Articles of soft question

Is any type of geometry $not$ “infinitesimally Euclidean”?

Question: Is there any (absolute) geometry which is not “infinitesimally Euclidean”? Context: All of the geometries listed on the Wikipedia page “Foundations of Geometry” (describing axiomatic formulations of geometry) seem to correspond to special cases of absolute geometry, and it seems like any absolute geometry is either hyperbolic, elliptic, or Euclidean (parabolic?) according to the […]

How to deal with misapplying mathematical rules?

I have noticed that though I understand the mathematical material in my classes nigh perfectly, I still make frequent careless mistakes. Roughly 55% (95% CI: (37, 72) of these errors are due to me misapplying mathematical rules. For example, I may in a derivation go from $a = 4(c + b)$ to $a = 4c […]

Weil conjectures – motivation?

Can anyone explain (heuristically, intuitively is fine) what the importance of the Weil conjectures is? I realize they have motivated much of recent algebraic geometry. I don’t really understand why or what one should do with them, now that they have been proven.

Reflections on math education

Why is there such a big difference in math education between The Americas and (Europe and Asia) ? except for a few privileged who have the opportunity to access to math much earlier than the ordinary people. The Question arises when in my first class in my topology class, my professor, who is russian, said […]

When do we use hidden induction?

If I’m correct, hidden induction is when we use something along the lines of “etc…” in a proof by induction. Are there any examples of when this would be appropriate (or when it’s not appropriate but used anyway)?

What is the motivation for complex conjugation?

I have been dealing with complex numbers for few years now. But when I’ve tried to think about the motivation behind complex conjugation, I was not sure. Let me write what I am working with. For a complex number $z \in\mathbb{C}$, where $z=\operatorname{Re}z+i\cdot \operatorname{Im}z$, we define complex conjugate of $z$ as $$ \overline{z} = \operatorname{Re}z-i\cdot […]

Find the Roots of $(x+1)(x+3)(x+5)(x+7) + 15 = 0$

Once I came across the following problem: find the roots of $(x+1)(x+3)(x+5)(x+7) + 15 = 0$. Here it is how I proceeded: \begin{align*} (x+1)(x+3)(x+5)(x+7) + 15 & = [(x+1)(x+7)][(x+3)(x+5)] + 15\\ & = (x^2 + 8x + 7)(x^2 + 8x + 15) + 15\\ & = (x^2 + 8x + 7)[(x^2 + 8x + 7) […]

How to fill gaps in my math knowledge?

Just finishing highschool, even though I am doing “well” (in the context of the math course itself), I have significant holes in my actual math knowledge. As I think many people who explore math outside the typical walls of highschool will realize, the math in high school is often very selective, taught with shallow depth. […]

A question of H.G. Wells' mathematics

H.G Wells’ short story The Plattner Story is about a man who somehow ends up being “inverted” from left to right. So his heart has moved from left to right, his brain, and any other asymmetries belonging to him. Then H.G Wells’ goes on a slight metaphysical exposition: There is no way of taking a […]

Derive or differentiate?

When the action is: Taking the derivative what verb should be used? to differentiate to derive I feel that deriving is not the correct word here. In my mind it’s more a synonym of deducing. Am I right or has the word derive got the same meaning as differentiate? Or perhaps differentiate is not a […]