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 Sep22 comment How to start with mathematics? If you're not fazed by sticking to only pen/paper or a blackboard/whiteboard for doing all the manipulative work (remember, math ain't a spectator sport!), then sure, you don't even need to learn how to use a computer to help you with math. One convincing reason to learn programming (at least for me) is that you can let the computer do all the grunt work so that you have time for the finer things in life (e.g. "Hmm, that's funny..." or "There's a pattern here somewhere...") Sep21 answered A singularity of hypergeometric functions Sep21 comment Find the range of $x$, given $y_{min} \leq y(x) \leq y_{max}$, where $y(x)$ can be any function ( Updated) That's assuming your range of interest is within one period of the sine function; if the range covers more than that, then the troubles with oscillatories I was alluding to comes into play. Sep21 revised Minimal set with subsets that sum to given values edited tags Sep21 comment Casio hand calculator vs Mathematica at (1+1/n)^n Even Timing[((1 + 1/#)^#) &[N[10^20, 5000]]] takes only a blink of an eye on my (relatively slow) netbook. Sep21 comment Find the range of $x$, given $y_{min} \leq y(x) \leq y_{max}$, where $y(x)$ can be any function ( Updated) Hmm, you should have mentioned that too; throwing an oscillatory function into the mix can present a different set of problems. Sep21 comment Active Contour Models - Lagrangian/Eulerian approches Mind mentioning where (e.g. book, website) you encountered these? Sep21 comment Given coordinates of hypotenuse, how can I calculate coordinates of other vertex? This is related to the geometric fact that an angle inscribed in a semicircle is a right angle. I suppose there are cases where the algebra with this approach is easier than Robin's, and conversely. Sep21 comment Find the range of $x$, given $y_{min} \leq y(x) \leq y_{max}$, where $y(x)$ can be any function ( Updated) Ah, so it's a rational function! Why didn't you say so to begin with? You can do the analysis by looking at the numerators and denominators; rootfinding of the polynomials in the denominators may sometimes be necessary, but this is certainly more tractable than what you were asking at first. Sep21 comment Branches of mathematics not having a general method to solve Even from the numerics viewpoint, it only takes a tiny change in the initial conditions for some DEs to see a change in the behavior of the solutions. Lorenz's set comes to mind. Sep21 comment Find the range of $x$, given $y_{min} \leq y(x) \leq y_{max}$, where $y(x)$ can be any function ( Updated) As mentioned already, solution strategies for such things are often tailored to the inherent structure of the problem. Not exploiting it can only be a profligate waste of computer time and effort. Please give more specifics about your function(s) of interest. Turning to the multivariate case, exploitation of structure becomes even more important, since you have much degrees of freedom. Sep21 comment How to start with mathematics? Heh, indeed, trying to break your algorithm and trying to poke holes in your proof require the exact same mindset. Building up the discipline to go over such things line-by-line is great practice. Sep21 comment Given coordinates of hypotenuse, how can I calculate coordinates of other vertex? Kevin: mentioning something along the lines of "I tried method X and it went to crap" wouldn't have hurt. Sep21 comment Given coordinates of hypotenuse, how can I calculate coordinates of other vertex? Geometrically, subtracting equation 2 from equation 1 is equivalent to finding the radical line (mathworld.wolfram.com/RadicalLine.html ) of the two circles represented by the two equations. Sep21 comment Why does $1/x$ diverge? This is the Oresme proof, and see math.stackexchange.com/questions/250/840#840 as well for how you might attempt a physical demonstration. Sep21 comment How is $\mathbb{C}$ different than $\mathbb{R}^2$? There's nothing corresponding to "maximum modulus" on $\mathbb{R}^2$. Sep21 comment Indirect proof that sum of first n even numbers is $n^2 + n$ ...but induction is the most elementary way of going about it, Ignorunt. Sep21 answered Indirect proof that sum of first n even numbers is $n^2 + n$ Sep20 revised Interesting results easily achieved using complex numbers removed cruft from Amazon link and converted it to a hyperlink Sep20 comment Interesting results easily achieved using complex numbers This is a great example; I remember T mentioning somewhere here that algorithms that have to be restricted to real variables are often clunky compared to algorithms that allow complexes.