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Jul
27
comment Why is the derivative of a circle's area its perimeter (and similarly for spheres)?
I hadn't thought of this. Does this also explain why it works in 3 dimensions?
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 Is $1$ a prime number?
You can still edit though, can't you?
Jul
24
accepted Is $1$ 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 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 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
answered Explanation of method for showing that 0 / 0 is undefined
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 Why isn't reflexivity redundant in the definition of equivalence relation?
At the risk of making another embarrassing mistake, I think I've realised why I got the last one wrong -- I was using the definition of Euclidean that I got from your link to the wikipedia article, rather than the one you gave above. I think my counterexample works with that definition of Euclideanity.
Jul
22
comment What is the meaning of this 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
comment Why isn't reflexivity redundant in the definition of equivalence relation?
@Charles: ouch: I typed it into a program that does a brute-force search for counterexamples, but I misrepresented the Euclideanity as 'Rxy & Rxz -> Ryz'. Perhaps I should have read the definition you gave! Sorry.
Jul
22
comment Why isn't reflexivity redundant in the definition of equivalence relation?
Not to sound as though I'm picking on you, but I think this revised claim is also false. My first clue was the fact that the wikipedia article you referenced said that Euclidean+Reflexive => Equivalence. Since being reflexive is a stronger requirement than being serial I sought a counterexample, the smallest of which I can find is on the set {1,2} and is defined as Rxy <-> y = 2.
Jul
22
awarded  Nice Question
Jul
22
comment Why does the series $\frac 1 1 + \frac 12 + \frac 13 + \cdots$ 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 $0^0 = 1$.