hi everyone in this video i'm going to show you how to teach yourself mathematics from start to finish starting with just like basic sets and proofs so we'll start with actually learning to write proofs and learning some logic and a little bit of discrete mathematics and then from there in theory you could teach yourself tons of mathematics and i'm going to show you the exact order that you should follow or at least a highly recommended order let's get started right away okay so here i have four books that i think are a really really good starting point so the first two are discrete math books and the bottom two are actually proof writing books so these are books that actually math majors would use so let's go ahead and take a look at the first two so this first book i already did a review on actually just recently it's discrete mathematics with applications by susannah epp this is an extremely good beginner book and you can get a used copy for a few dollars this will let you get started with the theory of logic so you'll learn about like logical implications truth tables all kinds of things regarding you know basic mathematical logic and the beautiful thing is you need zero algebra to learn this now you will have to learn algebra at some point and i'll show you later in the video but you can start with this you can start by learning like the core foundations of mathematics it's really really quite beautiful this other book is just another book on discrete math that i have and i have like you know 10 of these but these two are really good for beginners and that's what this video is about right you want to get started and you want to learn this is the book by coleman busby and roz and this is also a very very beginner friendly book on discrete math so in discrete math you learn all kinds of stuff like logic sets and you learn how to write proofs you also learn other more advanced topics that you typically don't see in other classes like calculus and stuff like that so you learn a lot of really cool math that oftentimes math majors don't even see so it's a good place to start and a good place to keep you motivated you know to start at the beginning because you've probably done algebra in the past maybe you were in school before maybe you're in school now and we just do so much algebra so i think that this is a really good starting point for people discreet math especially this book the one by epp and the one by coleman busbian ross is also pretty good this one is better though this is probably a really good choice so here we have two proof writing books this is mathematical proofs a transition to advanced mathematics this is the chartrang palomini and zang book this book is awesome now this is way more advanced than the previous book so you know you pick up these books you read what you can and you try to understand as much as you can and then you move on to the next topic you're not expected to master everything in any of these books by the way and this is another book on proof writing this is the one by bonding keen this one is also good uh but this one is better and by the way every book i'm showing so far here you can probably get it for less than ten dollars on the internet if you get a used copy so these books are good for math majors because they teach you to write proof so this is this is a big deal you know you go to college for four years to get a math degree and one of the key things you come out with is learning how to write proofs and these two little books basically teach you how to do that so again you're not expected to you know completely master everything in these books so just make an effort why are there pre-algebra books in this video well remember this is from start to finish so once you've attempted those books and you've learned some logic and maybe that's all you've learned maybe your math is really bad so maybe you need some pre-algebra so these are just two pre-algebra books that i have that i like this is the one it says ags this one's good because it has solutions and this is firon's pre-algebra this one doesn't have solutions but i actually like this one better than this one so if you're just trying to like refresh really basic math these are two pretty good pre-algebra books i have about eight pre-algebra books and i think these are maybe the best ones so maybe you decide you don't need pre-algebra or you've already worked through it and you feel good enough about the material then college algebra is a good next choice so college algebra is a course where you learn just basic algebra there is a course in between these called intermediate algebra i decided to skip that one because most intermediate algebra books just contain like just thousands of problems and it's just too much these books i think are better and they're better written so this is the one by kaufman it's a little bit more beginner friendly than this one this is the one by blitzer i think both are good and both are worth having and you will learn all of the basic algebra skills that you need really to go further in mathematics once you get through college algebra or like you feel like you've mastered enough of the material from college algebra you can jump into this this book is typically used to teach a course called precalculus and also trigonometry uh in the u.s precalculus is like algebra it's like college algebra but it's harder so you do like more things you do more things with inequalities you do some stuff with matrices it's just a little bit more challenging than algebra so in theory you could start with a book like this this is the one by hornsby lyle and roxwoold and this is good because it's the instructor's edition so it actually has answers to every single problem right next to the exercise which is really convenient and this book has everything right it's got conic sections it's got matrices it's got algebra stuff it's an awesome book so again this is the book a graphical approach to algebra and trigonometry and it's the one by hornsby lyle and roxworld and again you could in theory skip all the pre-algebra books skip all the college algebra books and start here so if you know some basic algebra this is a good place to start and we have finally arrived at calculus so the book on the bottom you see here is calculus by james stewart this is a common book it's probably the most popular book in the united states that's used to teach calculus and in canada james stewart was a canadian mathematician he passed away several years ago so this is a good book this one calculus for people who are going like into engineering or if you just want to learn just basic calculus it's got tons of problems and tons of pretty good explanations a lot of people don't really like this book but it's hard to find a book comparable to this one another good one is the book by larson so again worth owning a copy of this book you can get a used copy and typically in the u.s at least the courses that are taught with this book are calculus one two and three so you could use this book to teach yourself those things so what about this other book this is a famous book this is called calculus by michael spivak and this has less material than the stewart book but it's more advanced and the beautiful thing is if you follow this video so far up to this point you already know some logic and you already know some proof writing and that is the number one thing that gets people hung up with this book they'll pick up this book because they'll read about it on the internet but like oh this book is so good and they're like i don't understand what's going on what is michael's did that talking about it's because those people lack the proof writing skills which we covered at the beginning of the video so if you jump into this and you follow this order you'll actually be able to read a book like michael spivak's calculus and you'll understand the trig that's in here because you've learned some trig so awesome book worth owning especially if if you know some some sets and some proofs once you get through calculus it's time to learn some differential equations which are super awesome the main thing for learning differential equations is that you know some integration so if you go through a book like the stewart book and you pick up like all the integration techniques that will help you a lot in a course like this so this is the book that i use to teach differential equations so i'm biased this is the one by zill it's an okay read this one is a bit easier this is the one by larry andrews not a very popular book but it's pretty good for beginners it's called ordinary differential equations by larry andrews so again differential equations is a good next place to be there's many more books i just i'm trying to keep this video short so here's two good ones okay so now you know how to write really basic proofs you know some basic logic and you know a little bit of calculus and again you don't have to be a master to jump into this stuff so now it's time for linear algebra so this is the book by howard anton elementary linear algebra i like this book for beginners because it's very consistent with what you would learn in a college level course the exercises are very similar to like the test questions you would see in a college level course it's really really good for beginners it goes along very well with what's taught so that'll help you get an a in your class this one is a much harder book it's a drier book it's a much more rigorous book but it is still worth owning it is the one by friedberg insult and spence and this one is proof-based but remember you have proof writing experience even just a little bit because that's where you started so both books are worth owning and again you would just try to learn as much linear algebra as you can in this next step you don't have to master it before going to the next step let's keep going at this point it's like the world is open because you know how to write basic proofs you've made it through some calculus and you know some basic linear algebra so now in theory you can learn almost any math subject that's taught at the undergrad level so this is a book on mathematical statistics so if you're interested in statistics this is a really solid choice and this is an advanced book but you know what it's okay you're at an advanced stage you know calculus and this book requires calculus for some portions this one also requires a little bit of calculus for some portions and it's called a first course in probability but again a lot of this stuff can be done without calculus but the fact that you know it will let you get through you know giant portions of these books a very fun transition might be to jump into complex variables or complex analysis that's basically calculus with complex numbers so in theory you could actually skip differential equations and jump into this although i think it's better to learn differential equations first uh it's it's fun it's worth it but this again if you want to jump into this once you know some calculus you can just jump into this so and you already know some of you are at this point so this is the book by safin snyder and this is the book by brown and churchill both are pretty much the same they're both excellent beginner books and the study of complex variables real analysis is often considered the bane of math majors this is probably one of the hardest if not the hardest class math majors take you can actually learn that now if you're self studying so again you can just jump into this after linear algebra this is the book by terence tau this is analysis one and analysis two really good for reading really good examples in these books extremely enlightening much more readable and much more interesting than these other two books however these other two books are a little more standard and they're still recommended this is the one by fitzpatrick it's really good for beginners who are trying to learn advanced calculus and this is the classic book by rudin it's called baby rudin that's the nickname it has and it's principles of mathematical analysis so any of these books would be good choices for beginning to learn some real analysis one more sorry i forgot this is elementary analysis the theory of calculus by ross this one is very special because he spends a great deal of time going into the proof so again another solid choice for advanced calculus aka real analysis another possibility that you could jump into now is abstract algebra so extract algebra is like the study of groups rings and fields so you look at like a set with some like basic properties and it's really proof based so linear algebra is a good course to take before these because these are very very proof-based courses this is my favorite beginner book on the subject it's the one by sarah chino and this is the book by gallian another good beginner book for abstract algebra okay let's look at a couple more subjects that you can actually learn now so topology i just picked a book i have on topology this one has full solutions to all the problems this is the one by gamelan and green it's a great book and it's a dover books that means it's cheap it's only a few dollars on amazon let's see what's below it this is a book on combinatorics this is the one by alan tucker this is an okay book the good news is you will know some combinatorics because you started with those discrete math books so a lot of what's in this book will be review so even though this is a little bit more advanced book you have some background in it because you started with this stuff that makes it awesome what's below it naive set theory so this is something that could have been studied you know even near linear algebra set theory is not a very difficult subject this is the classic book by paul halmos who passed away several years ago this is a very famous book it's a little book um it's it's quite enjoyable again it's naive set theory by paul helmos this is a book on functional analysis and yes in theory you could learn this stuff you already know some advanced calculus aka real analysis you know how to write proofs this is probably the easiest book on the subject it's the crisis book on functional analysis and what's below this a book on graph theory this is the book by ronald gould and again you might already know some graph theory because you started with with that discrete math that's why i think it's such a good starting point that's why i picked this order uh for this video let's look at some more stuff if you really want to get into the higher level math you could you could do it so here we have two famous books this is the one uh called real analysis by royden this is a book on basically measure theory this is gradual level mathematics really hard stuff and this is the classic book nicknamed papa rudin by rudin it's called real and complex analysis and this is the book by rooting both of these can be gotten quite inexpensively perhaps i know the roydon book isn't so much money this one might have appreciated over the years you know it's kind of a cult classic but so is royden and i think royden is a little bit better so both are good books for attempting to teach yourself some graduate level math let's take a look at some more books so these are just books that i left out of the video that i had at the beginning in the video in in the first scene so let's just talk about them so this is one on linear algebra this is the one by serge lang and this could have been another choice for learning linear algebra i love this book it's an excellent book and you can get it very inexpensively one of my favorite linear algebra by serge lang what's below it ah linear algebra by huffman and cuns this is a classic this is a book that was used at mit i believe in the 60s this is more like the friedberg book it's a proof-based linear algebra book this next one is algebra by michael arton so this is a great book on algebra it takes a linear algebra centric approach to abstract algebra so this is an abstract algebra book with an emphasis on linear algebra what is this book this is calculus made easy by thompson this is a book you read when you're laying in bed um i should have mentioned this one before but this one's worth picking up this is an old book uh written i think over it's written a long time ago i think from the 1900s early 1900s and um yeah this is an awesome book for calculus and for understanding calculus so you don't want to use this by itself to learn calculus you kind of want to use the other book i recommended and then like lay in bed and read this one this will give you some more intuition here we have a geometry book that's right i had nothing in this video about geometry so i thought i should include a book this is just a basic book on geometry and this is the one by jurgensen brown and king so just really really basic geometry you can learn this at any point independently of any of the other topics here we have another book by paul holmos this is his linear algebra book it's called finite dimensional vector spaces and this is something again that you can use to learn linear algebra when you decide to jump to that stage and then another book on linear algebra this is the shams again just another resource to have and then another book on linear algebra this is the one by gilbert strang that's the guy who has the mit lectures and i think they're on youtube also really cool really cool guy so those are the suggested ways of teaching yourself mathematics from start to finish and again you don't have to follow this particular order and also keep in mind when you're trying to teach yourself math that math is hard and you're going to get stuck you might not understand everything and that's normal and that's okay and that's expected so let's say you're learning calculus and there's just some stuff you don't get it's okay feel free to move on and explore your interest right this video is about teaching yourself math it's about learning for the sake of learning and you know enriching your mathematical knowledge so i think it's the best way to do it you know you read what you enjoy read what you want to read you know if you're taking a class on linear algebra you don't want to study linear algebra you want to study something else it's like this human nature thing we always want to study what we don't have to study and so by teaching yourself you kind of get to explore what you really enjoy and what you really like in mathematics and hopefully this video has shown you all the different types of math that are out there that you can actually learn on your own good luck and take care