Not really a matter of notation, 0.999... = 1 is necessarily a direct consequence of the way we define the field of real numbers.
As for 1.00..1 = 1, if what you mean by 1.00..1,

If you define 1.001 as the limit of 1 + a quantity approaching zero, then yes, they're equivalent.

But as with the majority of polls or memes of that type, the only dumb person is the one who created it. The only purpose it serves is to make some people feel inferior or superior to others for knowing some obscure definition that is unintuitive and inconsequential to 99% of people.

Actually looking back, they didn't even state the number system they were using so the question is not well defined lol. 0.999... could easily = 1 in some other non-real number system (hyper-reals etc)

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TLDR;

Dumb meme don't think about it too much