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Jun
22
comment An easy example of a non-constructive proof without an obvious “fix”?
I guess what you mean is that the obvious classical proofs of this proposition can't be fixed. I think there may well be some entirely different constructive proof that hasn't been found yet.
Jun
1
comment Real world applications of category theory
@Niriel Transition went very well. You can read about the actual project here: theverge.com/2013/9/1/4680456/…
May
14
comment How many degrees of freedom would a rotation matrix in R5 have?
@loupblanc Yes, quite right. I'll add a comment to the answer.
Apr
27
comment Why is $x^0 = 1$ except when $x = 0$?
If you're in the combinatorics business, zero is natural. If your business involves primes, zero is not. I'm not sure what people do when they have a foot in both camps.
Mar
16
comment Domain of log and its derivative
Did you mean to write $(0,\infty)$? The derivative clearly can't exist where the function being differentiated doesn't exist.
Mar
12
comment How to explain brackets to young students
It sounds like this student has a very good understanding of brackets, something I've seen students struggle with. The problem here isn't with brackets, it's with addition.
Mar
12
comment Prove f(a)=f(b) if $\int_{-\infty}^{\infty}f(x)dx=1$
Very roughly: if f(a)>f(b) then shifting the region [a,b] left by δ would increase the integral by approximately δ(f(a)-f(b)) which would then allow you to shrink the length of the interval and still achieve the same integral.
Dec
15
comment Are the real numbers really uncountable?
@CarlMummert I'm not trying to say anything profound. Typically, in mathematics, when you want to pick a particular thing (in ZF, say) to talk about, you use a definition.
Dec
15
comment Proving $\int x \, \mathrm{d}y =xy- \int y \, \mathrm{d}x$
$\frac{dy}{dx}$ might look like a straightforward fraction, but it's not. You can't simply cancel $dx$'s without further justification.
Dec
3
comment Approximate solutions for quintic equation
See math.stackexchange.com/questions/309178/polynomial-root-finding
Nov
12
comment Cardinality of infinite sets - Help with intuition
"why can't I do the exact same thing when dealing with the mapping between odds/integers?" Have you tried?
Nov
7
comment Function 'result arity'
In my many years of mathematics I've never come across such a term.
Aug
13
comment If $ P(A) = 0 $ is $ A $ a null event?
@user159813 The fact that the "event is a non empty set" does not make the event "non-null". The set of pairs of integers that sum to 13 is non-empty but it's impossible to roll 13 with two ordinary dice.
Aug
5
comment In calculus, which questions can the naive ask that the learned cannot answer?
First, you don't need table look-ups to compute elliptic integrals. There are a variety of methods for computing them. Second, in what sense can you not exactly calculate the arc-length of an ellipse but can exactly calculate the arc-length of a circle?
Aug
4
comment using the same symbol for dependent variable and function?
This is a problematic area. Despite functions playing a central role in much of mathematics there isn't a clear unambiguous notation that mathematicians agree on. I think a good modern view is that $y$ is a function and $y(t)$ is the value of the function evaluated at $t$ in which case it (usually) makes no sense to say $y=y(t)$. But others take the view that $y=y(t)$ is a special use of notation that emphasises that $y$ is a variable that depends on $t$ and so is perfectly acceptable.
Jul
24
comment Can universal instantiation be used more than once?
If technique X can only be used once in a proof, and I use it to prove A implies B, and use it to prove B implies C, then I wouldn't be able to combine these proofs to show A implies C as that would require two uses, and mathematics would pretty well come to a halt.
Jul
14
comment Why is the height of a heap defined as $\lg n$?
See mathworld.wolfram.com/Lg.html for some discussion of what the notation $\mathop{lg}$ (as opposed to $\log$ or $\ln$) means.
Jul
14
comment Are the real numbers really uncountable?
I'm not sure what the issue is @frogeyedpeas . When a mathematician says "there exists an x such that..." they're not saying "there exists an x, with definition Y, such that...". If you want to, you can study numbers with this property but that has no bearing on the existence of the other numbers.
Jul
9
comment How do I know if I have imaginary numbers when using Newton Raphson Method?
I guess you're wondering what happens if you're trying to solve an equation that contains complex values, or from a complex starting point. The answer is that it gets complicated. Seriously complicated: chiark.greenend.org.uk/~sgtatham/newton
Jul
7
comment Integral of the square of a function
The integral can easily be non-negative even if the integrand is sometimes not real. Consider a function $f$ that takes a small imaginary value for a small region and otherwise takes large real values. Overall the real parts will dominate the integral giving a non-negative result.