Are numbers such as 3.89000.. With reoccurring 0s neither irrational numbers nor reoccurring numbers?

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I think the title sums it up pretty well. I asked my friend who has a degree in mathematics "To put it another way would you agree numbers such as 3.8900000.... With reoccurring 0s is not a reoccurring number nor an irrational number?" to this he said "Wittgenstein's method of defining things don't work with Math because math is the subject of rules and definitions strict rules and definitions so much so that we use symbols to convey meaning instead of words since words can be interpreted differently" hence me asking on here to see what mathematicians have to say on this. Is the answer a simple yes or are things more complicated than that?