length(1/p) 58
7 11 13
6
2 6
61 67 71
60 33 35
17
16
73
8
19
18
79
13
23
22
83
41
29
28
89
44
31 37
15 3
97 101
96 4
41
43
5
21
103 107
34
53
47
46
109
108
53
13
113
112
So far we have focused on unit fractions 1/b. Now we will compare the decimal expansions of
all reduced proper fractions with a common denominator (like 1/7, 2/7, 3/7, . . . , 6/7). In the tables
below, only purely periodic decimals are listed.
Fraction
Decimal
Fraction
Decimal
1/3
.3
2/3
.6
1/7
3/7
5/7
.142857
.428571
.714285
2/7
4/7
6/7
.285714
.571428
.857142
1/9
4/9
7/9
.1
.4
.7
2/9
5/9
8/9
.2
.5
.8
1/11
3/11
5/11
7/11
9/11
.09
.27
.45
.63
.81
2/11
4/11
6/11
8/11
10/11
.18
.36
.54
.72
.90
1/13
3/13
5/13
7/13
9/13
11/13
.076923
.230769
.384615
.538461
.692307
.846153
2/13
4/13
6/13
8/13
10/13
12/13
.153846
.307692
.461538
.615384
.769230
.923076
1/17
3/17
5/17
7/17
9/17
11/17
13/17
15/17
.0588235294117647
.1764705882352941
.2941176470588235
.4117647058823529
.5294117647058823
.6470588235294117
.7647058823529411
.8823529411764705
2/17
4/17
6/17
8/17
10/17
12/17
14/17
16/17
.1176470588235294
.2352941176470588
.3529411764705882
.4705882352941176
.5882352941176470
.7058823529411764
.8235294117647058
.9411764705882352
1/19
3/19
5/19
7/19
9/19
11/19
13/19
15/19
17/19
.052631578947368421
.157894736842105263
.263157894736842105
.368421052631578947
.473684210526315789
.578947368421052631
.684210526315789473
.789473684210526315
.894736842105263157
2/19
4/19
6/19
8/19
10/19
12/19
14/19
16/19
18/19
.105263157894736842
.210526315789473684
.315789473684210526
.421052631578947368
.526315789473684210
.631578947368421052
.736842105263157894
.842105263157894736
.947368421052631578
4
DECIMAL DATA
Fraction
Decimal
Fraction
Decimal
1/21
4/21
8/21
11/21
16/21
19/21
.047619
.190476
.380952
.523809
.761904
.904761
2/21
5/21
10/21
13/21
17/21
20/21
.095238
.238095
.476190
.619047
.809523
.952380
1/23
3/23
5/23
7/23
9/23
11/23
13/23
15/23
17/23
19/23
21/23
.0434782608695652173913
.1304347826086956521739
.2173913043478260869565
.3043478260869565217391
.3913043478260869565217
.4782608695652173913043
.5652173913043478260869
.6521739130434782608695
.7391304347826086956521
.8260869565217391304347
.9130434782608695652173
2/23
4/23
6/23
8/23
10/23
12/23
14/23
16/23
18/23
20/23
22/23
.0869565217391304347826
.1739130434782608695652
.2608695652173913043478
.3478260869565217391304
.4347826086956521739130
.5217391304347826086956
.6086956521739130434782
.6956521739130434782608
.7826086956521739130434
.8695652173913043478260
.9565217391304347826086
1/27
4/27
7/27
10/27
13/27
16/27
19/27
22/27
25/27
.037
.148
.259
.370
.481
.592
.703
.814
.925
2/27
5/27
8/27
11/27
14/27
17/27
20/27
23/27
26/27
.074
.185
.296
.407
.518
.629
.740
.851
.962