Self Study Map of Mathematics

[@SteamShinobi]

THIS IS THE VIDEO I HAVE BEEN LOOKING FOR. I couldn't stand math in high-school, but I love branching into every subject. So I got my degree going in liberal-arts, linguistics to be exact, and then I used the knowledge I gained there to go back and re-learn every other subject I could think of. The only issue was I knew most of the sciences very well in terms of where to go from point a to whichever destination, but math, I am illiterate af. I last touched polynomials, and I have had no idea where to go since then to keep studying math so that I can move beyond grasping the theory of sciences, and actually procede to learn formula, understanding equations better, having better overal mathematical literacy, and then hopefully also brancing into applied mathematics via the combination of theory & practice. I may just be a liberal arts student, but I am so grateful this video has provided the guidance I need to now move past a plateau that has kept me confusing myself and running in circles, hopefully I can actually come out of this as someone who is literate in math, a dream of mine, and I am super grateful.

[@arongil]

That is incredible to hear! If you'd like ideas about where to go when studying math, feel free to ask me here and I'll try to get back to you. If you last left off with polynomials, one place I'd start is the KZbin channel 3blue1brown. He does animations of math and is the *best* channel on KZbin (in my opinion). I can't recommend him more highly. He has a series called Lockdown Math which he started during COVID-19, and it covers some high school math topics in an interesting, engaging way. Check it out! The thing with math is you will get confused sometimes. It's part of learning -- I'm confused almost all the time when learning math, because the edge of my understanding is where I grow. So don't get discouraged at any point along the way. It's a wonderful journey you'll be glad you undertook. And do check out 3blue1brown. He is so, so good at making math accessible and beautiful. Thank you very much for your comment, and best of luck to you, SteamShinobi.

[@throwawayavclubber7269]

I second the recommendation of 3blue1brown.

[@goldenwarrior1186]

Other YT recommendations: numberphile and mathologer (I feel like you’ve heard at least one of these before)

[@ksilofo]

Hello! Did you see the progress you were looking for?

[@SteamShinobi]

@@ksilofo I did! I decided to go the route of going through Serge Lang - Basic Mathematics and using Real Physics as a mentor (They have a playlist of the entire textbook; you can find the textbook online). Basic math goes through: Algebra: numbers, linear equations, real numbers, quadratic equations; logic and set theory; Intuitive Geometry: distance & angles, isometries, areas and applications; Coordinate Geometry: coordinates & geometry, operations on points, segments, rays, and lines, trigonometry, intro to analytic geometry; Misc: functions, mapping, complex numbers, induction and summations, and determinants. Afterwards, I went to Sege Langs books on Calculus (First and Second lesson) and Abstract Algebra. It's actually made a tremendous impact on my personal life despite not being in a profession where I frequently am required to know math to this degree. I'm currently getting into topology, number theory, computer science, and analytical geometry. Thanks to y'all for responding to this and reminding me about this comment, this video made such a big impact on my life it's actually wild.

[@adrianarreguin1739]

I just started college at 27 and have to re-learn my high school mathematics(Forgot a ton of stuff)! This gave me a road map of where to begin! Thank you for the in depth explanation

[@arongil]

You got this! Good luck, Adrian. Let me know if there's anything else I can do to help.

[@cameronfirmin5715]

Well done, Mate! Back to school at 24 for me. Finished up my first year strong in college algebra and pre-calc I took another year off and I’m learning calculus through Khan Academy before I go back for the proper 4 years. 😁 Love math!

[@arongil]

@@educlarity Yes, it would be great to make more. I'll see if I have time soon.

[@corbinwilson660]

I remember watching this video at the beginning of my math journey. I now have an associates in math, I’ve taken calc 1-3, Linear Algebra and ODE. Currently I’m studying proofs so I can move onto even more mathematics like analysis and abstract algebra

[@arongil]

Your story is inspiring, Corbin! Thank you so much for coming back here to share your success. Abstract algebra is a beautiful branch of math, and very useful despite its name. Good luck!

[@Mynhassty]

You should make a video giving book recomendations for every subject in this self-study map!

[@arongil]

Hey, that's a good idea! I'll keep it in the back of my mind. For a couple subjects I know I've come across good books (i.e. Sheldon Axler for linear algebra), although I find the most useful recommendations come from people who've read at least three textbooks in the given subject. Then they can say that a book isn't just good, it's the best they've seen. I may not be there yet, but I love the idea so I'll keep it in mind.

[@elultimopujilense]

that would be just awesome!

[@hafeazomany2479]

one of best book is.... math hack ...it is amazing you can download it free from pdfdriver.COM....which i did

[@filipnoble5146]

do i take caculus ab before caculus 1 ?

[@egyptianplanner]

YES YES YES PLEEEEEEEEEEEEEEEEEEEEEEEEESE ??

[@Phylaetra]

So - 'linear algebra' is something you will see in algebra II, then a sophomore/junior level course in college, then _again_ as a senior/early graduate level course. You learn new things each time. I would _strongly_ recommend some linear algebra before differential equations. It's not necessary, but it is _super_ helpful. Once you start getting into 'proofs' courses, you need a tutor who can at the very least point out your errors in the proofs. Errors can be very subtle, and you simply cannot learn it on your own because you cannot know when you have made an error. You also want to work on your writing skills. A senior-level Real Analysis proof may be a couple of pages long and include a significant amount of prose writing (that goes for pretty much any of the non-computational courses). The higher levels of mathematics are largely about _proving_ things, and it is really a social activity. Try to find at least one other person to work on more advanced things with and bounce ideas off of each other. Even working on the same proof, you may find different approaches (and possibly different approaches from what an 'answer guide' might provide. You should also get used to reading mathematical literature - papers, books, etc.. I will say that statistics is good to know, especially if you start looking at more advanced analysis (which gets you into a field called 'Measure Theory', tying statistics into 'regular' mathematics. Galois Theory is interesting, but not actually that important if you are more interested in various aspects of 'finite' mathematics (like number theory, combinatorics, graph theory, numerical methods etc.). Not mentioned is the extension of Real Analysis into Complex Analysis (that is, calculus involving complex numbers - that is numbers that include the square-root of -1). Differential Equations extends into Partial Differential Equations, which can then lead to Functional Analysis and Harmonic Analysis. And at some point you start to see how all these 'different' things all tie back together (like with Analytic Number Theory, which uses techniques for Real and Complex Analysis to approach problems involving (for example) the distribution of prime numbers). Let's see - extension of Real Analysis into Topology. But then there is Algebraic Topology as well. Etc. Etc. Etc. But - really - find a math buddy and work on things together and have at least one person who knows a _lot_ who can check your work on proofs, once you get there.

[@blocksource4192]

I hate school. I hate the way I learn in school. I hate everything about school. But I love learning. I'm a self-taught programmer and definitely would like to know more about math.

[@arongil]

“I have never let schooling interfere with my education." -- Mark Twain Keep learning. KZbin is a great resource, and you can go beyond it by finding people in your community who care about math, programming, and technical problems. You've got this.

[@SharkyShocker]

I have to say, as someone that ended highschool taking Algebra II and then going through Trig, Pre-Calc, Calc 1, Calc 2, and now struggling with Multivariable Calculus... it was sobering for everything previous to be referred to as "normal high school student" level.

[@zeldamage001]

I'd say if you continue the branch of analysis, you end up at functional analysis (which links to differential equations), topology and differential geometry. The last one is the domain of Gregory Perelman. Different than abstract algebra, but certainly not less difficult!

[@vilkoos]

Excellent route to learn the math you need for classical physics and the engineering that is based on it (e.g. mechanical or electrical engineering) . . . this is probably not the best route if you want to end up in pure mathematics or computer/data science . . . then you might start at set theory and logic, linear algebra should be done as soon as possible, discrete optimization can replace calculus . . . so, before you plan a route you might want to consider where you want to end up.

[@parammaewal1497]

Would love if you could do this video again and add prob/stats, and discrete math

[@123ucr]

Here are the math classes that you need to take in order: Arithmetic/Pre-Algebra Algebra 1 Geometry* (*You can take this concurrently with Algebra 2) Algebra 2* (It is possible to pass Algebra 2 without knowledge of Geometry but difficult. I recommend taking Geometry first or concurrently enroll with Geometry). Trigonometry/Pre-Calculus Calculus 1 Calculus 2 Calculus 3* (This class can be taken concurrently with Linear Algebra and/or Differential Equations. That being said, I would now take two math classes at a time. I would recommend Calculus 3 and Linear Algebra before Differential Equations.) Discrete Mathematics/ (Proof Writing) (I would take this class with Differential Equations after completion of Linear Algebra and Calculus 3.) Elementary Linear Algebra* (If your Linear Algebra class is heavy on proofs, take a Discrete Mathematics class concurrently with Calculus 3). Elementary Differential Equations* (It is possible to pass this math class without knowledge of Calculus 3 and Linear Algebra but difficult.) After you are done with the college sophomore level math classes here are upper division and Graduate division math classes you can take: Partial Differential Equations Linear Algebra 2* (Linear Algebra with proofs) Abstract Algebra 1* (I would take this after or concurrently with Linear Algebra 2) Differential Geometry Number Theory/Combinatorics Real Analysis/Advanced Calculus Complex Analysis Vector Analysis and Tensor Calculus Galois Theory* (Which is one graduate level math course)

[@arongil]

Beautifully said, 123ucr. That's a practical roadmap to follow. For all the courses on your list, up to Calculus 3, learners can find world-class free resources on Khan Academy.

[@Mooseableg]

when do ppl learn non euclidean geometry?

[@holymotherofgodpleasenobut9626]

you don't know how helpful this is to me omg, I've been finding a way onto where to start, since online school ruined me with my math studies😭😭

[@ayafaustino4752]

I am starting a pre-medicine college course next year, and we are taking some advanced math classes there so I figured that I'd go over every lesson from the beginning to fill in knowledge gaps. Math has always been difficult for me but because of school closure, I found that I learn Math better at my own pace when I self-study. And this helps A LOT. Thank you for saving my high school academics, and now you're saving my uni grades

[@arongil]

Dear Aya, this means so much to me! It's wonderful to hear you find self-study more conducive to learning. If you're interested in another source to learn from, I like a free, online book called "Napkin" by Evan Chen. He's a PhD student at MIT who tried to write up all of undergraduate mathematics at the level of an advanced high schooler. It is pure math for sure, so maybe exactly what you're looking for or maybe not, depending on your goals. Sending my very best wishes from Massachusetts!

[@ayafaustino4752]

@@arongil Whoa, Evan Chen's metaphorical napkin is huge! Like 900 pages huge. Thank you for that recommendation, and I see myself coming back to it for years to come as a student (and a professional) in the STEM field. God bless you, sending my regards from the Philippines!

[@BiancaAguglia]

@@arongil Thank you for this video and thank you for the book recommendation. I checked Evan Chen's website and I saw that he has a few other resources in addition to the book. Interesting and fun reads. 🙂

[@TheThreatenedSwan]

While I enjoyed my pre-calculus and calculus classes, math in school was one of the most tedious areas where the traditional teaching/learning methods and time in between courses meant you were always annoyingly reviewing. It's also like there is a kid mode of teaching and then the normal mode where you're shielded from the actual language of the field and the resources to really dig into it.

[@arongil]

Agreed. There's a lot the education system could improve upon. There's a large literature on theory of education. For math, one particularly evocative one is "A Mathematician's Lament" by Paul Lockhart.

[@wofker4719]

this is criminally underrated! they say school teaches you beyond just subjects like life blah blah blah but they dont even teach you this

[@arongil]

Yes, it would be helpful if school explained the map of where we are all going early on. Like, "If you study for a few more years, here's where you can be." Knowing the incredible things we can do once we know math might motivate many children. See "A Mathematician's Lament" by Paul Lockhart.

[@wofker4719]

@@arongil exactly the map is helpful in any subjects even in gaming

[@jeanmarc7563]

Statistics takes the back burner since its mostly used for the social sciences, but its definitely something that should be included as it's often used in multiple mathematical and engineering disiplines. Most real world applications of mathematics will utilize statistics to some degree, because of the uncertainty of non ideal situations.

[@maxgriswold2141]

One point of note is that if you're looking to approach statistics from a purely mathematical angle as opposed to an applicatory one, it's based on the fundamentals of Measure Theory, a branch of Real Analysis. Understanding sigma algebras is key to having formal justification for probability theory. Statistics is wildly useful without these tools, but it can't be fully understood without them.

[@shortfusedynamite5166]

This is so nice thank you. Just went back to school after almost 6 years and took like a warpspeed level math class that I got totally lost in and have been scouring the internet for something like this.

[@arongil]

Glad to help out!

[@tamirlyn]

Propositional logic > First-order logic > Set theory > Abstract algebra > whatever you want.

[@epicmarschmallow5049]

You don't need to learn any serious mathematical logic or any serious set theory (beyond basic definitions) before abstract algebra

[@tamirlyn]

@@epicmarschmallow5049 You don't need to learn any mathematics to write a compiler either. And yet we teach everybody CF grammars and push-down automata. Do you want surface, or full understanding? Depends on what you do I suppose. Still, logic and set theory are considered the foundations, and taught at first year of every technical university around here, as a starting point to developing mathematical capability.

[@priestofbenism1775]

Yknow if I had even just this in middle/high school, I might have gotten through so much more of math - maybe I'm unique (I have no idea) but high level overviews with targets give me motivation in the form of just wanting to understand what's at the end.

[@arongil]

Yes! Math is such a beautiful subject, and schools don't do a good job of showing you where you're going with all the things they make you learn. Wouldn't it be so motivating to see the kinds of wizardry you can do with math once we have tools from linear algebra, or cryptography, or multivariable calculus as applied in countless fields? What if students knew that Google started out as a search engine based on Markov chains from linear algebra? I'd be way more motivated knowing that!

[@Baseds_Backup_Account]

This man singlehandedly teaches us better than our own teachers. God bless him.

[@connor9024]

It’s almost like my phone could hear me talking to recommend this video!!! I returned to school at 25 to get a degree in economics, I hadn’t done any mathematics past simple algebra since high school, so the school required me to take many refresher courses. So far the math has been my favorite aspect of the degree and I want to take my understanding farther!

[@arongil]

Woohoo! I'm so happy for you, connor9024. There are so many beautiful ideas in math. Economics is a great track to pursue. Keep learning the math. At some point the math will feel hard (banging head on wall for 6 hours on one problem...), and at these moments, remember the reason you first loved math. Especially if you're working with friends, you'll make it through. Good luck!

[@tinaaaaaaaaaa3651]

this is really what I was looking for. Thanks a lot dude😊

[@TaimTeravel]

Oh myyyyyyyyyy goshhhhhh This is exciting. My grades are so bad I couldn't apply to college, but I love math and am determined to learn it. Thank you so much.

[@arongil]

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.

[@brylanjacobs9139]

I would definitely advise learning discrete math before university level math. You’ll learn to be abstract and how proofs work

[@victorgscardoso]

I am engaging a “self-study” path to try to “get back” the college math level I had some 20 years when I took my Mechanical Engineering masters degree. My question is the following: I have four time slots per week (1-2 hour’s/slot) to learn math. The topics I want to learn are Algebra, Geometry, Discrete Math and Calculus. There is no deadline pressure. Is it better to make one session of each topic per week or to focus on one or two at a time? E.g. is it better to do an Algebra and a Geometry textbook for some months before moving to the others, instead of “pulling the four wagons” simultaneously?

[@arongil]

Hi, Victor! I'd go with 1 to 2 subjects at a time so you can focus on them more deeply. Four subjects at once might be overwhelming. I'd also dedicate some time to math exploration -- there are some wonderful KZbin channels (like 3blue1brown for example) that can excite you about math and introduce new areas to explore. For me, the best way to learn algebra was to learn calculus, the best way to learn calculus to learn multivariable calculus, etc. Since you're free from school-imposed constraints and deadlines, I'd approach your learning with exploration and fun in mind, i.e. feel free to go to more advanced topics in order to motivate the earlier ones. Maybe you're all set here already since you learned everything 20 years ago already. Projects are also really great since they'll force you to put your ideas into practice. Programming physics engines in particular was a favorite of mine. All ideas above are suggestions. But to your original question, 1 to 2 subjects at once is what I'd go for. Wishing you luck!

[@victorgscardoso]

@@arongil Thank you for the your detailed answer and insight on the subject. I believe that your approach makes sense and will try to set it in place. Also the programming part is another passion of mine. I made my specialization in Thermodynamics and Heat Exchange and, at the time, I got real pleasure out of developping some simulations of dynamic systems using Fortran. So my first step is to get the rust of the math and then (or in parallel) go into the programming bit. Once again thanks and keep on the great videos!

[@jacksonh2083]

@@victorgscardoso It is up to you, but a brief overview of basic discrete mathematics can make it easier to understand the rest. Of course I’d recommend revisiting it to understand it more, but ye, just a thought. Still, no matter how you approach it, make sure you do practice problems/quizzes and revisit old subjects every now and then.

[@matteogirelli1023]

hi, nice video. I think you should have added another "pathway" for geometry --> topology that somehow connects to set theory and proofing, which connect back to real analysis. Moreover it would be important to highlight that real analysis very much related to calculus but much more rigorous and at the same time fundamental in statistics, which is another "pathway" you could have added (combinatorics -> probability -> stats, that also math students often take).

[@Selftaught_ai]

Thank you! Finally, a resource that get's straight to the point

[@johncipolla8335]

I have always wanted to re learn everything I learned about math from the beginning. to have a better understanding of mathematics now that I am older

[@arongil]

Yes, that's a great goal. I wish you luck in your journey. There are so much great ways to learn math now with the internet. I'd recommend 3Blue1Brown if you haven't found him already.

[@GM-xk1nw]

Thanks dude, really helpful.

[@letsgamepc2544]

Algebra 1 -> Geo ->Algebra 2 -> Pre calc -> Calc1 -> Calc2 -> Multivar Calc -> Liner algebra is my current history for math!

[@arongil]

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!

[@matthewzeller5026]

I don't know why this was recommended to me but it was really informative, thank you 👍

[@circushonk9839]

Thank you, very helpful, was looking for this

[@douglasstrother6584]

Start with Geometry: it's visual, tactile, and teaches one how to think like a Mathematician. Algebra I was rough for me (I felt like "Patrick".) Geometry turned-on my Math Lightbulb; then the following courses were difficult, my overall retention and skill was much higher than Algebra I.

[@fleunaflicfleunaflic6386]

Can u please elaborate more on how geometry would make us think way better than to first study algebra I?

[@douglasstrother6584]

@@fleunaflicfleunaflic6386 Great things about Geometry: it's visual (you draw pictures), tactile (pencil, straight-edge & compass), proof by construction, and based on axioms that young kids can understand. Geometry preceded Algebra by centuries as well.

[@hindelkihal3804]

@@douglasstrother6584 you wrot a boock about cancer i think

[@kichigan1]

Trigonometry is intellectually fulfilling to study. It's very visual and practical and more extensive that Geometry. However, there are some awesome problems that are more practical to solve them with Geometry. Algebra is an awesome language.

[@arongil]

Yes! The history of how we arrived at the language of algebra is also fascinating. Many, many developments occurred in the 1700s - 1800s. You might enjoy reading into the history.

[@kichigan1]

@@arongil Thanks. If you have some book suggestions let me know.

[@joshuajose8598]

I think a basic combinatorics course is also necessary.

[@arongil]

Good point, I'm taking combinatorics now!

[@withermax1234]

Started taking AP Calc this year, can’t wait for higher level college math.

[@arongil]

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

[@IAmAwesomeSoAreYou]

What section does probabilities and statistics apply in?

[@arongil]

At a high school level, you could fit basic probability and statistics in anywhere at or before calculus. At a college level, you can pursue these subjects with ever more finesse until they become graduate level topics, for instance if you do probability with measure theory or if you explore how statistics applies to AI or other advanced examples. So you can start anywhere, and there's always more to learn. Hope this helps :)

[@aa898246]

with this knowledge i will create the elundis core...

[@duckgo8755]

Thank you for this

[@thatkindcoder7510]

When studying math in Universities, do you get a really broad overview of the subject (I.E. covering all those subjects in some detail), or can you really go into one specific branch of math (like group theory and stuff)?

[@arongil]

Right now I am studying math in university. Usually, a student starts out more broadly. In the last year or two, a student can dive deeper into a specific branch like you said. A student can also do research with a professor if they want to go above and beyond. I hope to cover a variety of areas relatively deeply if I can!

[@travelingwithyoumyfriend6042]

Thank you for your video making me have a way to learn maths again.

[@theentrepreneur607]

Can you share what topics or concept should I learn from each or ATLeast what’s needed from each to be competent enough in the next course

[@arongil]

Yes! Khan Academy is a great resource (completely free). You can do each course here on Khan Academy. They have videos to learn and exercises to practice. Once you hit 90% mastery in a given subject, you know enough to move on to the next course. (Caveat: if you clearly know material in a course, you can just skip that course until you find the right level to start at.)

[@ibrahemtaha8177]

Many Thanks for sharing your amazing video with us, bro! could you please make another video in which you show us a big map, you know, include where Probability and Stats should be? and when to learn them? after which categories? Thanks in advance!

[@arongil]

Thanks so much for the comment, Ibrahem! I'm pretty busy so I may not get to another map of math video for a while, but I can give you my thoughts here. There are many levels of stats and probability. You can understand the basic ideas after you know a little algebra -- this would be intro college statistics. Intro probability you can also understand knowing algebra and some basic combinatorics (counting, i.e. how many permutations are there of the letters in the word "MATH"?). The next level of statistics and probability (my preferred place to start) would incorporate calculus from the very beginning. You'd learn about probability density functions using integrals and derivatives and all that cool stuff. The final level of probability at least, which gets into some serious pure math, is called measure theory. This strikes at the heart of what we even mean when we refer to chances, and it resolves some nasty paradoxes that can arise with a more naive approach. (See the Banach Tarski Paradox video by VSauce, for instance!) On the statistics side, there's a lot of depth I don't even know about. There are so many techniques. Some use linear algebra, multivariable calculus, and more. Bayesian Updating is a cool idea -- check out distill.pub/2020/bayesian-optimization/. There really is an endless amount to learn in any field. I'd say start learning now! Have fun and good luck :)

[@ibrahemtaha8177]

@@arongil Many Thanks for your amazing answer!! wow!!! just one last question brother: mmm.. could you please, one day in the future, make a full road map video answering "How to become the best Cryptography Cryptanalysis expert in the world" (both, cryptography & cryptanalysis) Many thanks in advance!

[@yashwantjodha818]

What courses should I take for Mathematics involved in Computer Science ? Any idea ?

[@arongil]

Hello, Yashwant. I would recommend you check out this lecture series by MIT called "mathematics for computer science." They are all quite well made, and I believe it covers exactly what you are looking for. Good luck with your studies! kzbin.info/www/bejne/gmSvfpWmj65-nrM

[@zeldamage001]

A more generally (and probably less useful) answer to your question is: mathematical logic and discrete mathematics. These are two topics that particularly applications to CS.

[@convexpolygon]

This is so useful! Im halfway through college algebra and college geometry. Could you please do a similar video on physics and chemistry? I would really like to know those too!! Great video anyways!😊

[@arongil]

Hey Convex Polygon! Glad this video was helpful for you. If you want some recommendations for physics and chemistry, here are some: www.youtube.com/@ChemistryButSimple/videos kzbin.info/www/bejne/kJrLqqqqnt2Ij9E

[@ChemistryButSimple]

Hey Convex Polygon! I just uploaded a chemistry self study map in your honor! Enjoy :) kzbin.info/www/bejne/h4G6npl4fsSFftE

[@pichirisu]

in the states a lot of region will consider calculus as an advanced course for high schoolers. I don't understand why this is labelled as "high school" math, when even now it still is not.

[@arongil]

You make a good point. Traditionally, calculus is a college math course. Now, many high school students take it through the AP curriculum. My own view is students should learn calculus whenever they feel like --- which could potentially be when they are very young, if they want to!

[@aralmariv1207]

Great video content! Where would Discrete Math fit in? What about Pure Math?

[@arongil]

I've answered somebody else's comment about discrete math here. If you ctrl-F for it you can see my response. For pure math, see my videos on how to self study linear algebra and abstract algebra. These are two foundational areas of pure math. The idea in pure math is to develop math for math's sake, that is, to further our understanding of the underlying structures and connections in math itself. That's as opposed to developing math for an application, such as applying linear algebra to machine learning.

[@maleforthedragon5177]

Referenced this multiple times; Thx a ton

[@arongil]

You're welcome! Glad you found value out of it.

[@gamma_dablam]

IMO it's possible to jump from Precalc or Algebra II to Real Analysis. You don't need Multivar or indeed any Calc to start Real Analysis imo. If anything it may well be easier.

[@arongil]

Perhaps. Certainly, it's worthwhile to ignore lots of standard advice about how long to spend studying courses and in what order. Self study liberates the student to go at his/her own pace.

[@minidupasan1520]

I am learning mathematics by myself because I really want to be good at it and I have a question about the order I should study. But I am expecting to study, 1. Algebra 1 2. Algebra 2 3. Precalculus (college algebra and trigonometry) * I am currently studying college algebra and completed algebra 1 and 2. 4. Calculus 1 5. Calculus 2 6. Calculus 3 7. Differential equations Please correct me if this is wrong or tell me If I am missing something here. And also please tell me when should I learn linear algebra in the above order?

[@arongil]

Hi Minidu, you have a good plan. Linear algebra would fit in nicely before multivariable calculus (Calculus 3). Even more than multivariable calculus, differential equations becomes more intuitive once you know linear algebra. I'd recommend to learn linear algebra before (7) and preferably before (6). I have another video with study resources for linear algebra, plus an important distinction between a computational and a theoretical course. I hope these thoughts get you started. Most of all, enjoy the journey!

[@minidupasan1520]

@@arongil I am very much grateful... I will check the video of study resources for linear algebra on your channel🙂

[@deathitself6703]

the tech tree for math

[@arongil]

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

[@Kyle_Bu]

Recommendations anyone?

[@elijaharulandu6597]

It seems like you are currently taking a Galois Theory currently. you explained it a lot more than the rest. Are you enjoying the class?

[@arongil]

Yes, I was taking Galois Theory at the time! I loved the course, and it also was maybe the most challenging course I had ever taken. You can see my video on self study strategies for Galois theory for more information.

[@jordankeith1821]

im 26 and its been quite awhile since studying any math, i still have a pre calc textbook from my first semester of college (that was my only semester lol) do you think it'd be fine to just go through that and then go to calc? or should i go back even further and find an algebra textbook?

[@arongil]

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.)

[@majidshahbazi68]

thanks

[@gone3394]

What do you use to write your notes?

[@arongil]

SmoothDraw3

[@jeffocks793]

These are great video overviews and guides - many thanks! btw is the MIT link expired?

[@arongil]

Thank you for letting me know. I've added an Internet Archive copy of the website so you can look at it again.

[@Meurantbotha]

Where does trigonometry fall in here?

[@arongil]

Trigonometry would fall between Algebra II and Precalculus. The video "tattoos on math" by 3blue1brown has an excellent visual introduction. Lots of memorization is not the way to go for trigonometry. Instead, I would try to understand a few key visual ideas and then derive the identities as needed from Euler's Equation. (There's another 3blue1brown video about it.) Hope this helps and good luck!

[@yenilmezoguzsilahtaroglu3776]

YOU ARE THE BEST

[@arongil]

Thank you!

[@TheSdog9]

I'm from Australia and the link appears to be broken. A quick google fixed it but just posting in hope that it may assist in a fix. Thanks.

[@soygolpista]

Wish there was another calculus class to take Stats is boring

[@arongil]

I agree. I prefer statistics when it is covered using calculus. You might like a KZbin channel called Mutual Information. His videos do a wonderful job explaining statistics visually, often using calculus along the way.

[@leube13]

Are there any groups ( reddit, discord, etc. [ only groups I know that why I put etc. ] ). That actively communicate about math and can help you understand better?

[@arongil]

I'd recommend the community around 3blue1brown, for example the SoME Discord. There are also some math circles such as Euler Circle that you might enjoy.

[@timhotty4837]

it was rlly weird taking calc in junior year of HS and then straight to Group Theory in senior year

[@arongil]

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

[@kentozapater8972]

Maybe i missed it but besides learning this in a mathematics degree, does something exchange in an engineering? (I know it is very broad)

[@arongil]

Not sure what you mean by "exchange." If you mean are these topics relevant for engineering, then I'd say absolutely for differential equations and linear algebra and multivariable calculus... maybe less so for abstract algebra.

[@Bbb78651]

would complex analysis usually only require real analysis, or are there any other dotted branches that help?

[@arongil]

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

[@Bbb78651]

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

[@rahul.s.r5485]

What is the map of pre algebra, I’m bad at math

[@MattHudsonAtx]

I really like this but red on black? I can't even see it

[@arongil]

Yeah, as a stylistic choice I might have opted for a different one if I'd done it today.

[@eddycolon1986]

Where's logic position?

[@arongil]

Logic is good to study at a basic level as early as possible. This means proof techniques, and, or, not, xor, De Morgan's laws, and so on. There is a lot more to logic later on which I haven't studied yet myself, but the rabbit hole goes quite deep. That would come after courses like calculus and linear algebra. Math is a lot like a tree later on though, so there is nothing stopping you from exploring one branch before others. Have fun learning!

[@EpicIntegral]

What about computer/data science? I barely passed high school maths but now I’m determined to start again and finally do something with my life. I’m taking some programming and data science courses and would love to know how can I learn the maths in those subjects. Any ideas?

[@arongil]

Khan Academy has an intro to computer science in JavaScript that is quite good. For data science, I could recommend some KZbin 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!

[@EpicIntegral]

@@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.

[@BrunoRainJoku]

Question I have basically no knowledge of math, where should i start? how much time i will take to learn stuff i should know for my age ? (16) Sorry if you not understand english is not my first language

[@faridaysa7726]

Hi, Arithmetic is a great starting point that will prove useful in every aspect of algebra and calculus.

[@arongil]

I'd also recommend Khan Academy very strongly. Search it online. It is free, always.

[@georgesadler7830]

Thank you for the video.

[@drewbryk]

Where if at all does discrete math fit?

[@arongil]

Great question! Discrete math can fit in a lot of places. I'd say it's generally good to study calculus first, but really you can learn discrete math at almost any time. You can check out 6.042 on MIT OpenCourseWare.

[@arthurvictorgama3952]

Thanks a LOT! This was helpful and objective. (The MIT's link was also a good info).

[@arongil]

Glad the video was helpful for you :)

[@banenabedalrhman7785]

I still in school but I love math so much , and I start read calculus . I can understand it .. the problem is I am Arabic and my english not very good . and in my country there are no to much good free math courses .I hope you put translations so I can understand. ❤️❤️

[@arongil]

حظًا طيبًا! ChatGPT هو أداة أخرى يمكنك استخدامها لتعزيز تعلمك للرياضيات. يمكنه التحدث إليك بالعربية. قد طلبت منه ترجمة هذا النص الإنجليزي إلى العربية. إذا كانت الترجمة جيدة، فها هو يمكنه التحدث بالعربية. يعرف الكثير عن التفاضل والتكامل. اسأله أن يشرح الأمور ببساطة، ثم مع أمثلة، أو بواسطة مقارنات، أو أن يعطيك مشكلات للتمرن. يمكن أن يصبح معلمك الخاص. مرة أخرى، حظًا طيبًا!

[@arongil]

@@banenabedalrhman7785 Good luck! I'm thinking of you and wishing you the best.

[@jenromeave4793]

omg are u organic chemistry??

[@ChemistryButSimple]

He isn’t The Organic Chemistry Tutor, but I’m his sister and I make videos all about chemistry! And hey, if you like TOCT, I do too and he was SUCH a great resource for me in Organic Chemistry 🩷🕯️

[@jobpatyahoo]

why no trigonometry?

[@arongil]

Many precalculus or algebra II courses cover trigonometry.

[@hzb8889]

Do u think that if someone not good at math starts with this topics u mention could be good at math? or I have to learn more things before I start this trip?

[@arongil]

Hi! Any starting point is a good starting point. Khan Academy has lots of content ranging from elementary school to college level which can help to build up from ground zero if that's what you need. Really math is just experience and inspiration, so if you want to learn more, keep watching videos about math on KZbin and start practicing some. You got this!

[@Tonal.Harmony]

Link is dead

[@arongil]

Fixed! Thank you for letting me know.

[@stickmanbattle997]

What software are you using?

[@stickmanbattle997]

Where you are writing and what recorder are you using?

[@arongil]

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

[@cookies_-zy2sr]

when do you recommend learning probability and statistics?

[@arongil]

I'd recommend it even before learning calculus. There's a FANTASTIC resource called "Seeing Theory" by Brown for probability and statistics. Once you're learning calculus, you will become able to explore deeper reasons why phenomena in statistics do occur. It's a fun journey.

[@ZWADAN]

Peace be upon you, I am a high school graduate. In primary and secondary school I was one of the best students, but because I had a teacher in other subjects I felt a bit discouraged and hated mathematics until I graduated from high school. Now I want to major in data science, and I have difficulty with post-algebra. I can understand, but when I reach some topics I cannot pass them and reach the next stage. Should I start with basic algebra, pre-algebra, or where do I start? plz

[@arongil]

I'd recommend trying out Algebra I on Khan Academy. If you're getting the topics (try the exercises), move on to Algebra II. Same for that and move on to Precalculus if the problems are making sense. And if the problems aren't making sense, check out the videos that explain things. You should be able to pinpoint the right level quickly this way, then you are ready to learn.

[@theesecretchannel]

Start with Pre algebra

[@TepsiMorphic]

You sort of need more then calculus to take real analysis, like logic, set theory and proof writing experience.

[@arongil]

I agree those are important. Real analysis can also be the class that teaches the student logic, set theory, and proof writing.

[@josephschroeder3194]

What about trig?

[@arongil]

Trigonometry is usually covered in precalculus or algebra II.

[@watchf]

I have seen an trig course before. They might have a removed it from a lot of schools due to it being covered clearly in other courses

[@EFE_sopita]

👍👍👍👍👍👍😮

[@renatoigmed]

How far should I learn to master the course I intend to take in the next year (or perhaps the next semester of this current year) in computer science? my intention is to make projects of simulations and games with graphic engine with ultra realistic physics. I am tired of games that simply CANNOT faithfully reproduce Newtonian physics.

[@arongil]

Hi Renato, I'd say it's always good to master course content as thoroughly as possible before moving on. Exception: sometimes it can be helpful to go further ahead than you understand to really challenge and immerse yourself for a short while; then the original material seems easier. Also, cool that you're making physics simulations. That's how I got into math / programming at first. If you tell me more specifics, I can give more detailed advice regarding the course you'll take next year.

[@renatoigmed]

@@arongil One step at a time. First I will try to pass the entrance exam ... Then after a few semesters I will see if it will be enough time to know what the future of my career will be like. I'll keep in touch .

[@gokhalesadan]

what about Probability and Statistics??

[@arongil]

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.

[@brixflores5121]

what about statistics and probability?

[@arongil]

Good point, those are important, too! You can begin to learn them at a basic level anytime. Calculus, linear algebra, and real analysis are key subjects to learn if you want to understand statistics and probability at a deeper level. Best of luck!

[@brixflores5121]

@@arongil thanks , noted

[@pipam1061]

Thank youuuu

[@PochaProprio]

Hi, where would Discrete Math fit in here?

[@arongil]

It could fit in anywhere after algebra. Discrete math is also a lot about logic and sets. You can look up "6.042 MIT" for a good MIT course called Mathematics for Computer Science, which largely covers discrete math. Good luck!

[@PochaProprio]

@@arongil Appreciate it!

[@demohub]

Good 👍

[@roc7880]

u cannot study math except self study, I cam to the conclusion recently.

[@aaronkaw4857]

What's the path to category theory?

[@arongil]

Great question. I'd recommend acquiring a strong foundation in proofs first. From there, the best is to see lots of examples of mathematical structures in all sorts of contexts. The more times you see similar structures appear in different fields, the more prepared you are to learn category theory, which will tie them all together. For example, you might learn linear algebra, abstract algebra, topology, and then category theory. If you're coming from a programming standpoint, you might be able to just jump in and see. There are lots of resources online, like the paper Rosetta Stone by John Baez. Feel free to jump in and see how you like it. math.ucr.edu/home/baez/rosetta.pdf

[@epicmarschmallow5049]

Most courses only have basic point-set topology as a prerequisite. For that you'll probably do a bit of real analysis and linear algebra, study metric spaces which generalises to topology. It's also good to have a good understanding of commutative algebra and algebraic topology; so learn some group theory and ring theory and then study those

[@Loots1]

how is there no proofs on your map?

[@arongil]

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

[@Loots1]

​@@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!! :)

[@mahesaguervo]

I applied for part time degree for engineering course and one of the first module on first sem is engineering math. Welp i never touch any math for atleast 5 years so this helps alot, as i don't know where to start back from. Anyway which category does laplace, ODE and PDE falls under? I'm quite confused 😅

[@arongil]

Laplace transforms, ODEs and PDEs fall under differential equations. I have some videos on that which might be helpful! Another great resource is MIT 18.03 on ocw.mit.edu, or Herbert Gross's classic lectures: ocw.mit.edu/resources/res-18-008-calculus-revisited-complex-variables-differential-equations-and-linear-algebra-fall-2011/part-ii/ Best of luck to you, Mahesa!

[@mahesaguervo]

@@arongil thanks a lot for your reply. Subscribed 👍

[@paublotosod6358]

what happened to the mit course viewer

[@arongil]

Seems like they removed it, unfortunately.

[@gam7880]

Complex analysis?

[@arongil]

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.

[@Edokun90]

NOOOOOOO the link doesnt work anymore :((

[@arongil]

I know, it's really too bad :/

[@user-zz3sn8ky7z]

The fancy interactive map is down, but you can still find MIT's degree charts online in their raw form!

[@xninja2369]

In Android phone in recent manager you can long press app and copy somthing Ot just take screenshot and google lens > click on photo that you take immediately and Google lens through it and delete it Or Open chrome> turn on desktop mode>open exact video on KZbin the link will work

[@orthoguy1366]

One question, what about arithmetic

[@arongil]

It would be helpful to learn arithmetic before beginning starting on Algebra I.

[@nelsonberm3910]

ty

[@arongil]

You're welcome!