Dan Piponi
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# 75 Comments

 Mar16 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. Mar12 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. Mar12 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. Dec15 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. Dec15 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. Dec3 comment Approximate solutions for quintic equation Nov12 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? Nov7 comment Function 'result arity' In my many years of mathematics I've never come across such a term. Aug13 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. Aug5 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? Aug4 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. Jul24 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. Jul14 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. Jul14 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. Jul9 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 Jul7 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. Jul5 comment Axiom of Choice and finite sets "How do I choose such an x?" is often a good question if you're writing a computer program. But in mathematics, if you want to use the existence of $x$ then tautologically you need only a proof that $x$ exists. Jul4 comment Understanding matrices. That code isn't right. Maybe you mean newX = cos(angle)*oldX-.... Jun30 comment Given a histogram, programatically, how do I find the normal distributions that comprise it? Jun30 comment What's wrong with having the same variable in the integrand as in the limits? @QiaochuYuan You point to the real issue. In principle $\int_0^x(\int_0^xx^3dx)x^4dx$ is unambiguous. But mathematicians like the abuse of notation that allows them to write what you just wrote and they find it worthwhile to sacrifice $\alpha$-equivalence to get it. (See en.wikipedia.org/wiki/Lambda_calculus#Alpha_equivalence .) I suspect that modern mathematics students with some computing experience may have slightly different ideas of what's reasonable.