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Jul
20
revised Elementary theorems with several proofs?
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Jul
20
comment Elementary theorems with several proofs?
Formula for the sum of the first $n$ positive integers: trans4mind.com/personal_development/mathematics/series/…. See also the beautiful visual proof that's the top answer at mathoverflow.net/questions/8846/proofs-without-words and read all the comments to find where the diagram first appeared.
Jul
20
answered Elementary theorems with several proofs?
Jul
15
comment can quaternions be expressed in terms of tensor products?
Comparing the representation of quaternions as $4 \times 4$ real matrices in the Wikipedia page on quaternions with the formula for Kronecker products (that is the concrete matrix expression of tensor products of matrices -- see the Wikipedia page on Kronecker products), the connection with your notation $A$, $B$, and $C$ is that $1 = I \otimes I$, $i = B \otimes A$, $j = A \otimes I$, and $k = C \otimes A$.
Jul
15
comment can quaternions be expressed in terms of tensor products?
Without checking your calculations, I just want to point out that your tensor products are in ${\rm M}_2(\mathbf R) \otimes_{\mathbf R} {\rm M}_2(\mathbf R)$, which is isomorphic to ${\rm M}_4({\mathbf R})$. It is well-known that the quaternions can be realized inside ${\rm M}_4(\mathbf R)$; see the section about matrix representations on the Wikipedia page for quaternions. Therefore it would not be surprising that you can find a realization of the quaternions $i$, $j$, and $k$ inside ${\rm M}_2(\mathbf R) \otimes_{\mathbf R} {\rm M}_2(\mathbf R)$.
Jul
13
comment What exactly is a number?
@AsafKaragila: Ordinal and cardinal numbers for infinite sets were introduced by Cantor before abstract algebra was systematically developed, so I presume the fact that they can be added and multiplied (albeit with less structure than in $\mathbf Z$ or $\mathbf R$) in a way that extends the operations with those names on the nonnegative integers is what led them to be called numbers. But don't ask me why matrices aren't then also called numbers. :) We both know this question is inherently incapable of having an exact answer.
Jul
13
revised What exactly is a number?
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Jul
12
comment What exactly is a number?
The elements of $\mathbf C$ were called numbers simply because they continued the progression from $\mathbf N$ to $\mathbf Z$ to $\mathbf Q$ to $\mathbf R$ one additional step, and everything previously had been called a number. There's nothing deep about the use of "number" in that development of ideas. I think you might find parts of the book "Numbers" by H.-D. Ebbinghaus and 7 other authors to be interesting. The development of number systems, even containing $\mathbf Q$, can go off in directions very different from the real numbers (namely, the $p$-adic numbers for different primes $p$).
Jul
12
comment What exactly is a number?
Perhaps you might want to look up the definition of a "field". I think if you consider a "number" to be an element of some field then this might make you happy, at least for a little while. It doesn't include the quaternions (they're not commutative for multiplication), but it's pretty good for many purposes, e.g., much of basic linear algebra works over a general field. An even more encompassing definition might be "an element of an algebra over a commutative ring", but division by nonzero elements breaks down at that level, and you might not be prepared to consider such things as numbers.
Jul
10
comment How to integrate $\int_{-\infty} ^\infty \frac{\cos(xy)}{x^2+1}dx$
You have to try something, so just go ahead and use the contours you've used in the past for integrals over the whole real line. At least try a contour before asking others what to use. This stuff is learned by experience.
Jul
10
comment How to integrate $\int_{-\infty} ^\infty \frac{\cos(xy)}{x^2+1}dx$
While using the residue theorem is probably the standard method, for someone who knows complex analysis, the integral can also be evaluated without complex analysis by using differentiation under the integral sign. See section 11 of math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf.
Jul
10
comment How to integrate $\int_{-\infty} ^\infty \frac{\cos(xy)}{x^2+1}dx$
Your formula is only valid for $y \geq 0$. Note the integral is an even function of $y$, but $\pi{e}^{-y}$ is not. Or you could say the integral is $\pi{e}^{-|y|}$ for all $y$.
Jul
10
comment Complex Numbers: Im$(\frac{12}{z-7})=1$
It is still wrong: you want to multiply by the complex conjugate, which is $\overline{z}-7$, not $\overline{z}+7$.
Jul
10
comment Complex Numbers: Im$(\frac{12}{z-7})=1$
This suggestion is false: multiplying $z-7$ by $z+7$ gives $z^2-49$, which is almost never real. The correct factor to multiply top and bottom by is $\overline{z}-7$, if you want to write down explicit formulas.
Jul
7
comment What resources are there for learning Russian math terminology?
I've wondered for a while where the "elsewhere on this site" is that Alex wrote his remarks about motivation, and finally I stumbled onto it, so I just want to record it here for posterity in case anyone else is curious about that in the future: meta.math.stackexchange.com/questions/1617/…
Jul
6
comment Who named “Quotient groups”?
@MartinBrandenburg: In what way is Quotientengruppe more precise that Faktorgruppe? Do they not mean exactly the same thing? If you use the terminology Faktorring and Faktorraum as well, then as Serge Lang would say the terminology is functorial with respect to the ideas. As a contrast, the similar-sounding label "quotient field" is completely unlike quotient group, quotient ring, or quotient space, which might be why the name "fraction field" is used too. (While "factor group" and "factor ring" in English sound archaic but not too weird, "factor space" in English sounds very bizarre.)
Jul
6
revised Who named “Quotient groups”?
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Jul
6
comment Do these definitions make sense?
@blue: It presumably means "is it also true" about these other constructions. I fixed the English.
Jul
6
revised Do these definitions make sense?
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Jul
6
revised References for mathematical theory of summability of divergent series
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