Question 1097476
To find a rational number, note that

6.21744 < 6.22000 < 6.22746

and 6.22 can be written 622/100 or reduced to  {{{ highlight(311/50) }}}.
EDIT: I had accidentally typed out an extra zero to make 1000 in the denominator.  Edited to fix it.  Sorry about that.      
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For the irrational number, let's call it x

{{{ 6.21744 < x < 6.22746}}}   and x irrational (so x^2 is not an integer)

{{{ 6217.44 < 1000x < 6227.46 }}}
{{{ (6217.44)^2 < 1000000x^2 < (6227.46)^2 }}}

Now if we restrict our range to:
{{{  (6218)^2 < 1000000x^2 < (6219)^2 }}}

We can just pick an integer between {{{ 6218^2 }}} and {{{ 6219^2 }}} (excluding endpoints) and 
set {{{ 1000000x^2 }}} to that:

For example, one solution is:
{{{ 1000000x^2 = 6218^2 + 1 = 38663525 }}}
{{{  x^2 = 38663525/1000000 }}} 
{{{ highlight(x = sqrt(38.663525)) }}}   is irrational and is approximately 6.2180001, which falls between the limits given.   This is just one of many, many solutions.
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The reason it is irrational is every integer between {{{ (6218)^2 = 38663524 }}} and {{{ (6219)^2 = 38675961 }}},  excluding endpoints, can not be a perfect square, and hence its square root is irrational.