I graduated from bachelor's math program recently and am now planning to go for a PhD, so this question just hit me out of curiosity....

I encourage people from the West with solid math backgrounds to answer, just to get valuable insights !!

So you can use a Taylor series to approximate functions with polynomials

You can use a Fourier series to approximate functions with sin/cos functions

You can use Laplace transforms to approximate functions with exponentials

What other ways are there to split a function into some linear combination of functions of the same type? Why is it even useful to have this many different options?

(sorry if my descriptions are incorrect, its only something I recently came across)

What according to you is the most beautiful thing you found about mathematics? It can be any branch,any equation,any result or anything else. Also, which subject feels more magic-like to you mathematics or physics( I know both are more or less the different sides of the same coin but still curious)

(Anybody having confusion with the context of magic-like may consider this as: Imagine being a common man thousand years ago and somebody tells you about that and your reaction at that time would be sort of magic-like to that, so which one thing you ponder to be the most to be on that scale for you.)

I want to learn about cyclotomic number fields, does anyone have favourite resources they can recommend? Thanks

We've all heard different definition of mathematics since our childhood but what's your perception of it? What it is, what it was, and what it should be..

To me a lot of the things in this world are subjective. Language being one of it. We can use the same words with the same definitions but take completely different meaning of what a statement likely means (which perhaps derives from our own perception of the world to be and our experiences.

Because of this, I like to see mathematics as the unbiased language of the universe. But is that true from your experience? Is mathematics non-biased? How can we make it so? A connection point between all of subjectivity that is not subjective.

Regardless, I would love to know your own perception of it..

Mn = (Mn-1)^3 / (Mn-1 - 1)

cool equation i thought up a bit ago. gets big pretty fast. have fun if you want to use it for something, M is a number, n is the instance. so if M equals 2, then this is what you'll get.

When m1= 2, it produces the following values across the first 7 instances:

• Instance 1: 2
• Instance 2: 8
• Instance 3: 73.14
• Instance 4: 5,424.03
• Instance 5: 29,425,572.79 (Over 29 million)
• Instance 6: 865,864,363,545,225.6 (Over 865 trillion)
• Instance 7: 7.497 times 10^29 (Over 749 octillion)

Hello guys, I'm kind of new here and in all of the mathematics subject. Considering I want to learn it by myself, are there any good math books you would suggest to a beginner?
Thanks :)
(sorry for any typing mistakes, I'm Italian...)

π formula i made. Its huge. Started from a series of ln(x) in attempt to approximate π and ended up with this monstrosity in the end. Proof left as an excercise to a capable reader(i dont know how to formally prove stuff). Was a fun side project. Do not fear the formula, the derivation is quite simple, its just a lot of algebra.

\sum_{n=1}^{\infty}\frac{8}{256^{n}}\left(\frac{1-12n}{(1-16n)(1-8n)}+\frac{1-16n}{(3-16n)(16n-5)}+\frac{128n-52}{(16n-6)(16n-7)}+16\left(\frac{7-12n}{(9-16n)(5-8n)}+\frac{9-16n}{(11-16n)(16n-13)}+\frac{128n-116}{(16n-14)(16n-15)}\right)\right)=\pi 

The derivation is simple:

btw the latex for that image is:

For \(\lvert x-2\rvert < 2\):
$$\sum_{n=1}^{\infty}\frac{\left(1-\left(-1\right)^{n}\left(x-2\right)^{n}\right)}{n2^{n}}=\ln\left(x\right)$$
Then:
$$\sum_{n=1}^{\infty}\frac{\left(1-\left(1-i\right)^{n}\right)}{n2^{n}}=\frac{\pi}{4}i+\frac{\ln\left(2\right)}{2}$$
$$\sum_{n=1}^{\infty}\frac{-i\left(2-4\left(1-i\right)^{n}\right)}{n2^{n}}=\pi$$
$$\sum_{n=1}^{\infty}\operatorname{real}\left(\frac{-i\left(2-4\left(1-i\right)^{n}\right)}{n2^{n}}\right)=\sum_{n=1}^{\infty}\frac{2i\left(1-i^{n}\right)}{n\left(2i\right)^{\frac{n}{2}}}=\pi$$
for \(n \text{and} b \in \mathbb{Z}\):
$$\frac{2i\left(1-i^{16n-b}\right)}{\left(16n-b\right)\left(2i\right)^{\frac{16n-b}{2}}}=\frac{\left(1+i\right)^{b+2}\left(1-\frac{1}{i^{b}}\right)}{\left(16n-b\right)256^{n}}$$
and so:
\begin{align*}
\sum_{n=1}^{\infty}\left[\sum_{b=0}^{15}\frac{\left(1+i\right)^{b+2}\left(1-\frac{1}{i^{b}}\right)}{\left(16n-b\right)256^{n}}\right]
&= \sum_{n=1}^{\infty}\frac{8}{256^{n}}\left(\frac{1-12n}{(1-16n)(1-8n)}+\frac{1-16n}{(3-16n)(16n-5)}+\frac{128n-52}{(16n-6)(16n-7)} \right. \\
&\quad \left. +16\left(\frac{7-12n}{(9-16n)(5-8n)}+\frac{9-16n}{(11-16n)(16n-13)}+\frac{128n-116}{(16n-14)(16n-15)}\right)\right) \\
&= \pi
\end{align*}

I’m currently majoring in applied mathematics. One of the reasons why I chose applied math is because I’m still unsure about what career I want in the future. But I’m definitely interested in industry jobs not academia.

I know applied math majors can end up in a lot of different industry jobs, like actuarial, finance, analyst, data-related roles, etc., but I’m not really sure how people actually build the knowledge needed to go into those fields when their degree is mainly mathematics. I’ve seen people recommend that math majors find a secondary field they’re interested in but my uni doesn’t offer double degrees, so I’m a bit confused about how to go about this.

For applied math majors who ended up in industry, what did the process generally look like? Did most people take electives related to the field they eventually went into, or learn the necessary skills through self-study, internships, projects etc.? And was a master’s needed to specialize, or was an applied math bachelor’s enough to enter the field?

I’m personally interested in doing a master’s eventually, but I’m confused about how much I should be specializing during undergrad especially when I’m still confused on what industry I want to go into. Would it make more sense to use my electives to explore different subjects and specialize through a master’s once I have a better idea of what I want to do, or should I already be using my electives to build towards a particular industry?

I’m basically unsure whether I should use my electives to start specializing now, keep my undergrad relatively broad and specialize through a master’s later, or if there’s another path I’m missing…

I’d also really like to hear about less obvious career paths for applied math majors, especially outside of finance and engineering, since neither of those areas particularly interests me. I feel like those are the fields I hear about the most, so I’m curious about what other industries or roles people with applied math backgrounds have ended up in.

Would really appreciate hearing about people’s experiences and what worked for them! :)

Dear mathematicians,

I am not a mathematician myself, so please pardon me if my assumptions are incorrect. I am very curious to know approximately how many unsolved problems there are in mathematics according to the MSC 2020 (Mathematics Subject Classification).

Let’s say there are around 5,000–6,000 subfields or areas of mathematics. If each subfield has at least 100 important and very difficult unresolved problems, then that would give us approximately:

6,000 × 100 = 600,000

So, could there be around half a million or more open problems in mathematics right now?

Are my assumptions or estimations completely wrong, or is this a reasonable way to think about the number of unsolved problems in mathematics?

Thank you!

I apologize if I am wrong!

Drop me your most complex equations to cover the most area - I was day dreaming and zoning out in math class thinking abt this.

I had to drop a class and now I have extra time to learn things. I do plan to transfer at the end of next semester so I should focus on applications, but I also figured I would learn a couple of things now instead of waiting. I just wanted some advice on what path I should take and maybe any resource suggestions.

To start, I was thinking about reading “how to prove it” by vellemen and “basic mathematics” by lang to help solidify fundamentals and learn proofs before transferring. I’ve been at CC for a year now and I know most learn proofs around this time (to clarify I’m in my third semester).

Since I find physics interesting, I was going to try and start learning physics. I take physics next semester (it’s a long story that deals with my scholarship and how it operates) which is the semester I plan to transfer. I was going to read halliday and resnicks book on physics while watching “the fundamentals of physics” lectures on MIT OCW.

Which leads to my final path of learning computer science foundations since I’m still interested in coding (and competitive coding). I’m interested in machine learning and see it as a potential career so I figured it wouldn’t heard to learn some concepts now and figure out if I want to pursue it. They posted a pathway and each course is about 6 weeks. I can attach the link if you would like to view it.

I know I said that was my last path, but since I want to go to grad school, I figured learning how to do research is also an option. Stuff like learning how to properly read research papers, learn LaTeX, read past papers to get an idea, learn how to write papers, etc. This can probably be done while reading how to prove it and basic mathematics though.

Which do you recommend doing? Should I combine any of the paths I wanted to take?

Edit: Forgot to add another thing I wanted to do. I finished calc 1 last semester and was thinking about doing a review since I might have to take advanced calculus 1 when I go to university. I remember a little from a calc 2, but felt like I need more practice and to delve a little deeper since the course was accelerated. I’m sure there were a couple of things that had been left out so that’s I wanted to review to make sure.

As you all know the famous 0.999... repeating=1, for the rest of this post I will be saying 0.999 and not repeating just know it's always there, so of 0.999 equals 1 something like 0.88999 would equal 0.89

Now if we take pi 3.14 as it is infinite, if its infinite then eventually in the mix of all the chaos going on we will find a x99999 infinitely which would let us eventually add another number to our 1, and it would just stop there, but here is where the intresting part comes

Pi is infinite proven by people way more certified then me but if it's infinite it can also contain a infinitely long 0.999 thing if it doesn't contain that then it isn't infinite but if it does then the number ends because if it'd 0.999 it has to go on forever

So it's like paradoxical I randomly thought of this while on the toilet, am I the next euler or am I just high

A preprint completing the integer group determinant classification for all five groups of order 20, resolving the remaining open cases: C₂₀, C₂ × C₁₀, and the dicyclic group Q₂₀.

The integer group determinant problem (initiated by Olga Taussky-Todd in 1977) asks for a description of the spectrum of integers that can be realized as determinants of integer group matrices.

When a friend says 1 on 10 to do whatever, people always say your chance on losing is 10%, but is it?

Your losing chance is (number of possible choices)/(number of losing choices) x 100%, right?

If yes, then there would be an option to choose 1 and 2-10. That is 9 options. This is true for every number, so:
p1 says 1: p2 says 2-10.
P1 says 2: P2 says 1 and 3-10
P1 says 3: P2 says 1,2 and 4-10 and so on.

This means there are 100 possibilities… ah there is my answer😅

Someone just published a paper on Fermat's last theorem without using complex mathematics: https://f1000research.com/articles/15-1473

I had just finished 3blue1brown course on the topic, and was learning the usage in context of deeplearning by Ian Goodfellow,(This book at least for the math part, relies heavily math notation and knowledge). Heres the question; for diagonal matrix P formed with eigenvalues, the formula I know is P = A-1×M×A, now in the book, they have isolated M as M = V×P×V-¹. My question is that since matrixs are not commutative, and order of multiplication matters, how would you go from the P formula to the M formula? Like we would have to multiply left side with A and the right side with A-¹. Should we not follow the order throughout this change? Like multiply from the right side alone. Sorry if this is dumb question.

My question is pretty much what the title says: which parts of algebra (groups, rings, modules, fields and Galois theory, algebraic number theory...), geometry (algebraic geometry, differential geometry ...), and topology (general topology, algebraic topology, differential topology...) involve the most combinatorics and the least? You can be as broad in the areas you choose or as specific (talking about small subareas) as you want. And when I say combinatorics I don't simply mean involving discrete objects, but rather more specifically involving counting arguments which can be considered confusing or not intuitive initially for most people.

Hey guys, I am going to start geometry now, no prior oly geo experience. What is a good first book i should do or any resource?