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 Jul 6 awarded Popular Question Feb 4 accepted Why does $i^2 = -1$? Feb 4 comment Why does $i^2 = -1$? I see. That makes sense. In the case of √a√-b = √-ab, it really is the same as multiplying two positive square roots and then multiplying that whole dealie times i. As in: √-1√a√b = √-ab = i√ab. The rule "if and only if a and b are non-negative real numbers" applies, we're just throwing a √-1 into the mix as well. Feb 4 comment Why does $i^2 = -1$? @amWhy Not exactly. "However: √ab =√a√b if and only if a,b are non-negative real numbers." We just showed above that √ab = √a√b when a OR b is negative, no? As in: (√a)(√-b) = √-ab Feb 4 comment Why does $i^2 = -1$? Ah, that's clearer. I undertand it, for sure, taking i^2 = -1 as a first principle. It just seems inelegant to have to that and I wanted to understand it from the perspective of the first principle i = √-1 Feb 4 comment Why does $i^2 = -1$? It seems that if either a OR b were negative, though, the usual rule would apply, as in: (√5)(√-3) = √-15 ? Feb 4 comment Why does $i^2 = -1$? Then why is √1 by definition greater than 0? √25, for instance is ±5, no? Feb 4 asked Why does $i^2 = -1$? Sep 13 comment Self-Teaching: Is Geometry the Nexus of all Mathematics? Thanks a bundle, Ben! This is, like, exactly what I needed to hear! I really appreciate the thorough response! Sep 13 awarded Scholar Sep 13 accepted Self-Teaching: Is Geometry the Nexus of all Mathematics? Sep 13 comment Self-Teaching: Is Geometry the Nexus of all Mathematics? Thanks, André. I certainly don't want to do anything boring! I'll keep thinking on it. Sep 13 comment Self-Teaching: Is Geometry the Nexus of all Mathematics? Henning: I think I don't know enough about geometry to know what I mean! Essentially, I'm thinking back to a biography of Donald Coxeter that I read a few years ago; he seemed to think that the entire world was understandable geometrically. That, in large part, I think, is where this idea is coming from. Sep 13 comment Self-Teaching: Is Geometry the Nexus of all Mathematics? I understand that, for sure. The problem is that it all interests me to the point that trying to figure out a starting place is too overwhelming. That's why I'm trying to think of it in terms of what would be the most practical. Sep 13 awarded Student Sep 13 asked Self-Teaching: Is Geometry the Nexus of all Mathematics?