The integers are {...-3,-2,-1, 0, 1, 2, 3,...}

The whole numbers are {0, 1, 2, 3,...}

So any of the negative integers

{...-3,-2,-1}

would be an integer but not a whole number.
So -1 would be the simplest example.

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Any non-terminating decimal which does not eventually repeat
one digit or block of digits forever is a real number which is
not a rational number. For example, this non-terminating,
non-repeating decimal:

1.101001000100001000001000000100000001···

is an example or a real number which is not rational.  
Other examples are  and  for their 
decimal forms can be shown to be non-terminating as
well as non-repeating.

Edwin