Reciprocal of 19
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Theorem
The decimal expansion of the reciprocal of $19$ is as follows:
- $\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$
This sequence is A021023 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
Performing the calculation using long division:
0.05263157894736842105... ------------------------ 19)1.00000000000000000000000 95 152 76 -- --- -- 50 180 40 38 171 38 -- --- -- 120 90 20 114 76 19 --- -- -- 60 140 100 57 133 95 -- --- --- 30 70 ... 19 57 -- -- 110 130 95 114 -- --- 150 160 133 152 -- --- 170 80 152 76
$\blacksquare$