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15h
answered Proving a Set is a Vector Space
1d
comment How do mathematicians find the underlying idea?
Even without the underlying idea, this particular instance seems straightforward: just write down the triangle inequality in full generality, and then figure out where the thing you want to know about has to fit in, then figure out where the things you already know have to go, and check if you get a result you're looking for.
1d
comment Why should quaternions exist?
$x^2 = x = 1$ shouldn't have any solutions either then?
1d
comment Recovering a basis from an isomorphism with the dual space.
$\phi^{-1}$ is an isomorphism $V^* \to V$. Also $\phi^*$ is an isomorphism $V^* \to V^{**}$. So this really isn't even a different problem!
2d
answered What allows us divide/multiply dx in calculus?
Aug
26
comment How to compute fraction sums?
Problems like this are usually rather hard. However, the problems you will be assigned will usually be ones that succumb to tricks, such as being arranged into a telescoping sum. Using those tricks, or ideas related to those tricks, are pretty much your only options for simplifying the sum.
Aug
26
answered Motivation for the Definition of Compact Space
Aug
26
comment Limits of cosine and sine
@CiaPan: Yes it does.
Aug
26
comment Splitting field as a terminal object?
@Rob: I think the idea is to make $E$ the largest field you can make by adjoining roots.
Aug
25
comment Higher infinities without Set Theory
And to drive home the point made by @ThomasAndrews that it's about complexity not size, it's easy to write down a computable partial function from the natural numbers onto the computable reals. The trick is that you can't compute the domain of the partial function.
Aug
25
comment nonstandard topology?
Every collection of disjoint sets contains a minimal element with respect to $\subseteq$, finite or not.
Aug
24
awarded  Yearling
Aug
23
answered Iterated square roots over finite field. When do we hit a nonresidue?
Aug
23
comment Solve $y^2 + 3xy - 10x^2 + y + 5x = 0$ for y in terms of x
If you can solve for $(y + (3x+1)/2)$ then you can solve for $y$.
Aug
23
comment What type of surface is it?
@Loki: Not identifying: he means to group the paths so that leftbottom is one path (obtained by following the left then the bottom) and righttop is the other path, and observe that the marked identifications respect the grouping (thus identifying leftbottom with righttop).
Aug
23
comment What happens if to introduce infinite and infinitesimel quantities this way?
$(1+x)^{1/x}$ is strictly descending in the vicinity of $x=0$, so there aren't any nonzero infinitesimals of nonstandard analysis that satisfy $(1+\epsilon)^{1/\epsilon} = e$. Formulations of infinitesimals with $\epsilon^2 = 0$ probably wouldn't apply either, since I think they would want $(1+\epsilon)^{1/\epsilon} = e(1 - \epsilon/2)$ if it is to make sense at all.
Aug
23
comment What happens if to introduce infinite and infinitesimel quantities this way?
Do you want that to be literally equal, or just that the difference of the two sides is infinitesimal?
Aug
23
comment Guessing how many times a smaller number goes into bigger number
Once you have a guess, you can always subtract it off and look at the remainder. e.g. if I guess $3$ goes into $100$ 30 times, the remainder is $100 - 3 \cdot 30 = 10$, and if I want can improve my estimate by seeing how many times $3$ goes into the remainder.
Aug
23
answered Conventions adopted for extended reals
Aug
23
comment Is there a name for complex numbers over affinely extended reals?
Note that your example is more like viewing $\mathbf{C} \subseteq \bar{\mathbf{R}} \times \bar{\mathbf{R}}$ rather than adding two extra points to the complex numbers. (where $\bar{\mathbf{R}}$ is the extended real numbers)