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Reciprocal
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
- ↑ 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 ).