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Reciprocal

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The reciprocal of a number is the number . The reciprocal of a reciprocal is the original number, which is to say that (but the term is generally used for the number that must be expressed as ). For example, the reciprocal of 2 is 0.5, and the reciprocal of 0.5 is 2.

The reciprocal of can also be expressed as . In most place-value numeral systems, 0.1 represents the reciprocal of the base.

If is a fraction expressed irreducibly as with ,[1] we will observe that . For example, .

Also, the continued fraction of is the same as that of but the terms are switched one place over to the left or the right. The continued fraction for is

while that for is

Therefore in the OEIS there isn't a separate entry for the continued fraction of the reciprocal of some important number the way there almost certainly is for the decimal expansion of that reciprocal (in the previous example we used Buffon's constant: see A053300 for the continued fraction of either that number or its reciprocal, but A060294 and A019669 for the decimal expansions of Buffon's constant and its reciprocal, respectively).

Some other important reciprocals in the OEIS include (see A049541) and (see A068985).

Table of the reciprocals of small integers

In the following table, the reciprocals are given to 12 decimal places, rounded. Where applicable, follow the link to the OEIS sequence entry for much greater precision.

A-number
1 1.000000000000 A000007
2 0.500000000000 A020761
3 0.333333333333 A010701
4 0.250000000000 A020773
5 0.200000000000 A000038
6 0.166666666667 A020793
7 0.142857142857 A020806
8 0.125000000000 A020821
9 0.111111111111 A000012
10 0.100000000000 A000007
11 0.090909090909 A010680
12 0.083333333333 A021016
13 0.076923076923 A021017
14 0.071428571429 A021018
15 0.066666666667 A021019
16 0.062500000000 A021020
17 0.058823529412 A007450
18 0.055555555556 A021022
19 0.052631578947 A021023
20 0.050000000000 A020761
21 0.047619047619 A021025
22 0.045454545455 A021026
23 0.043478260870 A021027
24 0.041666666667 A021028
25 0.040000000000 A166926 (*)
26 0.038461538462 A021030
27 0.037037037037 A021031
28 0.035714285714 A021032
29 0.034482758621 A021033
30 0.033333333333 A010701
31 0.032258064516 A021035
32 0.031250000000 A021036
33 0.030303030303 A010674
34 0.029411764706 A021038
35 0.028571428571 A021039
36 0.027777777778 A021040
37 0.027027027027 A021041
38 0.026315789474 A021042
39 0.025641025641 A021043
40 0.025000000000 A020773
41 0.024390243902 A021045
42 0.023809523810 A021046
43 0.023255813953 A021047
44 0.022727272727 A021444
45 0.022222222222 A007395
46 0.021739130435 A021050
47 0.021276595745 A021051
48 0.020833333333 A021052
49 0.020408163265 A021053
50 0.020000000000 A020761
51 0.019607843137 A021055
52 0.019230769231 A021056
53 0.018867924528 A021057
54 0.018518518519 A021058
55 0.018181818182 A021059
56 0.017857142857 A021060
57 0.017543859649 A021061
58 0.017241379310 A021062
59 0.016949152542 A021063
60 0.016666666667
61 0.016393442623 A021065
62 0.016129032258 A021066
63 0.015873015873 A021067
64 0.015625000000 A021068
65 0.015384615385 A021069
66 0.015151515152 A021070
67 0.014925373134 A021071
68 0.014705882353 A021072
69 0.014492753623 A021073
70 0.014285714286 A020806
71 0.014084507042 A021075
72 0.013888888889
73 0.013698630137 A021077
74 0.013513513514 A021078
75 0.013333333333
76 0.013157894737 A021080
77 0.012987012987 A021081
78 0.012820512821 A021082
79 0.012658227848 A021083
80 0.012500000000
81 0.012345679012 A021085
82 0.012195121951 A021086
83 0.012048192771 A021087
84 0.011904761905 A021088
85 0.011764705882 A021089
86 0.011627906977 A021090
87 0.011494252874 A021091
88 0.011363636364
89 0.011235955056
90 0.011111111111
91 0.010989010989 A021095
92 0.010869565217 A021096
93 0.010752688172 A021097
94 0.010638297872 A021098
95 0.010526315789 A021099
96 0.010416666667 A021100
97 0.010309278351 A021101
98 0.010204081633 A021102
99 0.010101010101
100 0.010000000000
101 0.009900990099 A021105
102 0.009803921569 A021106
103 0.009708737864 A021107
104 0.009615384615 A021108
105 0.009523809524 A021109
106 0.009433962264 A021110
107 0.009345794393 A021111
108 0.009259259259 A021112
109 0.009174311927 A021113
110 0.009090909091
111 0.009009009009 A021115
112 0.008928571429 A021116
113 0.008849557522 A021117
114 0.008771929825 A021118
115 0.008695652174 A021119
116 0.008620689655 A021120
117 0.008547008547 A021121
118 0.008474576271 A021122
119 0.008403361345 A021123
120 0.008333333333
121 0.008264462810 A021125
122 0.008196721311 A021126
123 0.008130081301 A021127
124 0.008064516129 A021128
125 0.008000000000
126 0.007936507937 A021130
127 0.007874015748 A021131
128 0.007812500000 A021132
129 0.007751937984 A021133
130 0.007692307692
131 0.007633587786 A021135
132 0.007575757576 A021136
133 0.007518796992 A021137
134 0.007462686567 A021138
135 0.007407407407 A021139
136 0.007352941176 A021140
137 0.007299270073 A021141
138 0.007246376812 A021142
139 0.007194244604 A021143
140 0.007142857143
141 0.007092198582 A021145
142 0.007042253521 A021146
143 0.006993006993 A021147
144 0.006944444444 A021148
145 0.006896551724 A021149
146 0.006849315068 A021150
147 0.006802721088 A021151
148 0.006756756757 A021152
149 0.006711409396 A021153
150 0.006666666667
151 0.006622516556 A021155
152 0.006578947368 A021156
153 0.006535947712 A021157
154 0.006493506494 A021158
155 0.006451612903 A021159
156 0.006410256410 A021160
157 0.006369426752 A021161
158 0.006329113924 A021162
159 0.006289308176 A021163
160 0.006250000000
161 0.006211180124 A021165
162 0.006172839506 A021166
163 0.006134969325 A021167
164 0.006097560976 A021168
165 0.006060606061 A021169
166 0.006024096386 A021170
167 0.005988023952 A021171
168 0.005952380952 A021172
169 0.005917159763 A021173
170 0.005882352941
171 0.005847953216 A021175
172 0.005813953488 A021176
173 0.005780346821 A021177
174 0.005747126437 A021178
175 0.005714285714 A021179
176 0.005681818182 A021180
177 0.005649717514 A021181
178 0.005617977528 A021182
179 0.005586592179 A021183
180 0.005555555556
181 0.005524861878 A021185
182 0.005494505495 A021186
183 0.005464480874 A021187
184 0.005434782609 A021188
185 0.005405405405 A021189
186 0.005376344086 A021190
187 0.005347593583 A021191
188 0.005319148936 A021192
189 0.005291005291 A021193
190 0.005263157895
191 0.005235602094 A021195
192 0.005208333333 A021196
193 0.005181347150 A021197
194 0.005154639175 A021198
195 0.005128205128 A021199
196 0.005102040816 A021200
197 0.005076142132 A021201
198 0.005050505051
199 0.005025125628 A021203
200 0.005000000000

Notes

  1. It almost goes without saying that must not be 0; we only mention it to make the point that 0 is the only number in that doesn't have a reciprocal. Along these lines, we should also remark that 1 and –1 are the only numbers that are their own reciprocals: the reciprocal of 1 is 1, and the reciprocal of –1 is –1. (The reciprocal of the imaginary unit is ).