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# Reciprocal

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

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

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

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

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

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

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

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

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

## Notes

1. It almost goes without saying that ${\displaystyle b}$ must not be 0; we only mention it to make the point that 0 is the only number in ${\displaystyle \mathbb {C} }$ 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 ${\displaystyle i}$ is ${\displaystyle -i}$).