Reciprocal of 99,990,001

Theorem

The reciprocal of $99 \, 990 \, 001$ is:

$\dfrac 1 {99 \, 990 \, 001} = 0 \cdotp \dot 00000 \, 00100 \, 00999 \, 99998 \, 999 \dot 9$

Proof

Performing the calculation using long division:

         0.00000001000099999998999900000001...
        --------------------------------------------
99990001)1.00000000000000000000000000000000...
           99990001
           --------
               999900000
               899910009
               ---------
                999899910
                899910009
                ---------
                 999899010
                 899910009
                 ---------
                  999890010
                  899910009
                  ---------
                   999800010
                   899910009
                   ---------
                    998900010
                    899910009
                    ---------
                     989900010
                     899910009
                     ---------
                      899900010
                      799920008
                      ---------
                       999800020
                       899910009
                       ---------
                        998900110
                        899910009
                        ---------
                         988801010
                         899910009
                         ---------
                          899910010
                          899910009
                          ---------
                                  100000000
                                   99990001
                                  ---------
                                   ........

$\blacksquare$