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comment Has there ever been an application of dividing by zero?
@jpmc26: Sure; in this setting, $1/x$ is continuous at $x=0$ so you can always replace it with a limit (and vice versa), but again, why go through all the trouble with limits when you can just do arithmetic?
1d
comment Has there ever been an application of dividing by zero?
@jpmc26: Why would you go through all the trouble with limits when you can just do arithmetic?
Feb
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awarded  Nice Answer
Feb
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awarded  Great Answer
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Nov
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comment Trivial Fundamental Group Notation (1 or 0)
I've even seen an exact sequence with $1$ at the left end and $0$ at the right end, because the left group was typically thought of multiplicatively and the right one additively.
Nov
23
comment Why does this matrix give the derivative of a function?
Very closely related, in fact, since the dual numbers $\mathbb{R}[\epsilon]$ are isomorphic (as a topological ring) to the matrix ring $\mathbb{R}[A]$.
Nov
21
comment Why does the fundamental theorem of calculus work?
Mulling over the details, I think going from the first to the second line of "combining" is actually a rather nontrivial step. The overall idea is fine, of course, just that the detail of justifying that step is tricky to get right.
Nov
20
comment Why does the fundamental theorem of calculus work?
@Arjang: That's an issue of setting up the integral, not the meaning of the integral once you've written it. If the "change in area" is not $f(x) \mathrm{d} x$, then integrating $f(x) \mathrm{d} x$ doesn't give you the area under the graph. For a graph $r = f(\theta)$ in polar coordinates, for example, the "change in area under the graph" is $\frac{1}{2} f(\theta)^2 \mathrm{d} \theta$. But if $\frac{1}{2} f(\theta)^2 = g'(\theta)$, then you still get $\int_0^{2 \pi} \frac{1}{2} f(\theta)^2 \mathrm{d} \theta = g(2\pi) - g(0)$.
Nov
20
answered Why does the fundamental theorem of calculus work?
Nov
20
comment Why does the fundamental theorem of calculus work?
@Arjang: There are no metrics involved here. $\mathrm{d} x$ is a "change in the real variable $x$" (one way to be more precise is a differential form), not any sort of measurement of distance.
Nov
20
revised Prove that there is no integer a for which $a^2 - 3a - 19$ is divisible by 289
added 265 characters in body
Nov
18
answered Confused about the definition of a group as a groupoid with one object.
Nov
17
comment integrate sin(x).
The relation between the constants could also be set by plugging in $x=0$.
Nov
16
answered Do trigonometric reduction formulae work for every angle?
Nov
14
comment Developing category theory inside ETCS
Is this approach more or less the same thing as positing the existence of an internal topos with a sufficiently nice functor to the ambient category? (I imagine the functor would have to be $\hom(1,-)$, and it would have to be logical)
Nov
14
revised Inner product equals for all vectors means
added 488 characters in body
Nov
14
comment Inner product equals for all vectors means
@SangchulLee: See my answer for a counterexample to your comment.
Nov
14
answered Inner product equals for all vectors means