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I have seen the Wikipedia article on this topic, it looks more explanatory. You didn't answer my first question well, probably why i was still confused.
The set of values you presented (2, 1.5, 1.42....) are actually for r belonging to Q. You said they belonged to Q but never stated if it was for r or r².
RE: Research diaries #10: The real line, gaps in the rational numbers and the axiom of completeness