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AL2006

The last four numbers ... -6,  -3/2,  0,  and  9 ... are rational numbers.

We can't be sure about  5.73773777..... .  You haven't said whether it
goes on forever, or ends after 100 or 1,000 or a million decimal places. 
If it ever ends, or if it repeats every  8  or  37  or  10,000  digits, then
this one is a rational number too.