Any decimal number z of n digits after decimal point is given. Does there always exist an irrational number q (or expression related to irrational number) which has same n digits of z after decimal point?

Question

Any decimal number   z of   n digits after decimal point is given. Does there always exist an irrational number   q (or expression related to irrational number) which has same   n digits of   z after decimal point?

Cristian Hamilton · Accepted Answer

Yes, and you can explicitly construct such number. Suppose your rational number is  d  =  0.      d    1        d    2        d    3    …      d    n   where       d    i    ∈  {  0  ,  1  ,  …  ,  9  } stands for a digit .Then   d is rational. Now take       d    ′    =  d  +      10          −      n      −      1        ⋅      2  . This number has digits      d    ′    =  0.      d    1        d    2        d    3    …      d    n    141421356  … where the tail is just the decimal expansion of       2  . Clearly       d    ′   is irrational and   d and       d    ′   have the same first   n digits after the decimal point.You can do something similar for any rational and with any irrational, not just       2   .

Garrett Black · Accepted Answer

Since the rationals are dense in the reals, so are the rationals times       2   (or any other irrational number), and these numbers are all irrational.

Any decimal number z of n

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