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 Jul 27 awarded Critic Jul 26 comment How do the Properties of Relations work? I think you have mixed up the definitions of asymmetric and antisymmetric in this -- could you edit your post to fix this? (or indicate a reference which uses the words in the way you have). Jul 25 comment Why 1 is not considered to be a prime number? You can still edit though, can't you? Jul 24 accepted Why 1 is not considered to be a prime number? Jul 24 comment Why is the derivative of a circle's area its perimeter (and similarly for spheres)? (I realise that it might not be clear what the $n$-dimensional generalisation is of this, but perhaps this would happen even in different geometries or metric spaces?). Jul 24 asked Why is the derivative of a circle's area its perimeter (and similarly for spheres)? Jul 24 asked Why is the volume of a cone one third of the volume of a cylinder? Jul 24 comment Is $0$ a natural number? @Justin, I know that there are mixed views (as indicated in the second paragraph of my question). But for the case of 1 being classified as a prime number, it seems the consensus view of the Mathematical community is that it should not count as a prime number. My actual question is 'Is there a consensus on whether zero is a natural number?' (although the question's title is simpler), so a suitable answer would be 'No, there is no consensus' combined with a quick demonstration from a few Mathematical dictionaries or articles that there are conflicting definitions. Jul 23 comment Explanation of method for showing that $\frac{0}{0}$ is undefined Neither 0/x = 0 nor x/x = 1 would be considered axioms though -- they're just observations that can be proved for any x other than zero, so I don't think it makes sense to pit them off against each other. Jul 23 comment Explanation of method for showing that $\frac{0}{0}$ is undefined Except that the 0^-1 is undefined (the original question didn't ask why 1/0 was undefined, but that seems acceptable), so there is no need for us to be disturbed. Jul 23 comment How would you describe calculus in simple terms? Fair enough. I think I choose them in this order because I the concept of area seems more basic to understand, whereas the concept of tangents/gradients at points seems a bit more complex. But I would never teach integration before differentiation! Jul 22 comment What is the meaning of the double turnstile symbol ($\models$)? @Noldorin, perhaps the simplest example of the distinction could be seen in classical propositional logic. A statement like |= p -> (q -> p) means that the no matter the truth values of p and q, the value of p -> (q -> p) is always truth (which can be seen by looking at the truth table). A statement like |- p -> (q -> p) means that we can prove this formula purely by using axioms/rules of deduction, with no mention of truth or meaning. Incidentally, I think you could ask this as a separate question on the site, since we don't want the comment strand to be where one has to look for answers. Jul 22 awarded Nice Question Jul 22 comment Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? Thank you for adding this answer. I was hoping to avoid an answer that involved integration, so I also prefer AgCl's answer. But I am happy to see more than one demonstration/proof. Jul 22 comment Why is $x^0 = 1$ except when $x = 0$? Unfortunately there's also a debate about whether 0 is a natural number (see math.stackexchange.com/questions/283/is-0-a-natural-number). But I'll take your usage to mean 'nonnegative integer', and that you're voting for $0^0 = 1$. Jul 22 comment What is the meaning of the double turnstile symbol ($\models$)? Sorry, I realise that I haven't actually answered your question (about the meaning). I've seen it used in logic to stand for semantic entailment, but I imagine it has other uses. Jul 22 answered What is the meaning of the double turnstile symbol ($\models$)? Jul 22 awarded Commentator Jul 22 comment Why is $x^0 = 1$ except when $x = 0$? @Neil, except that 0^x = 0 is not true for all non-zero x (e.g. x = -1). Jul 22 comment Why isn't reflexivity redundant in the definition of equivalence relation? Perhaps I made up the noun form. If you google 'serial binary relation' it will come up. Although if you google 'seriality of binary relations' it will come up there too.