Learn Mathematics from START to FINISH 2 nd Edition

in this video I'm going to show you how to learn mathematics from start to finish we're going to start with very very Basics and work our way up I'm going to give you three different ways to get started with mathematics they're all different and all of them are okay starting points for learning mathematics if you like you can start with all three at the same time which makes it really really cool one way to get started with mathematics is by starting with algebra and this is actually the most common way that people start learning math is by learning basic algebra so this book here is called pre-algebra mathematics and it's by Nichols and this is great book for learning just the very very basics of algebra now if you feel like you know some basic algebra and want to start with something little more advanced this one here is more modern book and it's called Elementary algebra and it's by Sullivan stru and mozarella and this one is really really nice it's got tons of examples and it should teach you all the algebra you need in order to proceed in mathematics I'll try to leave links to all of these books in the description of this video another way to get started with mathematics is to actually start with discret math and this is really unconventional Approach at most colleges and universities in the United States typically you have to take calculus 2 before jumping into discret math however lot of the math in discrete math books you can actually learn without knowing any calculus or without really having strong WR command of algebra so let's go through and look at each of these books and I'll tell you who they're for so this first one here is called discrete mathematical structures it's by Colman Busby and Ross and this is super beginner friendly book so this book starts with really really basic math and proceeds from there in particular this book will teach you logic which is basically precursor to understanding how to write mathematical proofs and once you know how to write basic math proofs you can go into math Beyond calculus so nothing says that you can't start with book like this again this is the one by Coleman Busby and Ross more advanced book is this one here it's called concrete mathematics by Graham kth and pachnik this is very famous book because it's written by very famous people in particular canth is super famous computer science professor at Stanford who created Tech which is what latch is based on believe which is what's used to create math symbols in modern books today so legendary book written by legendary computer scientist this book actually has answers to all of the problems in the back of the book which is really really cool however this book is way more advanced than this one so this one is not for beginners in my opinion then here we have two standard books on discret math both of these are also pretty tough this one is discret math by rosin and this one is discreet math by Grimaldi both of these books are widely used in the United States in discret math courses these books are tough reads they take lot of effort to get through but they're both solid discreet math books so another way to start again is to basically pick up one of these books and learn the logic and proof stuff in these books which will help you for higher math finally the third way to get started with math is to just jump into the proof writing which think is super fun so here we have couple books we have how to prove it structured approach by Daniel velman we have foundations of higher mathematics by Fletcher and Patty an introduction to abstract mathematics by Bond and Kean and how to read and Drew proofs by Daniel solo so all of these books are awesome let me just talk about each one briefly so this one here I've had for really long time several several years and think it's great book taught an independent study many years ago with just handful of students and this is book that recommended to them and they thought it was pretty good they didn't think it was amazing but they thought it was pretty good think it's amazing and I'm big fan of this book great for learning to write proofs another one very similar to this one is the one by Fletcher and Patty this is one that was recommended to me here on the channel by several people so finally bought it and think it's awesome this book will teach you how to write proofs this one was recommended to me by former Professor it's called how to read and Drew proofs by Daniel solo again another excellent choice when you want to learn how to write proofs and have to say that my favorite one now out of these four is this one how to prove it structured approach by Daniel velman this one is really affordable in fact most of these books should be quite affordable but this one might be especially affordable and it's especially good he spends lot of time explaining things and he always falls back on the logic which is absolutely key when you're trying to learn mathematics this book is awesome and cannot recommend it enough he explains things in more than one way so when you don't get it he has sometimes an alternate explanation which think makes this an Incredible Book for anyone who wants to learn how to write math proof okay so now that you know how to get started with math and again you have three choices you can start with basic algebra you can start with proof writing or you can really jump into it and start with discret math what's next what can you possibly learn next in math well let's keep going and we'll go through more math topics that you can learn once you know these so in order to learn calculus you really need to learn more algebra and some trigonometry so to do that people usually go to college and they take course called pre-calculus and they take course called trigonometry and with those two courses they learn enough mathematics to take calculus pre-calculus is basically just like blend of random topics that come up in courses like calculus so let's go through each of these books and talk about what's in them this is solid book on pre-cal and it has everything you need in order to learn calculus this is the one by Stuart redin and Watson James Stewart was very famous Canadian mathematician who passed away several years ago he's known for his book titled calculus which is probably the most popular calculus book in the US this book is great it has everything you need to you know study calculus and definitely recommend it this one is called college algebra by Blitzer and this is basically just an algebra book that is little more advanced than the algebra book showed you earlier so this will basically teach you the algebra you need in order to study calculus however having this book is not enough because you still need some trig not this book pre-calculus also contains the trig that you need so if you got something like this that should cover all of the math you need whereas this one just teaches you algebra but wanted to include it because it's definitely an option and it's probably really affordable I'll try to leave links in the description to all of these books really popular book that I've actually used to teach pre-calculus and trig and combined course in pre-cal trig is this one here it's called graphical approach to algebra and trigonometry and this is the one by Hornsby Lyall and roxwell and really like this book like the examples like the exercises and like the explanations again just another option so that you can go further and study calculus very similar one is algebra intrig and this is by Stuart Redland Watson very similar to this one just so you have more options and just so you have even more options because know sometimes books can get expensive here's one by Michael Sullivan which is also fantastic so this book is very similar to this one which is very similar to this one so any of these books should do the job here we have more specialty book this one is called Shams outline series of trigonometry and this is just trig so if you do get this book still think it's worth getting one of these big thick algebra and trig books because you just need one right once you have one that's usually good enough if you're wondering why have so many books it's because collect math books if you want an older book on algebra in trig that's not too hard the one by Cameron is pretty good it's kind of an interesting look at the way math used to be taught really like this book and it's much much thinner than the big newer modern books if you want thicker older book think this one is really cool and it has really hard problems it's called pre-calculus mathematics and it's by Shanks Brum fleiner and iols this book is way harder than lot of the modern books and it's probably pretty affordable I'll try to leave some links in the description for you but really like this book I'm just going to smell it here smells so good old school book on pre-cal math it's always fun to get these older books because you get different problems this one here is really unique this is actually workbook on trig so if you're taking trig now or if you decide to learn trig on your own this book is totally different from every book I've shown you because what you can do with this book is you can actually write in the book you see and it has answers to every single problem now this book doesn't have everything you need for trig in it like the other books do but but it has the hard stuff in it and it gives you tons of practice problems so feel like Chris McMullen who wrote This Book knows what trig students struggle with and he made an entire book to focus on those topics and the last one's kind of unique it's an entire book on just trig wanted to include it because you don't really see too many books on just trigonometry anymore most of the modern books are pre-calculus or algebra and trig books this one is just trig and it's by Bernard by rice and strange this is an older book and it's been around for super long time so now you're at the point where you know some basic algebra and you know some trigonometry it's time to jump into calculus and you shouldn't feel like you need to be an expert in either algebra andt trig to jump in sure if your skills are weak you're going to struggle more but don't let anything stop you from exploring the higher levels of math that we're going to discuss in this book if you feel like reading book on advanced math you should read it that's the whole point of self-study so let's start with this one because this is book I've never talked about here on the channel this one is calculus with analytic geometry and it's by swokowski great book don't know if it's still in print but I'll try to find it and leave information for you this one is little bit different than lot of the other books I'm going to show you but it's got tons of calculus in it pretty much everything you need but it is an older book so you get different take on the subject more modern book and one I've used to teach Calculus 1 2 and three is the one by Larsson Edwards I've read major portions of this book and I've done ton of the exercises great book for calculus so everything you need for calc 1 2 and three is in this book legendary book on calculus is this one it's called thas calculus and this book has been around for while this is the 13th edition actually have older copies of this book older additions rather of this book and think it's awesome this book is incredible it's got tons of exercises great explanations and again everything you need just to give you another choice and the third or fourth really big thick calculus book want to show you is the one by James Stewart this is perhaps the most popular calculus book probably in the world at least in the US and Canada this is the same James Stewart who wrote the pre-cal book which showed you earlier and just look how thick these books are so any of these books is awesome you really only need one of these again have more than one when took calculus used the early transcendentals version of this book and again used the Larson book here to teach calculus so any of these should teach you all the calculus you know however books are hard to read so wanted to show you these other books have here which are super affordable okay these are really really inexpensive you know these books are expensive right but these are super affordable so what are these so this is book that will teach you some of the stuff that you learn in calculus ones like some of the single variable stuff it also has some Cal 2 topics in it and this book is awesome because you can write in it you see and it has full Solutions in the back of the book to all the problems it's written by Chris McMullen and it's super popular and super amazing however think this one is better this book is not as popular and the reason think this book is better is because this book is on multivariable calculus and it has examples in multivariable calculus in it that will help you when you're using these books to learn one of the things that people struggle with with with these books is having enough examples you know like if you just buy the Stewart book you're going to get some examples but there's going to be exercises in the book for which the examples are lacking or you're going to feel like you need more help it's these books or other books that provide more help so awesome book definitely recommended for anyone who is taking multivariable calculus course or learning calculus 3 and super affordable compared to these books here now that you know some calculus it's time to jump into differential equations and while have lot of books on differential equations just chose three for this video here because these are some that I've actually spent lot of time with so let's start with this one here this is the Shams outline on differential equations and this is really good Shams outline this one's pretty good some Shams outlines are better than others this is one of the better ones it's not the best one still think the best Shams is the one on complex variables or the one on topology but this one's really really good it's got lot of examples and pretty much everything you need think you could in theory use this to learn de on your own but still think it's better just to use it as supplement now this one here is the one used when took differential equations as an undergrad long time ago this one is by Nagel saf and Snider great book solid examples tons of content has answers to the odd problems it's just really nice Modern Standard book on differential equations and the one used to teach differential equations for many years is this one here by Dennis zil first course in differential equations do think this book is easier than this one you'll notice it's lot thinner but that's because this one includes the boundary value problems but great book it's worth having more than one book because it's hard subject to learn on your own so all of these are totally worth getting let's talk little bit about probability and statistics so this is something you can learn and we could have learned this before but didn't want to talk about it until now because some of this does require calculus so this is the Shams on probability and statistics very affordable worth having if if you you know want to have it recommend it it's got tons of examples in both probability and statistics but still think it's better to have textbook so here's book that's perfect for beginners in statistics it's by Weiss and it's called elementary statistics this one is the eighth edition so this is like your standard modern book on statistics and there's other books like this on statistics just really pick this one at random because think it's pretty good so tons of examples and it will teach you statistics after you know basic statistics you can jump into something like this this one is mathematical statistics and data analysis by John rice and this is the India Edition bought this Edition CU was able to save some money but it was still pretty expensive this book has lot of math in it some of it does require calculus but it's an advanced book so you can expect to struggle to learn the material and then left this one for the end this is really nice little thin book which has some fantastic examples in it it's called first course in probability by Sheldon Ross and you can read lots of this book without calculus but as you can see here you know there's triple integral there so you see you will need some calculus for lot of this book so those are some options for learning statistics as for geometry book that is often used is this one here it's geometry by jurgenson and there's definitely newer Edition out than this one and I'll leave the information in the description but this is standard book on Geometry that has been used for very very long time so worth getting if you are looking for book on Geometry okay it's time to talk about linear algebra so first let's talk about the Shams this one is really nice because it has lots of examples and lot of topics that you sometimes won't find in other linear algebra books so it's very very comprehensive and it serves as good reference and is good source of additional problems still think it's better to have regular textbook so let's talk about the books have here so out of these three here the best one for beginners is this one this is Elementary linear algebra by Howard Anton and there's other really good beginner linear algebra books out there just wanted to pick one for this video because well we're going over lot of books so didn't want too many books so this one is really good for beginners because the problems in it are very standard so if you are taking class in linear algebra and you have test coming up if you open this book and you go to the specific section for example maybe you're studying subspaces and you go through and you do these examples and you do these exercises for practice there's chance that some of these questions might show up on your test and the reason is it's just very standard in the presentation so it has problems that you'll find in you know most linear algebra books definitely recommend it very affordable it's the one by Anton these books are more advanced this one is older this one is newer so this one is classic book on linear algebra and it's proof-based now while the ANTM book does have proofs this one is lot more sophisticated and this one is kind of like modern version of this one this one is really good because it serves as great reference and I've used the index multiple times to look things up whenever I've needed to look things up so worth having super affordable it's the linear algebra by fredberg so proof-based linear algebra beginner linear algebra supplement for linear algebra complex variables some people say that this is the most interesting course you can take as an undergraduate and so here have bunch of books and they're all really really good so the Shams for complex variables is considered by many people to be the best Shams outline ever written ever like it's so good because it has everything you need mean in theory you could just use this and learn everything that is taught in complex variables course it is such good book now having said that still think it's better to have book but if you're studying complex variables get the Shams and if you have to buy one Shams get this one because it's that good so this one here is perhaps the most popular complex variables book in the world in the sense that it has been used for so many years and so many people have used this book you know for decades this is the seventh edition I'm not sure if there's newer one it's by Brown and Churchill and it's an introductory level complex variables book so that means that you only need you know some calculus for this you should know how to compute like partial derivatives and stuff like that which you basically learn in calc 3 so beginner book for complex variables this one is also beginner level book on complex variables it's by saff and Snider and it's really hard for me to say that one is better than the other I'm going to just come out and say it and be biased like this one little bit better because this is the one that used when was an undergrad but honestly think I'm little bit biased in saying that because this one's also really good these books are very very similar in content and either of these books should help you learn complex variables so if you want to learn complex variables my advice is get one of these and get the Shams so what's under here let me show you this one actually bought before any of these books it's over book so it's super inexpensive and bought this book when was in Cal 3 because wanted to learn some advanced mathematics on my own it's by detman and remember reading this while sitting at the dentist thinking this is really cool can learn complex variables yes so fun little book very very affordable and it has lot of mathematics in it this one is way more advanced in these wanted to give you more advanced book on complex variables it's the one by abowitz and focus this has has lot of material that you won't find in these other books here let's look at some partial differential equations so in order to study partial differential equations you basically need some differential equations so you definitely need to know ordinary differential equations so those books showed you earlier are musthave and the good news is you will use lot of the things you learn in ordinary differential equations in order to solve partial differential equations lot of times you'll take pde which stands for partial differential equation and you'll use some technique and you'll end up with an ordinary differential equation and you'll use what you learned from those other books to solve problems in these books so put this one first because it's very affordable and it's do book it's by Zack manag Glo and th great book on pdes this is one that use for course really good book the only downside is there are some mistakes in the answers which can cause you lot of pain but it's really nice book with really fancy cover the Shams outline on PD had to include that because it does help little bit and then this one here is actually really affordable believe this is do reprint so you can find it through Dover and it's by Miller this is great book for getting started with partial differential equations abstract algebra is one of my favorite subjects and have over 40 books but I've picked only few here just to show you so let's talk about each of these and I'll tell you who they're for so this one is called first course in abstract algebra and it was written by John fle this was mentioned to me by my former professor and he thought that this was great book so bought it and also think it's great book there are some things in this book that are little bit different from other books but it's still worth having and it's got lot of abstract algebra in it it's intended for beginners and like it lot wouldn't say it's the easiest beginner book but it's definitely considered beginner book this next one is contemporary abstract algebra by Joseph gallian have friend who lives in New Zealand and he has PHD in math and when he was in college as an undergrad this is the book he used and so he told me about it so went online and bought copy several years ago this book is really good because it has lot of examples that you won't find in other abstract algebra books like the one by fet always think of this as the book of examples for that reason alone so great book it's got lot of abstract algebra in it and tons of examples another book which think is easier than both of these is this one here abstract algebra first course by Dan Serino this one is much easier than these books in my opinion so this one is better for beginners but again they're all considered Beginner Books these books have more content than this one does but again this one is better for beginners another one which really like is this one here again beginner book and this one actually used to do self-study taught myself field Theory using using this book started with the section on polinomial and then work my way up from there and did that while was in graduate school because use this book as an undergrad and didn't learn enough math from this book so had to supplement on my own and got it from this book and out of all the books have out of all the abstract algebra books have which are not even in this video this is the one pick for that so this book to me is really special and think it's really amazing and it's just yeah don't think it gets the attention it deserves this one is classic book it's considered graduate level by many people it's called algebra by Michael arton and so this one's way more advanced than any of these and this one takes very linear algebra focused approach Michael has really good examples and really good exercises and places heavy emphasis on linear algebra this next one want to show you is the one by dummit and foot this one believe in my opinion like this one better because this one has better index so as reference this one is better book than this one and it just has lot of math and lot of exercises in it my copy is falling apart let me show you you see and bought it used so maybe should have gotten new copy saved few dollars by buying it used looking back in hindsight just should have spent the money on new copy in fact after bought this book and was disappointed bought new copy of this one so this one has way more math than all of these it has more topics this is the one by the late Serge Lang and many people consider this to be like the Bible of abstract algebra this has ton of abstract algebra in it and it's recommended for anyone who is serious about abstract algebra okay now it is time to discuss Advanced calculus or real analysis and this is all undergraduate level real analysis here and we're going to talk about each of these books and the pros and cons and who they're for so this first one is super famous I'm sure you've heard of it and if you haven't it's called principles of mathematical analysis and it's by Ruden and this is known by many people to be very dry hard rigorous book had friend who read the entire book and he didn't do the exercises but he read the entire book and tried to work through all of the theorems and you know he got stuck and he did what he could but he worked from the beginning to the end and yeah so you can learn lot of mathematics from book like this but it does take lot of effort and there are other approaches that might make your road to learning analysis little bit easier so this next book I'm going to go out of order here this one is Advanced calculus by Fitzpatrick this is perhaps one of the easiest Advanced calculus books out there now when was using this book to learn certainly did not feel that way instead felt like Advanced calculus was consuming my life and that's because it's super hard to learn this stuff right this is not easy mathematics but this is one of the easier books it's the one by Fitzpatrick another really easy book that bought maybe only few years ago is this one here by and sheret initially had checked this book out of library when was student and thought what great book because it gave me an alternate explanation when trying to learn things while was using this book so thought that this was great supplement to this book so definitely worth getting if you can find it again I'll try to look for it don't know you know the status of this book if it's still in print if it's easy to find but it's really really fantastic very very happy with this book and of course had to include book by Lang this is his book on analysis one an excellent text and understanding analysis by Abbott also very very wonderful text for learning basic undergraduate analysis and I've left my favorite of these for last it's this one here it's Advanced calculus By Buck this one starts with the multivariable approach so it starts with like calcul of one variables and then it also talks about calculus of you know variables so you know you have statements regarding like dimensional space and stuff early on in the book also it's got ton of math in it so awesome book definitely recommend it Advanced calculus by buck so here we have three books for learning number Theory this one is Elementary number Theory by Underwood Dudley we have number Theory by George Andrews and we have Elementary introduction to number Theory and this one is by long so all of these books are excellent for learning number Theory you can just pick any of these and you'll be good to go I've had this one way longer than these and I've spent more time with this one and great book I'm just going to give it whiff here just cuz it smells so good there's newer additions of this out now yeah what wonderful book yeah awesome books as far as graph Theory here's pretty good choice it's called graph Theory by Ronald Gould now keep in mind that those Des math books that showed you at the beginning of the video those actually do cover graph Theory and they also cover combinatoric so you learn lot of math with those discrete math books that's why gave that as an option for learning math so you can see the different types of math that is out there so yeah this is really solid book on graph Theory topology is typically course that math students take their senior year as undergraduates so here we have few books Let's talk about each first let's start with the Shams and this again is one of the best Shams out there think so this one and the one for complex variables in my opinion are just rock solid and the reason think they're so good is because they're very standard in their topics they contain things that you're going to see in your course if you take course in college so really great resource for extra proofs and extra examples and extra knowledge so again it's the Shams on General topology this one here is an older book it's classic it's introduction to topology by Bert Mendelson and this is the second edition so it's an older book on topology it's probably quite affordable so and very very standard in the treatment this one is awesome because it actually has full solutions to all of the exercises in the back of the book including the proofs right they'll give you partial proofs or you know pretty pretty good hints to all of the problems so great book very affordable because do book it's the one by gamelin and green and this one's like the standard often used to teach Advanced undergrads or beginning graduate students it's called topology and it's by monkes great book it's not going to have all the answers like this one does but it does have lot of mathematics and it's worth having my copy is the Eastern economy Edition because was able to save some money by getting this version versus the other Edition so all of these are awesome recommend all of them and if you're taking topology now or you buy one of these books and you start reading it and you struggle just know it's normal didn't include topology until now in the video because feel like you definitely want to have some Advanced calculus before you jump into topology just so your proof writing skills are sufficient topology can destroy people if they don't know how to write proof so you need to have solid command on your proof writing skills before jumping into something as cool as topology want to finish this video with just bunch of random books on different things that thought were really really cool and think these are all amazing so this first one here is called all the Matthew missed but need to know for graduate school and it was written by Thomas gity so had the pleasure of actually interviewing Thomas gity while ago here on the channel and he is super smart super awesome and think he's great guy and he wrote this amazing book and this book covers tons of mathematics tons let me just show you really briefly all the math in this book so you can't learn all the math in this book but look these are all the topics here's brief summary linear algebra real analysis look at all that all of that math right in one book and he basically gives you the overview of each of the subjects in this one book so incredible book super affordable worth having it just opens your eyes to all of the mathematics out there thank you for writing such wonderful book Mr Tomas scity this next book is one that don't think I've ever talked about it's book on cryptography this is book that I've read almost 100% through I've almost read the entire book this is the international Edition bought that one to save little bit of money and used this book for course in graduate school and our teacher was really hardcore and he went through the entire book so spent many many nights eating pizza staying up late with my friend trying to understand all the algorithm and stuff in this book it's really good book and we covered lot of material let me just say that worth it if you want learn cryptography this next one is awesome for anyone studying engineering physics or math it's the legendary Advanced engineering mathematics by kig this book has so much mathematics in it it is completely insane let me see if can find the contents just to show you so it starts with some differential equations po more more Dees Applause transforms that's more Dees vect so de full de course Vector analysis so you got some multivariable calculus like calculus 3 so you've got differential equations some calculus 3 then he jumps on jumps into some linear algebra matrices and determinant so already you know you've got two full courses here 4A series and integrals partial differential equations then here you've got some complex analysis so mean calculus 3 complex analysis it's got some linear algebra it's got some regular Dees it's got some Cal 3 mean just so much mathematics in this wonderful book and it's just written in very clean way so worth getting definitely recommended for anyone in physics engineering or math this next one is graduate level book on measure Theory it's called real and complex analysis and it's the one by Ruden this is the Ruden book and again it's classic it's basically the followup to the principles of mathematical analysis book that showed you earlier by Ruden so topics here are much more advanced these are things that you would learn as first year graduate student and the next book want to show you is one that's also considered graduate level book it's the one by kig on functional analysis this one's really good for beginners before reading this book you're going to want to have strong command on proof writing maybe you've had some topology before you read this some exposure to metric spaces would be good the book does cover all that but having prior exposure always helps wonderfully written book on functional analysis perhaps the best introductory book on the subject and the last book is one that took me long time to buy just wanted to include it in this video cuz finally bought it it took me long time to buy this because it's expensive it's not inexpensive you can't find it inexpensively it's always expensive and it's by Serge Lang it's called basic mathematics and it has bunch of strange Topics in it like it doesn't really fit in anywhere else can't really like categorize it and it basically just has some basic mathematics let me just show you what's in this book so you can see the topic starts with numbers linear equations real numbers quadratic equations logic right so you wouldn't expect that some geometry segments rais and lines trigonometry functions mapping so just complex numbers just lot of like random topics and does have answers in the back of the book to the problems let me just show you the answers so answers to selected exercises so it does have answers so nice book worth getting if you decide to spend the money it took me long time to buy it because didn't want to spend the money especially on paperback that's really what bothered me I'm just going to give it whiff yeah very happy to finally have this book just love the color too the the yellow cover it's just very very cool so that's it just wanted to show you how to get started with mathematics and how to learn ton of mathematics and again you have three ways to start you can start by learning some basic algebra and it doesn't necessarily have to be pre-algebra if you feel like you know some algebra maybe just get one of those like algebra int trig books showed you at the beginning of the video if you want to learn to write proofs pick up proof writing book and if you want to learn some discret math pick up discret math book in my opinion the best way to start is to start with all of them right and if you feel like you're ready for calculus jump into it don't be afraid to dive into into more advanced mathematics even if you think you won't understand it you know if you buy one of these books and it's too advanced who knows maybe year from now you'll pick it up and we'll start reading it because you'll be ready for it then hope you've enjoyed this video until next time good luck and take care