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Huh, interesting. Seems like a circle is a "smooth regular apeirogon" to use an informal description.

@@arongil I personally don't see how apeirogons could exist theoretically. A polygon with infinite sides would would also have to be made up of 2d line segments (I think?) because it is indeed a polygon. But in a finite space would this not be theoretically impossible? As far as I am concerned, the sides would have to have length to them, but wouldn't this mean that there are a finite number of line segments? If a shape had infinite amount of sides, wouldn't those sides have to be points, or in other words have no length to them? But if this is the case, points are one dimensional so it wouldn't be a polygon because a polygon are made up of 2d lines, and therefore a polygon with infinite sides couldn't exist, and what you are left with is a circle. I can't find much information on this topic unfortunately.

Yes, I did. In fact, at MIT, 18.100B serves as a dual real analysis and proof skills class. Both skills are useful---likely that proof writing even is more useful as it is more foundational!

Thank you, glad it helped!

True! He does a "read along" of his textbook.

@@arongilHey didn’t expect you to respond haha. I’m watching your self study of multivariable calculus and essentially using it as a guide for my self study of single variable calculus, I’ll be doing the calculus w theory from MIT as I actually have Apostol’s calculus vol 1 textbook and utilising some of the supplements you mentioned like Prof Leonard. If you also self studied single variable calc were there any other resources you used outside of the ones you mentioned? Trying to find videos on calc w theory and haven’t yet

Hi there! If you set out a schedule and structure for yourself, you could likely finish a chapter per week spending about 10 hours per week. Best would be if you found a friend or mentor to keep you accountable while reading, and to add to the fun. Good luck!

@@arongil Thank you so much for your answer ! I did a CS Bachelor in France and i have not did so much maths and i'm trying to change this and your videos are what i needed to improve my self study I have the impression that I'm going too slowly. You say that 10 hours for a chapter is enough but, for example, for Gilbert Strang's linear algebra there are 30 exercises per sub-chapter on average which would take me maybe 10 hours at least :( do you have a discord or email to communicate please? :)

Real analysis is enough. Multivariable calculus would also be helpful. Happy learning!

@@arongil thank you for the quick reply! Very helpful, subbed!

Khan Academy has an intro to computer science in JavaScript that is quite good. For data science, I could recommend some RUclips channels like RitvikMath and Mutual Information. For computer science generally, MIT OCW has some nice algorithms courses such as 6.042 and 6.006. I'd say pick one you're most excited about and dive in. Good luck!

@@arongil Thanks for the recommendations. I watched the first videos of 6.042 and the new 6.100L that was just uploaded like 10 days ago and understood most of the things in there. The teachers there are much better than any I got. I’m gonna continue taking them and see what becomes of it.

Hi Taim Teravel, wishing you the best in your journey! I hope you find it exciting and beautiful. Let me know if you need any pointers along the way.

Yes! This is another way to see why the vertex is -b/2a. There are many ways to think about the result, as is the case often in mathematics.

Hey there! Sal Khan advocates for making sure you have no gaps in your understanding. Since it's been a while, you might have forgotten a few things from earlier math---that is totally normal, it happens to me sometimes, too!---and the key is to not get discouraged but instead go fix up those areas. I'd highly recommend learning on Khan Academy (free world-class education for anyone, anywhere). Pick a course, say pre-calculus, and try the quizzes. If you aren't getting it, go back a couple grades and try the quizzes there. Go back as far as you need to until things make perfect sense, and then work your way back up. You might go pretty quickly. The point is to figure out where your gaps are, not wasting time on what you already know. You'll progress really rapidly and have a solid understanding of everywhere you've been. By the time you reach calculus (in a matter of weeks or short months), you'll be prepared to master it in short order (another couple of months, perhaps). Then you can carry on from there to other beautiful parts of mathematics, like linear algebra or multivariable calculus. Best of luck and let me know if you want more pointers. (3Blue1Brown is another excellent resource.)

Great question! We can think of the law of cosines as an angle in [0, pi), or we can think of it as an angle in [pi, 2*pi). Imagine the obtuse angle formed by wrapping around the corner of the triangle, rather than filling in the sector. Note that cos(2pi - theta) = cos(theta). The rest of the proof follows for these angles through that geometric reasoning. More generally, you might be intrigued to learn that you can define other sorts of dot products. For example, what if (a, b) * (c, d) = ac + 2bd? In linear algebra, we actually say that the dot product defines the angle. We define angles in terms of inverse cosine of the dot product. Normal Euclidean geometry arises if we use the standard Euclidean dot product (a, b) * (c, d) = ac + bd. Anyways, nice thoughts. If you want to learn more, you might like the book Linear Algebra Done Right by Sheldon Axler. Happy mathing!

@@arongil thank you for your answer, and thank you for the book you suggested, I've downloaded it. Even though I'm not sure I've understood your answer, I mean: if we choose the standard Euclidean dot product for a*b, we have cos(tetha) = a*b / ||a|| ||b|| and with this relation we have, tetha = arccos(a*b / ||a|| ||b||), which implies that tetha is in [0, pi]. So it would seem that tetha is always bounded in [0, pi] and we could not consider it in [pi, 2*pi) as you wrote, otherwise the arccos function would not be defined. Furhermore, if in general tetha is an angle between two (possibly rotating) vectors, any restriction of tetha in [0, pi] or whichever other bounded interval does not seem to have any physical meaning to me because tetha is mathematically a real number (which means not bounded and can have any measure: positive negative or null) and physically represents any rotation (clockwise, counterclockwise or none) of any entity. How could this be explained? I'm sorry for my fastidiousness, but I'm dreaming one day to write my own book of mechanics, and I'm collecting, if I can, very precise notions. So no problem if you prefer not answering, thanks anyway! (I subscribed to your channel) ;)

Lovely!

True, these are important courses that I left out when I made this self study map. Now 3Blue1Brown has some nice videos on a couple intro statistics topics. You can also get quite far by learning some machine learning. Tons of resources online for that.

Yes, that would be an improvement. Thank you for the comment!

So glad to hear it, @gryffindor_wolveslaylah! There's no reason why we should arbitrarily hold students back from learning fascinating and beautiful math just because it's above their grade level, as if age were a good indicator. Keep on the journey, especially with series like Essence of Linear Algebra by 3Blue1Brown. Best of luck!

@@arongil thank you for the recommendation! I find true maths fairly interesting, and the concept of it, all of the aspects beyond my own age level of simple GCSE. In the classroom, Maths is simply boring and I don’t enjoy it as it is too repetitive and it takes forever for the rest of my class to understand. Not that it’s a problem that they’re slower - that’s just the way it is and that’s perfectly fine, perfectly normal and understandable. I do wish they would give more opportunities to show students at GCSE level (14-15, KS4) about proper maths instead of spending an age and a half on the simple and boring stuff.

Who is that?

Where you are writing and what recorder are you using?

@@stickmanbattle997 SmoothDraw3, recorder back then was OBS Studio if I remember right.

Good point! I had in mind that proofs are taught alongside real analysis and linear algebra.

​@@arongil and that is true, in both R.A and a linear algebra class that is theory based, but before one goes into either class, in my opinion one should take an introduction to proofs class in order to set oneself up for success in the 2 classes you mentioned, but thats just my opinion! nice video though!! :)

Hmm, it can be jarring to transition to a more proof-based class. I hope you liked it, though!

Indeed! You really can climb it one step at a time, especially with effective free resources like Khan Academy.

Go Withermax! Calculus is lots of fun --- you would enjoy 3Blue1Brown's Essence of Calculus series. It's gold.

Lovely! There's lots of beautiful math to learn. Linear algebra is especially a favorite: Axler's book is legendary, 3Blue1Brown's Essence series is art. Have fun with your future math!

Hmm, it can be challenging if school is rushing you along. While I'm not sure of the particulars of your situation, I'd recommend making math become part of your fun and relaxing normal life by watching RUclips channels like Dr. Peyam. It's fine if it doesn't all make sense at first. It's the "Dr. Peyam Show" after all, so enjoy it. But really I recommend this because learning math that's a little bit beyond what you're studying in school actually makes the school math MUCH easier. You see where things are finally used, and why you study them in the first place. So check out Dr. Peyam for example. Good luck!

Complex analysis is beautiful. I'd recommend studying calculus at a somewhat rigorous level first. Evan Chen's "Infinitely Large Napkin" is a phenomenal free PDF to learn about complex analysis (~chapter 30), along with just about every other subject on the undergraduate curriculum.