Articles of soft question

parallel postulate of Euclidean geometry and curvature

In elementary geometry, we have two standard examples which violate the (strong) parallel postulate of Euclidean geometry: in hyperbolic geometry, we have more than one parallel through a point which doesn’t lay on a given line, and in spherical geometry, we have no parallels at all. In both of these geometries, we have some kind […]

What is importance of the Bunyakovsky conjecture?

Bunuyakovsky conjecture states that: An irreducible polynomial $f(x)$ of degree two or higher with integer coefficients and property that $\gcd(f(1),f(2),……)=1$ generates for natural arguments infinitely many prime numbers. Wikipedia article claims that this is an important open problem. My question is: how important do you consider the answer to this problem, and why?

Is it possible to practice mental math too often?

I’m far from a mathematician, but the field I’m trying to break into (management consulting) requires a fair amount of mental arithmetic. I’m okay, but I’m not even close to as good as I need to be in terms of both speed and accuracy. I have math apps on my iPhone. I use online mental […]

Would nonmath students be able to understand this?

For a course, I am required to do a presentation. The topic could either be something mundane, like a career strategy report, or something more interesting, such as a controversial topic, or an exposition on something you find interesting. What I would like to do is to present math in a way that probably no […]

Why Study Critical Polynomials?

In dynamical systems, I often read about the post-critical orbits. As in take a moduli space of functions $f$ which are self maps. Find general critical points, and see where they orbit. They would then be polynomials in some variables if we allow $f$ to be parameterised. Those are called critical polynomials. It could go […]

Correct formulation of axiom of choice

In a paper, Asaf Karagila writes: Definition 1 (The Axiom of Choice). If $\{A_i \mid i ∈ I\}$ is a set of non-empty sets, then there exists a function $f$ with domain $I$ such that $f(i) ∈ A_i$ for all $i ∈ I$. Does this formally make sense? Shouldn’t it say If $(A_i)_{i\in I}$ be […]

Intuitive explanation for $\mathbb{E}X= \int_0^\infty 1-F(x) \, dx$

I can see by manipulating the expression why $\mathbb{E}X$ works out to be $\int_0^\infty 1-F(x)\,dx$, where $F$ is the distribution function of $X$, but what is an intuitive explanation for why that is true? If at each point we sum the probability $\mathbb{P}(X>x)$, why should we end up with the expectation? Thanks

Deeper meaning of “canonical”?

For years I thought “canonical” for isomorphisms just means something like “obvious”, “without arbitrarily choice”, “simple” depending on the context, and I accepted the idea that there is no deeper meaning of “canonical” and no clear definition. Reading about categories I noticed that “canonical” is sometimes used when there’s a “natural equivalence” in the language […]

What is the best way to tell people what Analysis is about?

What is the best way to tell people what Analysis is about? I am currently taking Analysis course. However, I am really having a big difficulty explaining to people what Mathematical Analysis is about. Does anyone have any idea how to do it?

If $i^2=-1$, then what about $(-i)^2=-1$?

This question already has an answer here: How to tell $i$ from $-i$? 6 answers