That's not relevant to what they said, which is that distances can't be imaginary. They're correct. A metric takes nonnegative real values by definition
Specifically, the thing that exists is a regular homotopy of immersions from the standard embedding to its opposite. The "rules" aren't supposed to be self evident, they're part of a broader context in topology
The eigenvalues of a diagonal matrix are the values on the diagonal. Diagonalizable matrices' eigenvalues can be determined by diagonalizing them and looking at the entries on the diagonal.
It's interesting because it's highly counter-intuitive that such a thing is possible. It's not supposed to be useful except as an example of a false intuition, which can remind us to be careful in our reasoning.
Microsoft blocks people from downloading stuff all the time for unknowable reasons. You have to either reset your IP or go through customer support to fix it. I did the latter and they did not tell me why I was blocked in the first place.
No. sqrt(2) is an irrational number characterized as the positive solution to x^2 - 2 = 0. It's described by a very small amount of data. Even its decimal expansion can be determined up to any precision by a simple algorithm.
But something has to be written on the birth certificate and social security card, and that's what everything else will expect you to use. I think just due to technical limitations (e.g. of the printer/template for those things) it wouldn't be allowed, but I dunno about legally
Metric, not measure. Metrics are real by definition.