Learn Real Analysis With This Excellent Book

someone left comment in one of my videos few weeks ago mentioning that should write math book and think it's great idea but at the same time there's books like this that exist and this book is amazing really don't see how person mainly myself could improve on book like this because it's so well written so this book is called real analysis long form mathematics textbook and it's by Jay Cummings in this video want to show you some of the things that are really interesting about this book this book is not like other analysis book when first got this book was little bit shocked by how big it was and at first don't know if liked it but after opened it up and started flipping through the pages and looking at the book itself realized that it's really good size so that's something that's very different about this book just to give you an example have big calculus book over here this is calculus by Earl swakowski it's an older edition and you can see that even though this is thicker book this is actually bigger right so this distance here and this distance here are larger than this book here and this is about the size of like Stewart's calculus or Larson's calculus so you're getting very very large book the price of this book is also phenomenal this is not an expensive book most new books that you buy on real analysis are going to cost you lot more money than this book here as always will leave link in the description to this book and other books by Jay Cummings which are just as good really like the dedication in the book to my students without whom would have had to edit this whole damn book myself yes that would be very very big task so one of the things about writing books is it takes ton of work and time and you need someone else to help you because it's so easy to make mistakes especially when it comes to type setting mathematics if type set just few pages and don't triple check it chances are there's gonna be small error so can't imagine how long and how much work it took Jay to write this book let's take close look at the contents it starts with the reels and then it goes into cardinality sequences these are all things that you would study in real analysis course at the undergrad level series The topology of continuity differentiation integration sequences and series of functions then there's some appendices he talks about the construction of the real numbers and like this peculiar and pathological examples that's really fun thing that you know he put in the book that he really didn't need to you know at this point you know the book is done but no Jay decides to add extra content that is extraordinarily interesting think that's awesome and this is something really really like when saw this was really kind of Blown Away list of results so he actually has list of the results in the textbook I'm going to turn the page so you can see and it's kind of cool because you can go through and say that's that's something need to know or that's something interesting and you can find it it's just nice that he's done this really really like it yeah wish other books would do this this actually reminds me of some of my personal notes have notes that I've made from multiple abstract algebra courses and have page where just have list of results in abstract algebra and it's really really big list so I'm really happy that someone else in particular someone else who wrote book decided to do that and they put it in book this is definitely book that you can buy and read and learn from on your own here's where he talks about convergent sequences so he gives the definition and then he says the following this is an important definition this is the first of many weird looking definitions in this textbook and at this stage it is really important that you fully understand it so brew pot of tea find comfortable thinking chair and ponder it carefully don't move on until that pot is emptied the better you understand it now the easier the rest of the course will be here are few more comments to help so he spends lot of time actually trying to to explain this is completely the opposite of book like ruden's principles of mathematical analysis this is book for learning it's book for students and Jay Cummings has done an incredible job am very very happy with this book here's beautiful example of Jay Cummings showing that sequence converges to three so we have sequence sub we have to prove that the limit as approaches Infinity is equal to three and like how he breaks up the work for this problem into two parts you have the scratch work and then you have the proof and the way he does it is pretty much exactly the same way do in my videos and think that's great because do it that way because think it's clear at least to me and so hope it's clear to other people and Jay Cummings does it the same way which makes this book extra cool in my opinion here's an example of some of the exercises you must prove all of your answers unless stated otherwise remember the proof techniques that you learned in your intro to proofs class Direct by contradiction by using the contrapositive by induction and or by cases and you can disprove something by exhibiting counter example super clear super to the point super awesome and then he provides hints and solutions to select exercises at his website so more on that in second first let me just show you how many problems you get in chapter one so 16 35 yeah 35 no there's more 37 problems you get 37 problems in chapter one however I've been to the website and there are some hints and solutions there's not that many so have couple theories as to why that's the case people always ask me why do so many math books not have you know answers in the back of the book so think there's there's two possible reasons and I'm not saying that this is the reason that he didn't include them so one it is ton of work to actually you know go through all of those exercises and write up Solutions I'm thinking of it myself if was writing this book I'd write the whole book I'd have to come up with all the exercises and then not only do have to solve all the exercises have to typeset solutions that are 100 correct to all of the exercises that is an extra amount of work that is huge another reason that think sometimes books don't have Solutions and this is reason that some people might not like don't know don't necessarily agree with this reason it's that these books math books are often used in college courses so people take classes and students are assigned homework problems and it's always better if you spend considerable amount of time trying to figure out the problems yourself before looking at the answers you know have topology book that has all of the answers in it has full proofs but you gain nothing from it if you just jump to the solutions because you're not solving the problems yourself you're just basically trying to understand the answers which in itself is Meaningful but it's not the same it's more important to spend considerable amount of time working on the problems and teachers know that and so lot of times lot of teachers prefer that books don't have Solutions so that could be reason that this book doesn't have Solutions and also it's just ton of extra work but you get tons of problems in this book Crystal Clear explanations it is written in incredibly let me show you something here just saw something wanted to remind you of or tell you about open questions so Jay Cummings has done something amazing here after each exercise set you have open questions basically these are unsolved problems in mathematics and they're in this book so an amazing book that's great to read and you have open questions really cool and he does that for every single chapter after the exercises he also has that really cool appendix with peculiar and pathological examples for example here he discusses the devil's staircase really wish book like this existed when was first learning analysis think it would have made my journey much cleaner it would have been much easier for me to learn analysis had had book like this when was taking Advanced calculus as an undergraduate used the book by Fitzpatrick which is way different from this one right way different not written in the same tone much harder to read and it's considered probably one of the easiest to read analysis books but this one is actually easier to read it's actually that good so think if you're student and you're trying to learn analysis on your own for whatever reason maybe you're doing it for fun maybe you're going into another field like economics or maybe you're taking class or you want to get ready for class before you take it there's nothing like preparing ahead of time you know if you're like in high school or in college and you're not even at the point yet where you can study this stuff still think it's worth trying and this book is good choice it's big paperback really really big really well made fantastic thank you Jay Cummings for bringing this book into existence and again I'm getting Goosebumps here this is one of those books that whenever people say to me hey you should write math book think about all of the wonderful math books have collect math books and have math books written by brilliant mathematicians and math books written by people you've probably never heard of and lot of them are really good and this is an example of one such book anyways just wanted to show you this video because people always ask about analysis books this one is super affordable it's big it's well made very easy to read compared to others like it before forget the number one prereq for this is that you know how to write proof so definitely learn proof writing before you jump into this if you're thinking well how do learn to write proofs actually have it right here and this was not intentional check this out have his other book it's called proofs long form mathematics textbook so great book on proof writing this has its own unique characteristics and probably should do video on this at some point they're both worth getting and I'll leave links in the description to both of these hope this video has been helpful to you good luck