hey! I'm a 4th year math undergrad in the States and I am astounded by your knowledge of algebra. it's my favorite branch of math and I know a lot more than my peers but not nearly as much as you. where did you learn? any textbook recommendations?
keep up the great mathematics and posts!
haha, well, I don't know that much algebra to be honest (me using a fancy word in a joke means i have heard of it before, not that I actually know how to work with it!)
But yknow I could give out some resources, so here they are (so far I have mostly learned from classes but yknow i'm at that point where i'm starting to need to transition from listening to someone ramble to reading someone's ramblings and then rambling myself)
For basic linear algebra I didn't learn through a textbook, but I have heard good things about Sheldon Axler's Linear Algebra Done Right and it seems similar to what the classes I had did (besides the whole hating on determinants part, though I kinda get it).
For some introductory group theory, I also had a class on it, but the lecture notes are wonderful. I would happily give the link to them here but since they're specifically the lecture notes of the class from my uni I would be kinda doxxing myself. Also they're in French. I will give out some of the references my prof gave in the bibliography of the lecture notes (I have not read them, pardon me if they're actually terrible and shot your dog): FInite Groups, an Introduction by Serre (pdf link), Linear Representations of Finite Groups also by Serre (pdf link), Algebra by Serge Lang (pdf link). Since our prof is a number theorist he sometimes went on number theory tangents and for that there's Serre's A Course in Arithmetic (pdf link). I'm starting to think our prof likes how Serre writes.
For pure category theory and homological algebra I have read part of these lecture notes. I think a good book for category theory is Emily Riehl's Category Theory in Context (pdf link). For homological algebra, a famous book that I have read some parts of is Weibel's An Introduction to Homological Algebra (pdf link). Warning: all pdfs I found of it on the internet all have some typographygore going on. If anyone knows of a good pdf please tell me.
For commutative algebra, A Term of Commutative Algebra by Altman and Kleinman (pdf link). I haven't read all of it (I intend to read more as I need more CA) but the parts of it I read are good. It also has solutions to the exercises which is neat.
For algebraic geometry (admittedly not fully algebra), I am currently reading Ravi Vakil's The Rising Sea, and I intend on getting a physical copy when it gets published because I like it. It tries to have few prerequisites, so for instance it has chapters on category theory and sheaf theory (though I don't claim it is the best place to learn category theory).
For algebraic topology (even less fully algebra, but yknow), I have learned singular cohomology and some other stuff using Hatcher. I know some people despise the book (and I get where they're coming from). For "basic" algebraic topology i.e. the fundamental group and singular homology I have learned through a class and by reading Topologie AlgĂŠbrique by FĂŠlix and TanrĂŠ (pdf link). The book is very good but only in French AFAIK.
For (basic) homotopy theory (does it count as algebra? not fully but what you gonna do this is my post) I have read the first part of Bruno Vallette's lecture notes. I don't know if they're that good. Now I'm reading a bit of obstruction theory from Davis and Kirk's Lecture Notes in Algebraic Topology (pdf link) and I like it a lot! The only frustrating part is when you want to learn one specific thing and find they left it as a "Project", but apart from that I like how they write. It also has exercises within the text which I appreciate.
For pure sheaf theory, a friend recommended me Torsten Wedhorn's Manifolds, Sheaves and Cohomology, specifically chapter 3 (which is, you guessed it, the chapter on sheaves). I only read chapter 3, and I think it was alright (maybe a bit dry). I also gave up at the inverse image sheaf because I can only tolerate so much pure sheaf theory. I will come back to it when I need it. The whole book itself actually does differential geometry, but using the language of modern geometry i.e. locally ringed spaces. I have no idea how good it is at that or how good this POV is in general, read at your own risk.
Also please note I have not fully read through any of these references, but I don't think you're supposed to read every math book you ever touch cover to cover.
thanks for the kind comments, and I hope at least one of the things above may be helpful to you!
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hey y'all so I have a degree in math, and have been tutoring college level math, stats, physics, and study skills for a few years now and I have several certifications as a college level tutor and in my years tutoring I've picked up a lot of absolutely essential things you NEED to know to pass math classes (and others!) if you have a hard time in math class, which unfortunately most schools will not teach you
so if you're in college (or high school, but some of these will be college specific) and you would like to hear my Professional Tips for how to survive math in college I have compiled a list of things I think you should know below the read more! I'm going to put a couple general tips at the top though for people who aren't taking a math class
alright, welcome to the read more! let's start with some general things
ok I'm starting with a kind of tip before the tips: if you are disabled, talk to disability support services at your school. a lot of times they provide services to attain a diagnosis if you don't have one, but if you do it's typically pretty easy to get accommodations and if your teachers try not to follow them, they will get in A LOT of trouble (I'm in the us so idk how disability laws are in other countries but here this is a big deal for universities bc public funding etc and if you make a complaint they will be FORCED to comply). I'm not saying bad experiences never happen, I've not been to every school in the us and I'm sure there are places that suck abt this, but I literally would not have been able to get a degree in math without accommodations (I'm autistic and have dyscalculia + memory issues) so it was worth every annoyed sigh by a dumbass teacher (which honestly rarely happened. most of them were cool). some possible accommodations include, extra time on tests, separate testing locations, formula cards, ability to use notes etc etc
I work in a free tutoring center at my local community college. while I won't say these are definitely universal, every college I have encountered (in the US, where I live) has one of these. Google the name of your school with tutoring. there is a very high chance you have free tutors available in a variety of subjects who are ready to help you. you should utilize the fuck out of this bc ur already probably paying for it in ur tuition
my favorite resource ever is one you may have heard of but I'm reminding you of it anyway. the Crash Course YouTube channel! it has free comprehensive videos about various subjects (including anatomy, physics, biology, economics, statistics and lots lots more). the videos are about 10 minutes each and they're incredibly easy to understand. they're an amazing way to study for finals. trust me on this one. they actually now have real introductory college courses that you can take for credit (tho I think you have to pay for that?) through YouTube so check into that!
take notes!!!!!!!! for real. seriously. even if you've never had to before. trust me. and don't just copy exactly what the board says, write what your teacher says out loud as well, that is often the most important stuff. I highly recommend investing in a few colored pens and/or highlighters. anytime the teacher says something important (such as formulas in a math class or a grammar rule in an English class) either write it in another color, or highlight it. color coding your notes even the tiniest bit will help you tremendously when you're studying and doing homework later. this doesn't mean having a color code so elaborate and strict that you're spending more time and energy figuring out the right color than listening and writing. I usually do the bulk of my notes in black then things like formulas or whatever in one bright color and extra bits of info from the teacher in another, this way it's easier to find the important stuff later on
if you've got a couple extra bucks, invest in a mini stapler. you'll be shocked how often those lil bitches come in handy
DO NOT EVER purchase your text books before the classes start (unless you get like an email before class starts telling you you need the book, but this is pretty rare tbh). half of your teachers are going to tell you on day 1 that they don't use the book at all. and honestly almost all of your books can be found by googling the name + pdf. just triple check that you have the correct edition!
speaking of emails, CHECK YOUR FUCKING SCHOOL EMAIL. I am so dead serious about this one. set up notifications on your phone. if you do not check your email you are going to show up for cancelled classes, miss assignment corrections, and generally not do nearly as well in your classes. I know this sounds fake but holy fucking shit please for the love of all that is good and holy check your fucking email. seriously.
the best way to study for your finals is to look at past tests and homework! if your teacher isn't a total dick they'll give you back your tests and homework. when studying for your final, go through and redo any that you got wrong, and look over the ones you got right. teachers usually take final questions from old tests and homework so if you do this it's very possible you'll study your exact finals questions. if they won't give you these back, reread your notes (in a way I'll describe in a moment)
reread your notes the same day you take them or very soon after so the lecture is still fresh in your mind! when you do this, grab a colored pen and take notes on your notes. I know this sounds ridiculous but it's actually a very important study tool. if you come across something you wrote that is confusing, write out an explanation. write down extra things that will help you understand the material. if there's something that you don't understand or don't remember PLEASE ask your teacher. some of them won't answer email so catch them in office hours or after class if you can. at the end of the semester when you're studying for finals, do this all over again but through your whole notebook for the semester (not all at once. pls take breaks lmfao)
if your teacher doesn't offer it up at the end of the semester, ask "can you please tell us some of the topics we should emphasize when studying for our finals?" (you can also say this in a far less pretentious way but I've found that professors are more likely to give you a real answer if you talk like this ÂŻ\_(ă)_/ÂŻ) I'm ngl, some of em will be assholes about this. they may laugh at the idea and snarkily say "look at your homework" or some shit equivalent. roll your eyes at these old bitches and move on. but many if not most will at least give y'all some idea of what to expect. and crucially: write down what they say and use it as a study guide
okey dokey!! that concludes the general section. now I'm gonna talk about some math specific stuff that will help you a lot if you struggle with math!
starting with an easy one: get a good calculator. please for the love of GOD do not get the TI-30X IIS unless you love it and are EXTREMELY familiar with all of the different operations. I'm sorry but this calculator sucks ass and it will hold you back. for about the same price you can get my personal favorite the Casio FX-115ES Plus (1st edition only, I haven't tried the 2nd edition bc I don't like change) or an FX-991 EX. if you're a Texas instruments guy get a TI 30XS or if ur doing calc and shit I'd get a 36X pro. I just prefer Casio personally lmao. in all likelihood your math teacher will be a calculator nerd who can teach you how to use any of these but there are also lots of videos made by calculator nerds on YouTube as well
so next I just want to emphasize how important your notes are. you cannot pass a math class without good notes unless it's a class you've already taken or something, and honestly even then I'd recommend you write some stuff down because the thing about math is there are a lot of different ways to do the same thing. which brings me to my next point
pay attention to the process your teacher uses to solve problems (I'll give an example in a sec) and especially to how they write the process down. if you're like me and you have trouble with the whole "show your work" thing this will help a lot, because you can use what your teacher writes down as a guideline for what you should write down. for instance, you may have learned about a math concept like permutations and combinations in high school one way, and then be taught a completely different way of performing the calculation in college. if this happens, ask your teacher about your way. sometimes their way is better for a specific reason and it's really important that you learn it. sometimes they'll say it doesn't matter just do what makes the most sense to you. sometimes you will also not get full credit if you do things a different way too so be careful and pay attention to what your teacher says abt it. you may have also been taught to show some steps in an operation but not others and your new teacher may want all of them. or none. or different ones. unfortunately math has a lot of variations
similarly, if there's a concept you don't understand, start by asking your teacher about it first bc they may want you doing a specific thing. if they're not helpful and you don't have access to a tutor turn to the internet. here are my recommendations for resources: Khan Academy has videos and examples explaining concepts in pretty much all types of math. usually really helpful because they'll show you several different methods and use different explanations, MathWay for classes that come before trig/calc. you can use it for those but it's a little more annoying lol specifically in regards to graphing and solving integrals and shit. this tool has A LOT of stuff in it but is best as a calculator to check your work on stuff and for showing you graphs that have transformations and shit like that from college algebra. it's got settings for different math types and even chemistry tho!, for more complicated graphing I'd use Desmos. you can use this in all classes but it's just a bit more complicated imo and it's more made for complex operations so I prefer to use MathWay as much as possible bc it's just more user friendly., there's also Symbolab which a lot of my fellow tutors really like but I personally tend to use the others more, Wolfram Alpha is a pretty well known one. tbh I find it kind of hard to use sometimes so i usually use it as a last resort lmfao but it is really good!, this last one is calculus specific (including things like calc based physics ofc) but derivative calculator and integral calculator are everything to me. could not have gotten through calc 2 and 3 w/o these mfs
OK THIS PART IS REALLY IMPORTANT!! we're going to talk about how you can make your math tests WAY easier on you and massively improve your chances of passing. here's what you're going to do for every test
1. when studying for your test, go through and find ALL formulas that you used in the unit(s) your test is over (this includes formulas you learned previously but used in this unit as well!!)
2. commit them to memory. easiest way to do this (besides practicing using them!!) is to rewrite them a few times including what you use them for and what all of the symbols and letters stand for
3. when you go in to take your test, spend like 5-10 minutes beforehand, right up until they make you put everything away, studying and rewriting these
4. the SECOND the test hits your desk, flip it over and write down every single formula immediately (including as much extra info like when to use and variable definitions as possible). now you won't have to try to remember them 30 minutes in when your brain is frying!!
5. go through the test and read each question carefully. if you can't remember how to solve it within 30 seconds skip it. you might only do 4 or 5 questions (maybe less depending on the length of your test) after the first pass, but just go back to the beginning and do it again, giving urself a little more time w each pass. this will ensure that you're not spending 45 minutes on problem 4 and having no time to get to the rest of the test. additionally with math it is extremely common to basically find the answer to how to do one problem while you're doing another problem. doing the ones you know first will also boost your confidence and help prevent anxiety from wiping ur brain. this is a really really important part of math tests
6. before you hand your test in, make sure you've written SOMETHING down on EVERY SINGLE QUESTION. even if you have absolutely 0 idea whatsoever what to do, there is always a chance for partial credit. a lot of times, you also know more than you think you do. so even if you can only do half a step of the whole process, half a point is better than 0!! if you really have no clue what to do, make something up. I know this probably sounds ridiculous but I'm so deadass. once I was taking a physics test and could not for the life of me remember what formula to use so I just made one up based on my vague idea of how it worked lmao. I wrote off to the side "I know this isn't how physics works, but I can't remember that so just pretend I'm god for a second" and I got like 75% of the points bc the teacher appreciated the effort!! there were fucking countless times when I was taking a test and I got to a point where I knew the steps of what I was supposed to do but could not figure out HOW to actually DO the math. so I wrote in words my understanding of what the next steps were. even though I didn't finish answering the question, I always got points for trying. this is what teachers are wanting from you. effort. so PLEASE write literally anything even if you're just making that shit up (just explain your reasoning in words to the side, as long as you're using logic you're really getting the essence of math anyway). you would not BELIEVE how fast your grades will improve by doing this. I tutored a girl who went from Ds to Bs within literally 2 weeks of starting tutoring just because she stopped leaving any questions blank and started getting partial credit
that's all the important stuff I can think of for now though I'm sure there's much more so I may update this in the future!
of course everyone is going to have a different experience and relationship with math. so adjust all of these tips to fit how you learn best. please try to remember that learning math is a very important part of developing your critical reasoning and logistical analysis skills. these are essential to understanding and interacting with the world and math is just a way of exercising those muscles. trust me when I say I know how infuriating math can be. I have dyscalculia and a math degree. I've spent so many hours crying over math you probably wouldn't even believe me. but it's worth it! and frankly, if you're in college, you're paying a lot of money for this class. you deserve to get everything you possibly can out of it
above all, if you're having trouble ASK FOR HELP. ask your teacher, ask your classmates, ask the head of the department, ask student services, ask Google!! and if you need help you can always ask me! :) I love helping people with this stuff and even if I don't know the answer to your question I'm pretty good at knowing where to look for them!
Hello! for the longest time I've thought that I'm just "not good" at math, and that I'm just a creative type or something, but for the past few years I've been trying to brute-force my way through math drills and stuff to improve because I want to go into a STEM field. I've found that it doesn't work. I have, however, gained a kind of Stockholm syndrome for math, and I find myself wanting to know *more*. I feel like I'm missing out on something way beyond myself but idk where to even find that. So here i am. Asking a blog on tumblr for the secret math knowledge. If you have any resources that i could read about math that isn't just "here's how to do this" i would really appreciate it!!!
First of all, complete props to you for giving something that doesn't initially appeal to you, in this case, maths, a chance. Mathematics can be frustrating and even annoying, being an oftentimes nit-picky field. You can quickly realize that these are also the traits that grant the satisfaction and euphoria of mathematics. So don't forget to be proud of yourself for not disregarding mathematics as a whole and actually giving it a go.
Second of all, creativity is very necessary for maths. It is a shame that the educational system doesn't do maths justice (A Mathematician's Lament by Paul Lockhart is exactly about this). Mathematics is presented as something mechanical, force-feeding formulas and having students repeat them to find adequate results. That is not a fair portrayal of mathematics.
Different people will have different conceptions of what the learning of mathematics is or should be - I mean, for god's sake, mathematics hasn't even been defined properly and one expects people to know what to do with it. I can only tell you what I conceptualize mathematics, and its learning, to be. For me, mathematics is the boring work of examining multiple results and cases of the same formulas, it is the finding of patterns and attempting generalizations, it's the excited scribbles of formulas and the necessity of looking at a particular scenario from multiple perspectives. Most likely, the generalizations won't come easily, a minus sign will be forgotten making the following calculations obsolete, an approach to a problem will prove fruitless, and laborious work will be done only to find out there was a specific theorem that would have shortened the whole process. Patience is required for maths, and hopefully, you can now see that creativity is too, there is no shortcut to knowing where to look or what technique to utilize. As is the case with most worthwhile pursuits, and as you know from experience, mathematics is endlessly frustrating, but that also means it is endlessly fun.
Now, I do not have any secret math knowledge (or do I), what I can do is present you with some things that fascinate me and have led me to love mathematics as much as I do, outside of the conventional mathematics curriculum.Â
Maths really comes down to practice, practice, practice... as many other things. I heavily encourage you to start playing around with mathematics a little bit. Olympiad mathematics kind of do that, (I, very conveniently, have a lot of posts about that.) as do many other math competitions. Maybe try out some exercises from your national olympiad and, with no judgment, because this is just to have fun, play around a bit. Disregarding the conventional maths you know, test out your logic, laugh at mistakes and losses of time, and feel that happy rush when a conjecture you reach, or part of it, is correct.
In terms of the resources, I opted for a mix of funny math history moments and some actual mathematics, in no particular order. Prepare for confusion. â
0. How Mathematics is enjoyable
No better way to start than with the reasons mathematicians decided to dedicate a good portion of their lives to the subject, or why some people just enjoy mathematics. This is something that greatly interests me, I always love to see peopleâs passion for their interests, hence this post, where youâll find many peopleâs reasons for enjoying mathematics.Â
Andrew Wiles answers âWhat does it feel like to do maths?â - (video) - Andrew Wiles is a mathematician who proved the theorem which had been taunting mathematicians for centuries, famously known for being the subject of the third point of this same post, explains peoplesâ distaste for maths and his own experience with the subject.
YouTube Stories: Learning Mathematics with Wootube by Eddie Woo - (video) - Eddie Woo is a wonderful mathematics educator as well, who videotapes his lessons (available on Youtube). This serves as an introduction to him and his channel.
Mathematics is the sense you never knew you had by Eddie Woo - (video) - In his TEDx Talk, Eddie starts by explaining his distaste for maths in his school years, mentioning his low scores and lack of interest, and how, despite this, he ended up as a maths teacher.
Reddit asks: âWhy do you love maths?â - Another compilation of peopleâs explanations for their mathematical interest. Wholesome content ahead.
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1. Intro to a new perspective on mathematics
Taming, claiming and reframing the beast that is mathematics by Vinay Kathotia - Small article related to the portrayal of mathematics by most and by mathematicians themselves.
On proof and progress in mathematics by William P. Thurston - Mentioned in the previous article, 17-pageÂ
Anyone Can Be a Math Person Once They Know the Best Learning Techniques by Po-Shen Loh - (video)
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â2 . The Pythagorean Cult
Iâm sure youâre aware of Pythagoras and his fetish for right triangles, but do you know about his weird cult? Thatâs right, Pythagoras was the founder of Pythagoreanism, a cult based on the teachings and beliefs held by him and his followers, the Pythagoreans. - very original naming ik.
Hereâs and article about them. A really popular myth related them is the throwing of Hippasus into the sea as a punishment for âbelievingâ (i guess) in irrational numbers.
The unreasonable man by Vinay Kathotia, further explores the irrationality of the square root of 2, allegedly identified by Hippasus, providing some proofs of it.Â
Proof: â2 is irrational by Khan Academy (video) - Because Khan Academy rarely misses.
The Madness of Pythagoras by Kayla Mahoney - Did you know Pythagoras refused to eat fava beams, because he claimed they contained the souls of the dead? Now you do, and thereâs no going back >:).
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2. The base tools of mathematics
Introduction to Mathematical Thinking by Alexandru Buium - Pretty self-explanatory title, simplistic and general apprach to mathematics starting with the essential, logic.
The Five-fold Path to Mathematical Wisdom by Vinay Kathotia - Article about five different approaches to mathematical problems/representations.
How to think like a Mathematician by Kevin Houston - Really useful book, starts with certain study skills and develops to the logic involved in mathematical thinking, explains theorems, definitions and proofs! Highly reccomend.
How to Think Like a Mathematician by Eugenia Cheng - (video)
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3. "I have a truly marvelous demonstration of this proposition that this margin is too narrow to contain."
Yup, this is about Fermatâs Last Theorem, which took 350 years to be proven.
Iâm guessing youâve come across some jokes about it, how could one not. Essentially, in the 1630â˛s, our man Pierre de Fermat jotted down the conjecture (fight me, no proof no theorem) that states that no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2, along with the sentence "I have a truly marvelous demonstration of this proposition that this margin is too narrow to contain.". The paper which contained this was only found after his death, and, as far as we know, he never got around to actually writing the proof. Because of this, we call this statement Fermatâs Last Theorem.
Andrew Wiles was who ended up proving Fermatâs Last Theorem, in 1993, after six years of working secretly on the problem. (Imagine your dirty little secret being working on revolutionizing an entire field of study, what a guy honestly.)
Fermat's Last Theorem - The Theorem and Its Proof: An Exploration of Issues and Ideas [1993] - Documentary about the theorem and its proof, as told by Andrew Wiles himself.
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4. Functions
Honestly functions are just a very needed side of mathematics that can be applied to pretty much everything. Youâll find them in your average mathematics curriculum and in your everyday life, if you know where to look.
Introduction to functionsÂ
Introduction to Mathematical Reasoning - Numbers, Sets and Functions by Peter J. Eccles
Functions in the Real World - Education World
- This is definitely a more theoretical part of the list, I am sure youâve encountered functions before and the resources you have musnât be terrible. Look into these if you find the necessity, but again the âmath secretâ really is practice. They can be interesting reads though.
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5. MĂśbius strips and Klein bottles
The wondrous world of â¨one-sided objectsâ¨. Learn about these cool looking objects and the field they originated, topology.
The Mathematical Madness of MĂśbius Strips and Other One-Sided Objects by David Gunderman and Richard GundermanÂ
Intro to Topology by Alex KĂźronya - Decently advanced maths, I donât know what level of maths youâre learning, but honestly only read this if you have a pretty thorough backgroung in maths and youâre enrolled/very interested in the field.
MÜbius Strip - Cool graphics, cooler maths.
Klein Bottles on Numberphile - Because Numberphile never misses.
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6. The Chicken McNugget Theorem
Yes, you did read that right, No, this is by far not the only theorem with a weirdly funny name*. This theorem derives from a 1800â˛s math problem.
Explanation of the theorem by Xavier Lien
Chicken McNugget Theorem - Art of Problem Solving
Another explanation, if needed by Mike Beneschan
*Proof: Reddit asks: â What's the funniest theorem name you know? â - Literally a gold mine, read this.Â
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7. GodĂŤlâs incompleteness
One of the essential works of Modern Logic. His theorems destroyed the search for a mathematical theory of everything, at the mere age of 25.
GĂśdelâs Incompleteness Theorems - Stanford Encyclopedia of Philosophy
The paradox at the heart of mathematics: GĂśdel's Incompleteness Theorem by Marcus du Sautoy @ Ted-Ed (video)
How GĂśdelâs Proof Works - Quanta Magazine
GĂśdel: His Tragic Life Story by Aimee Lamoureux - (TW: Sensitive themes)
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8. Mathematics is an art, actually
Iâve seen many people stray away from the sciences because theyâre âcreativesâ or âarts peopleâ. Hopefully youâve gotten at least a glimpse of the creativity needed in mathematics - if you didnât, that is totally on me and Iâm sorry.Â
I donât doubt that you may be a âcreative typeâ. We actually need creatives, more than anything, in this field. I thought the perspective of mathematics as an art mignt be a good ending note.
Why Math is an Art, Not a Science by Peter Flom
Why the history of maths is also the history of art by Alex Bellos - He also wrote the book âAlexâs Adventures in Numberlandâ, which I own and have read. Itâs a pretty sweet book, it has my stamp of approval if youâre interested.
Math and Movies (Animation at Pixar) - Numberphile - On the mathematics behind the art of animation.
Art and Math: Aesthetics of Calculations by Rute Ferreira (DailyArt Magazine)
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Besides this Iâm also leaving a list of youtubers/authors that are worth looking into. :)Â
â. General resources:
Math-related Youtube:
Numberphile - Have different concepts explained to you in a seemingly personal and almost always funny manner.
3Blue1Brown - Maths explanations paired with excellent visual representation of the concepts.
Reducible - As they put it âReducing problems to their simplest form.â. Also paired with really good animations.
Websites:
Art of Problem Solving
Blogs/Articles:
Cantorâs Paradise - Iâve linked a few articles published there throghout this post. Essentially, different authors publish math articles in this online space, varied and generally well-written about topics.
Alex Bellos for The Guardian - Puzzles and mathematical articles.
Mathblr Blogs - Check the notes of this post to find a lot of them.
Games:
Nerdle - Math Wordle
(...) - Please reccomend more resources below, Iâm not the most well-versed on these subjects.
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Hopefully, you now find maths a little sweeter, funnier, or comprehensible. As with most things, people lead math discovery, and thereâs absolutely no reason youâd be unable to be part of that group.Â
Mathematical problem-solving is something really frustrating but also immensely fun and rewarding. Please look into it, be it olympiads or general problems you find online, donât take yourself too seriously and have fun with maths! Mathematics doesnât judge, people do. I wish you well on your math journey! Hopefully, youâll love it.
P.S: Feel free to add to this post with any math stuff you think is interesting. The more, the merrier!
Anya is live and ready to show you everything. Watch her strip, dance, and perform exclusive shows just for you. Interact in real-time and make your fantasies come true.
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Free to watch ⢠No registration required ⢠HD streaming
As some of you may know, Iâm a math major, so over the past two years, Iâve found a few resources that have literally saved my grade.Â
Khan Academy -- Very helpful videos that go step by step through problems. I think these are the only reason I passed Calc 2. They also have practice problems and written lessons. They cover the majority of calculus and differential equations.Â
Paulâs Online Math Notes -- These are notes that my calc teacher linked to us and they help me so much. They are easy to read, have practice problems with the solution (every math majors dream), and go over several topics from algebra to calculus 3 and beyond. I have used this site religiously this semester because Khan Academy doesnât have cylindrical coordinates. I use the problems to prepare for my tests and make sure Iâm doing everything correctly. http://tutorial.math.lamar.edu/
Symbolab -- This wonât necessarily help you learn, but I use it to check my problems. This gives step by step solutions to the problems that you insert. I have premium so I can see all step by step solutions. Do not use this website just to get answers. Use it to check yourself and if you get completely stuck, figure out where you went wrong in the problem.Â
These have helped me so much, but I have also used resources my school provides me such as tutoring and office hours.Â
I am going through last semesterâs notes on introductory game theory and am loosely considering rewriting them all in one place; however, if there is enough interest here, Iâd be willing to share my detailed notes on a routine schedule. This is just a hypothetical and Iâd decide based on how much interest (or lack thereof) my followers show.
My semester notes start at the very introductory level, starting with defining extensive form games and the Nash Equlibrium as a solution concept. The notes should be accessible to absolute beginners, so no background theoretical knowledge is required.
If this interests you, consider reblogging this post to signal-boost because, as I said, my commitment to posting these detailed lecture notes would be dependent upon how many people demonstrate interest in this idea.
this might be a dumb question but like. how do you learn math without a class/curriculum to follow. i have a pretty solid calculus understanding and I want to pursue more advanced math but like im not sure where to start. what even is like category theory it sounds so cool but so scary???. do you have any recommendations on specific fields to begin to look into/whether its best to learn via courses or textbooks or lectures/etc.? any advice would be super appreciated!! dope blog by the way
thanks for the compliment!
first of all it's not a dumb question. trust me i'm the algebraic-dumbass I know what I'm talking about. okay so uh. how does one learn math without a class? it's already hard to learn math WITH a class, so uhhh expect to need motivation. i would recommend making friends with people who know more math than you so you have like, a bit more motivation, and also because math gets much easier if you have people you can ask questions to. Also, learning math can be kind of isolating - most people have no clue what we do.
That said, how does one learn more advanced math?
Well i'm gonna give my opinion, but if anyone has more advice to give, feel free to reblog and share. I suppose the best way to learn math on your own would be through books. You can complement them with video lectures if you want, a lot of them are freely available on the internet. In all cases, it is very important you do exercises when learning: it helps, but it's also the fun part (math is not a spectator sport!). I will say that if you're like me, working on your own can be quite hard. But I will say this: it is a skill, and learning it as early as possible will help you tremendously (I'm still learning it and i'm struggling. if anyone has advice reblog and share it for me actually i need it please)
Unfortunately, for ""basic"" (I'm not saying this to say it's easy but because factually I'm going to talk about the first topics you learn in math after highschool) math topics, I can't really give that much informed book recommendations as I learned through classes. So if anyone has book recommandations, do reblog with them. Anyways. In my opinion the most important skill you need to go further right now is your ability to do proofs!
That's right, proofs! Reasoning and stuff. All the math after highschool is more-or-less based on explaining why something is true, and it's really awesome. For instance, you might know that you can't write the square root of 2 as a fraction of two integers (it's irrational). But do you know why? Would you be able to explain why? Yes you would, or at least, you will! For proof-writing, I have heard good things about The Book of Proof. I've also heard good things about "The Art of Problem Solving", though I think this one is maybe a bit more competition-math oriented. Once you have a grasp on proofs, you will be ready to tackle the first two big topics one learns in math: real analysis, and linear algebra.
Real analysis is about sequences of real numbers, functions on the real numbers and what you can do with them. You will learn about limits, continuity, derivatives, integrals, series, all sorts of stuff you have already seen in calculus, except this time it will be much more proof-oriented (if you want an example of an actual problem, here's one: let (p_n) and (q_n) be two sequences of nonzero integers such that p_n/q_n converges to an irrational number x. Show that |p_n| and |q_n| both diverge to infinity). For this I have heard good things about Terence Tao's Analysis I (pdf link).
Linear algebra is a part of abstract algebra. Abstract algebra is about looking at structures. For instance, you might notice similarities between different situations: if you have two real numbers, you can add them together and get a third real number. Same for functions. Same for vectors. Same for polynomials... and so on. Linear algebra is specifically the study of structures called vector spaces, and maps that preserve that structure (linear maps). Don't worry if you don't get what I mean right away - you'll get it once you learn all the words. Linear algebra shows up everywhere, it is very fundamental. Also, if you know how to multiply matrices, but you've never been told why the way we do it is a bit weird, the answer is in linear algebra. I have heard good things about Sheldon Axler's Linear Algebra Done RIght.
After these two, you can learn various topics. Group theory, point-set topology, measure theory, ring theory, more and more stuff opens up to you. As for category theory, it is (from my pov) a useful tool to unify a lot of things in math, and a convenient language to use in various contexts. That said, I think you need to know the "lots of things" and "various contexts" to appreciate it (in math at least - I can't speak for computer scientists, I just know they also do category theory, for other purposes). So I don't know if jumping into it straight away would be very fun. But once you know a bit more math, sure, go ahead. I have heard a lot of good things about Paolo Aluffi's Algebra: Chapter 0 (pdf link). It's an abstract algebra book (it does a lot: group theory, ring theory, field theory, and even homological algebra!), and it also introduces category theory extremely early, to ease the reader into using it. In fact the book has very little prerequisites - if I'm not mistaken, you could start reading it once you know how to do proofs. it even does linear algebra! But it does so with an extremely algebraic perspective, which might be a bit non-standard. Still, if you feel like it, you could read it.
To conclude I'd say I don't really belive there's a "correct" way to learn math. Sure, if you pursue pure math, at some point, you're going to need to be able to read books, and that point has come for me, but like I'm doing a master's, you can get through your bachelor's without really touching a book. I believe everyone works differently - some people love seminars, some don't. Some people love working with other people, some prefer to focus on math by themselves. Some like algebra, some like analysis. The only true opinion I have on doing math is that I fully believe the only reason you should do it is for fun.
