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Aug
14
comment Why is $\pi r^2$ the surface of a circle
@euclid that's because it's only a hexagon, so half the outer perimeter is only $3$. It gets better if you use more pieces. Going into the complexities of the number $\pi$ itself is probably beyond the question being asked.
Aug
13
comment The puzzle of billiard geometry
@LukeAllen I'm not hoping for points.;Very few people on this site work that way. If you take 10 minutes to learn the basics of trigonometry any of the very first things you see will be able to give you a direct answer. What I'm not going to do is spoonfeed you. I've given you everything you need to know beyond the formula itself.
Aug
13
comment The puzzle of billiard geometry
@LukeAllen do you know trigonometry? You now know all 3 side lengths of both triangles. You can figure out the angles using those and trigonometric functions.
Aug
12
answered The puzzle of billiard geometry
Aug
12
comment Do matrices have a “to the power of” operator?
@paul23 fair enough. Exponents like these are normally only defined for square matrices for that very reason.
Aug
12
comment Do matrices have a “to the power of” operator?
can I ask why you were convinced that there were different operators involved in that first expression? There's only one kind of matrix multiplication.
Aug
5
comment How to Garner Mathematical Intuition
@BrianM.Scott part of my point is that your average high-school math student probably wouldn't say that much of it was obvious at all. Or at least that it only seemed obvious in retrospect, once it was clearly stated. You're clearly beyond the point where it could tell you anything new. Are you saying that typical gradeschool math students would also find the book obvious/useless?
Aug
4
comment How to Garner Mathematical Intuition
Saying this isn't well defined seems odd to me. How you go about finding proofs is also a personal matter, but no one has a problem with Polya. It's absurd to say that there aren't general guidelines and ways to think about problems that lead to better intuition. This is vitally important for math education; dismissing the idea of teaching it because it isn't well-defined is precisely the problem in the first place. Can we take a more productive approach?
Aug
3
comment I can't get satisfied with such 'so on' type logic. Is there a better way to solve it?
It is rigorous though, at least with a couple extra steps. All you have to do is show there will always be another point available to choose. Or you might be a finitist, in which case may god have mercy on your soul ;)
Aug
2
answered Guess the functional form of a graph
Aug
2
comment Areas of math that can be “gamified”?
I'm familiar with material on game design; I spent a lot of time on it in highschool and it is my real priority. A lot of my ideas in the direction of "uncover things they're intrigued about" (robotics, biology, crowd dynamics, Escher, music, fractals, social choice, etc.) would probably be used for a presentation or project instead. While I 100% agree with the perspective you're describing, my issue is mainly about finding which areas games are ideal for. I consider 'game Euclid' superior to 'presentation Euclid', and am looking for other examples in this vein.
Aug
2
asked Areas of math that can be “gamified”?
Jul
31
answered What's the importance of the trig angle formulas?
Jul
31
answered A geometric property of the graph of $y = x^2$
Jul
31
comment Why are topological spaces interesting to study?
Have you studied things like algebraic topology and cohomology yet? There are cases where you want to study things that are only dependent on how the space is connected. Even though a metric may still exist, dragging it along when it isn't "respected" by anything you're doing would only make things more difficult. Topology cleanly isolates the important features in such problems. In particular, I take some issue with point 3: the metric isn't automatically useful or relevant just because it's there.
Jul
24
comment Calculus over $\mathbb{Q}$
@RGB Wow, I've been using Cauchy sequences for a long time and that never really clicked before for some reason. I worry though that this simply means that I've asked my question poorly. While saying "point 1 is equivalent to calculus in $\mathbb{R}$" is certainly very enlightening, it somewhat serves to reinforce the point that mathematical calculus depends on $\mathbb{R}$ (more aggressively than I thought!) while "real-world calculus" functionally does not. It is this apparent gap that I really wanted to get into.
Jul
24
asked Calculus over $\mathbb{Q}$
Jul
22
comment Some theorems in euclidean geometry have incomplete proofs
@ReekMaths if the written words talk about $A,B,C,P,Q$ but only do so in abstract, without making reference to this particular diagram, then the diagram is not being used in the proof. If it says "$|CP|/|CA|=|CQ|/|CB|$, that is not using the diagram. It doesn't depend on how $P$ and $Q$ were drawn, but solely on their formal definitions.
Jul
22
comment Do groups, rings and fields have practical applications in CS? If so, what are some?
can we not do the whole "I don't need real life (sneer) applications" routine? Good for you if you don't need them, but the applications to other academic fields have always been a huge source of inspiration for mathematics and play a large part in making it so wonderfully rich. Acting like "I don't need them" $\implies$ "we should all ignore them" is just as ignorant as the reverse view.
Jul
19
answered What calculation shortcuts exist to help or speed-up mental (or paper) calculations?