Is the following decimal rational or irrational?

0.00000**1**00**2**0 0**3**00**4**00**5**00 **6**00**7**00**8**00**9** 0**10**0110120 1301401501 6017018019 0200210220 2302402502 6027028029 0300310320 33034............etc. Please show steps. Thanks.

Guest May 19, 2018

#1**0 **

If you have an arbitrary precision calculator and take the reciprocal of the decimal, you get this:

1 / [0.0000010020 0300400500 6007008009 0100110120 1301401501 6017018019 0200210220 2302402502 6027028029 0300310320 33034............etc.] =998,001.

So, the decimal is rational in this case.

Note: The reciprocal of this number is very UNIQUE in that its decimal expansion has ALL the natural counting numbers 1, 2, 3,4........up 2,997 digits and then it repeats from the beginning again. However, it misses 998 itself and jumps from 997 to 999!. Very interesting number!!.

Guest May 19, 2018

edited by
Guest
May 19, 2018