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Jul
16
comment What makes induction a valid proof technique?
And how do you show that $k^2<k$ if $0<k<1$?
Jun
22
comment What is more important in Mathematics, Theorems or its Proofs?
@IttayWeiss: Never mind.
Jun
22
comment What is more important in Mathematics, Theorems or its Proofs?
@IttayWeiss: (a) It doesn't matter. (b) If so, we will probably never know. They probably wouldn't have written in their article "One day we woke up and thought about these random axioms ..."
Jun
22
comment What is more important in Mathematics, Theorems or its Proofs?
@IttayWeiss: And I never claimed you said that. My comment above was a reply to the question (specifically referring to "View 1"). If it had been a comment to your answer, I would have attached it to your answer.
Jun
22
comment A normal chicken's egg is 6cm tall. An ostrich's egg measures 15cm.
@SanathDevalapurkar: You should have omitted the last sentence from the copy/paste because it doesn't apply here: Sammy didn't use "Prove" or "Solve", but "Could you please teach me how to to solve" — I cannot imagine anyonr considering that one rude.
Jun
22
comment A normal chicken's egg is 6cm tall. An ostrich's egg measures 15cm.
The given information is not sufficient to answer that question; I guess there's an implicit assumption that the eggs have all the same shape (an assumption which doesn't generally hold for eggs of different species).
Jun
22
comment What is more important in Mathematics, Theorems or its Proofs?
Non-Euclidean geometry came into being exactly by deducing consequences if the parallels axiom were not true (actually in the hope of finding a contradiction). Only after that had been done, people realized that this indeed led to a valid theory.
Jun
22
comment What is more important in Mathematics, Theorems or its Proofs?
"Nobody wakes up in the morning, randomly chooses some axioms and starts deducing theorems." That's a pretty strong statement. Are you really sure not a single person has ever done exactly that?
May
17
comment Extending the set of complex numbers
There is no reason to restrict to non-constant polynomials. A non-zero constant polynomial has degree $n=0$ and $n=0$ complex solutions, so the statement is just as true for those. Only for the zero polynomial you need an exception.
Apr
27
comment In a group of 26 people, is it possible for each person to shake hands with exactly 3 other people?
And it's also easy to prove that for an odd number of people, it's not possible for everyone to shake hands with exactly three other people (because the total number of shaken hands would have to be odd, but in each handshake two hands get shaken), therefore that condition is both necessary and sufficient.
Apr
26
comment How can I write the numbers 5 and 7 as some sequence of operations on three 9s?
@Awesome: Care to explain? (For me, it just gives an overview of the windows on the current desktop ...)
Apr
22
comment Is it possible that “A counter-example exists but it cannot be found”
I forgot the @TrevorWilson in the previous comment.
Apr
22
comment Average Line of a Set of Lines
I'd say that completely depends on what you need the average line for (that is, what you want the average line to tell you), and/or where the original lines come from (indeed, this may decide whether the concept of an average line is actually meaningful in your context).
Apr
22
comment Prove that if $G$ is a group and $H$ is a subgroup of $G$ generated by all elements of order $N$ in $G$, then $H$ is a normal subgroup of $G$.
But $H$ is generated by the elements of order $N$. So for each $h\in H$, $h=h_1h_2h_3\dots$ where each $h_i$ is of order $N$. And $ghg^{-1} = (gh_1g^{-1}) (gh_2g^{-1}) (gh_3g^{-1}) \dots$, with each factor being of order $N$.
Apr
22
comment Prove that if $G$ is a group and $H$ is a subgroup of $G$ generated by all elements of order $N$ in $G$, then $H$ is a normal subgroup of $G$.
You're welcome.
Apr
22
comment Prove that if $G$ is a group and $H$ is a subgroup of $G$ generated by all elements of order $N$ in $G$, then $H$ is a normal subgroup of $G$.
And if $h$ is an element of order $N$, then this is ...
Apr
22
comment How to prove that the velocity field $u$ is always $\nabla\cdot u=o$ except at the origin?
What about simply calculating $\nabla\cdot \mathbf u$?
Apr
22
comment Prove that if $G$ is a group and $H$ is a subgroup of $G$ generated by all elements of order $N$ in $G$, then $H$ is a normal subgroup of $G$.
What is $(ghg^{-1})^N$?
Apr
22
comment Proof that $e^x$ is the eigenvector or the derivative operator
"You can (illustratively and not very well defined) understand a function as an infinitely dense vectors with values at $y(x)$. But then it makes no sense to talk about vectors at all." Of course it does, and it is actually very well defined. It's probably not very useful, and you'll certainly not be able to write down a basis, but it is well defined (indeed, it's exactly the definition of a function!) and a valid vector space. The only problematic term in your description is "infinitely dense" because that requires a topology; I don't know if a reasonable one can be defined in that case.
Apr
22
comment 2.71828. And then another 1828.
Here's another thing to "worry" about. Look at the first 30 digits of the decimal expansion of $\pi$. And search for the pattern "aba". What do you find? $3.\color{red}{\,141\,} 5926 \color{red}{\,535\,} 8 \color{red}{\,979\,323\,} 84 \color{red}{\,626\,} 43 \color{red}{\,383\,} 279$. What are four digits repeated once compared with that? :-)