Reciprocal

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The reciprocal of a number $x$ is the number ${\frac {1}{x}}$. The reciprocal of a reciprocal is the original number, which is to say that ${\frac {1}{\frac {1}{x}}}=x$ (but the term is generally used for the number that must be expressed as ${\frac {1}{x}}$). For example, the reciprocal of 2 is 0.5, and the reciprocal of 0.5 is 2.

The reciprocal of $x$ can also be expressed as $x^{-1}$. In most place-value numeral systems, 0.1 represents the reciprocal of the base.

If $x$ is a fraction expressed irreducibly as ${\frac {a}{b}}$ with $b\neq 1$, we will observe that ${\frac {1}{\frac {a}{b}}}={\frac {b}{a}}$. For example, $\left({\frac {4}{3}}\right)^{-1}={\frac {3}{4}}$.

Also, the continued fraction of $x^{-1}$ is the same as that of $x$ but the terms are switched one place over to the left or the right. The continued fraction for ${\frac {\pi }{2}}$ is

$1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{3+{\cfrac {1}{31+{\cfrac {1}{1+{\cfrac {1}{145+\ddots }}}}}}}}}}}}\,$

while that for ${\frac {2}{\pi }}$ is

$0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{3+{\cfrac {1}{31+{\cfrac {1}{1+\ddots }}}}}}}}}}}}\,$

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 ${\frac {1}{\pi }}$ (see A049541) and ${\frac {1}{e}}$ (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.

$n$	${\frac {1}{n}}$	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