Any number you can write completely is a rational number. Any decimal number with a repeating decimal fraction is also a rational number.
If the number of digits goes on forever without repeating, it is an irrational number.
Of the numbers listed, √3 and π are irrational. The remaining numbers are rational. ("The symbol π" is not a number; it is a symbol. It conventionally stands for an irrational number.)
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Is -6.55555 An irrational number? NO
Is -2 an irrational number? NO
Is 4/9 an irrational number? NO
Is the square root of 3 an irrational number? YES
Is the pi symbol irrational number? NO (it stands for an irrational number. You have asked about the symbol, not about pi. Pi is an irrational number.)
Is 3.14 an irrational number? NO
Is 23 an irrational number? NO