5.5 Math Books For Self Made Mathematicians

in this video we are going to look at five and a half math books for self-taught mathematicians that's right if you are trying to learn mathematics on your own after watching this video you'll have an idea of what some good math books are I'm going to show you five and a half books that you can use now you might be wondering what is this half book at the end of the video I will show you what I mean by this half book and it's actually it's actually a book that anyone can read I mean you do need to know some mathematics but most people can buy this book and read it and actually see how math is used in the real world in any case let's get started right away we are going to go in order from the easiest book to the most advanced book so this is the easiest book in today's video it is a basic calculus book so we're starting with Calculus so I've completely neglected algebra and trig and that's what's required in order to learn Calculus if I remember I'll try to leave a link in the description to some algebra books that you can use if you want to get your algebra skills up to par before jumping into this book in any case this is a great book it's called calculus it was written by James Stewart it is the most popular calculus book in the entire world there is no book that is more popular than this one James Stewart was a Canadian mathematician he passed away several years ago and I'm pretty sure he was a millionaire and there was something about his house being built in the shape of an integral you can read all about this famous Canadian mathematician on the internet James Stewart in any case this book is for people who need to take the calculus sequence in college typically calc 1 calc 2 and calc 3 the prere for this book would be of course in pre-calculus and a course in trigonometry at the college level now for those of you self-studying that basically means you need to know some basic trig and some basic algebra and if you know that you can jump in it's not a perfect book you're still going to struggle as you you know try to learn the mathematics that's in this book but honestly there's a reason it's so popular it's because it's really really good let's take a really brief look at the contents of this book so you can see exactly what it contains here's a brief look at the contents by the way the addition doesn't matter too much and there's also an early transcendentals version that just changes the order of things uh it's equally good so it starts with functions and models limits and rates of change and then it goes into derivatives applications of differentiation so all of this stuff here is calc one stuff okay so if you were to take a cal one course in college this is the stuff that you can actually learn on your own this is still calc one stuff typically and then this is typically where Cal 2 starts depending on your school um but for self-study people it doesn't really matter right there is no course right you learn all of it at once this is Cal 2 stuff techniques of integration further applications of integration it's got some differential equations which is pretty cool parametric equations and polar coordinates this is definitely C two stuff here infinite series really fun that was my favorite in Cal 2 and then here you get into multivariable calculus this is typically where a calculus 3 course would start starts off really easy by the way so like you could just jump into this you can actually start here right nothing says you can't pick up this book and start reading it right that's the nice thing about self-study um you decide what the prxs are right you make the decisions as to what you want to study partial derivatives multiple integrals Vector calculus and then we have some differential equations here this book has Answers by the way to all of the odd-numbered problems in the back of the book which is really really good you can probably find answers on the internet too or videos uh for some of the exercises because it a popular book and it's used all over the world you can see here you have answers in the back very very nice of Mr James Stewart to include that for us yeah awesome book uh and it has a lot of exercises so like here's 13.4 on the cross product this is a cal 3 topic so if you go to the exercises you can see that there are tons of exercises which is really nice right because you can practice and when you're doing self-study what you can do is just do the odd ones and check your answers that's probably the easiest way you could do the even ones but you don't have answers so it's kind of hard to check your work I mean you can do them anyways but I think it's better just to do the odd ones unless there's an even problem you really want to tackle then go for it as far as the examples and the layout of the book it's very very modern and very professional um nothing negative to say I used the early transcendentals version of this book when I took Cal 1 Cal 2 and Cal 3 and then I bought this one later uh just just as a collection because I collect math and science books but yeah awesome book again this is the one by James Stewart and it's a great book all you need to know is algebra and trig and you can jump in by the way I'll leave links to all of these books in the description of this video okay let's jump to the next book this next book is one that I've talked about in the past but I think it's been a long time since I've mentioned it this is actually a book on proof writing it's called a transition to advanced mathematics and it's by chartran palamini and Zang and these are the authors of the book I want to emphasize that because I believe there is another book on Amazon that has the same title and has different authors so I I'll leave a link to this exact one in the description in case you want to check it out so this is a book that was recommended to me by my friend many many years ago he was a really big fan of shart trans books and yeah here's the copyright see when this was 2003 Pearson Education got the preface and this is a book for writing mathematical proofs so this is a book that will open the door to more advanced mathematics it's pretty much a requirement you can actually start here by the way you can you can skip the calculus and start here because um I mean sure it helps to know some calculus and the more math you know the better but this starts with pretty much no prere just you just have to have a strong desire and some knowledge of math U because you see it starts with sets right starts from the very beginning it teaches you everything need to know so the prxs are low logic again what's the prere for logic being able to read that's pretty much it right and you can just jump in right here direct proof and proof by contrapositive if you can teach yourself to write proofs with this book The World is Yours you can learn so much math on your own it is completely ridiculous and I think a lot of people uh don't know that so it's really important to realize that if you know how to write proofs you can jump into some of the other math that I'm going to show you later in this video equivalence relations fun functions all of this stuff is super super key in mathematics mathematical induction cardinalities of sets proofs and number Theory proofs and calculus this is really cool this is not typical in proof writing books and then it even has proofs in group Theory and then you have answers and hints to selected odd-numbered exercises let's go there first because that's very important and you can see here he does have answers and hints or they because it's several authors in the back of the book which which is super super useful especially when you're doing proofs another thing I really like about this book are the examples I've worked out several of the examples here's a good example of what I mean LMA to prove so we have to prove that the limit as n approaches Infinity of n / n + 1 is 1 and see how it goes through it the proof strategy for a given Epsilon greater than zero we are required to find a positive integer n such that if little n is bigger than big n then we have that inequality there and it goes through and they just go through and they say okay we need n to be greater than that so then you choose n bigger than that and then you go from there and then there's your proof and so it's just a really nice way to um write the proof right it does does a really good job does a really really good job yeah and and that's throughout right that's just one example I just opened the book and that we found that one but it's it's a reoccurring theme uh in this book okay I want to emphasize that it's not just like oh that happens to be a good example no pretty much all the examples are pretty solid I've worked out a lot of the exercises from this book and I've worked out a lot of the examples I haven't read the whole book but I feel like I've spent a considerable considerable amount of time with this one and um it's it's a great book I don't know how much this book cost typically these proof writing books aren't super cheap but in my opinion they're worth every penny you should get as many proof writing books as you can because proof writing is hard okay uh if I'm being honest this is going to take you a lot of work to actually learn but if you can learn this if you can learn this stuff you can get a math degree right because if you can write proofs then you can take the more advanced courses and learn more advanced mathematics like I'm about to show you right now here we have another calculus book and you might be wondering why are we looking at calculus again if we already looked at a calculus book well it's because this book is different okay I bought this book when I was a student after I took HK 3 and I still had a hard time reading this book okay I struggled so much I struggled incredibly and I thought wow how how can this be everyone says this is like the best book in the world how am I not able to understand this book and the truth is I understood some of it but I got hung up on all the proofs so if you know proof writing it will help you tremendously uh in using this book for selfstudy this book is beautifully written okay it's called calculus by the legendary Michael spivac and it's pretty much a cult classic uh in the world of math books it doesn't cover everything as you can see here it covers different things here part one part two and part three um and then here we have part four and part five so it covers just some calculus some series stuff so we've got some series some complex numbers derivatives and integrals limits it covers the main main things in calculus it's got these fun little quotes at the beginning to be conscious that you are ignorant is a great step to knowledge absolutely yep and then I've read big portions of this book I remember reading all of chapter one when I was a student and going through and trying to go through these little proofs and understand just these little basic things and I had such a hard time with these little basic proofs and little basic logical arguments and they're not hard but you have to understand how mathematics works and the structure of proof writing so yeah I think I think that is something that can help you greatly or maybe you already own this book there's a good chance if you're watching this video you might even own SPX calculus I feel like it's one of the more popular books and many people own it simply because it's such a legendary book um if you already own it and you're having a hard time I think proof writing can help you greatly the exercises in this book are notoriously difficult uh there is a Solutions manual you can bu I have it um I actually own the solutions manual and I'll try to leave a link in the description to that as well I definitely recommend getting the solutions manual uh spitback provides answers in the back of this book for a lot of the exercises as you can see and he does a really good job and he does an equally good job in his Solutions manual I feel like he kind of provides the rest of the exercises it seems in these Solutions manuals so there is a new addition of this book out as well uh this is the I think this is the third edition I think there's a fourth so I'll leave a link to that one but you can see here Michael goes through everything really carefully theorem one if f is a constant function f ofx equals c then the derivative is equal to zero for all numbers here's the proof just applying the definition of the derivative you get C minus C in the numerator which is just zero boom done yeah so pretty easy proof and stuff like that you know I didn't understand really I had a hard time understanding that as a student but it's very very clean uh very you know easy to read in some sense and he gives he gives explanations that you won't see in other books like the way this book explains the concepts is very different from the way that things were explained in this book um so it's at a different level it's at a different level and this book again has legendary exercises some of them are super hard the ones that have a star are really really tough so if you're looking for challenging problems look no further actually out of all the calculus books I have like out of all the standard ones this one has the most interesting problems so that alone right there makes this book uh very very special and probably why it's one of the reasons why it's a cult classic all right let's keep going and let's go to something more advanced next up is Advanced engineering mathematics so this basically is a survey of math this book is going to give you insight into all areas of math and again in theory you could try to start here but you're going to get hung up this book contains a lot of mathematics uh that you may have never seen and don't let the name fool you right say I'm not going into engineering ah it doesn't matter right this is a math book this is a solid math book Irwin kig this is a wonderful book I'm sorry I have to smell it here I just have to give it with oh smells amazing here's the copyrights on this one 62 and 67 yeah beautiful I love this book this is this is one of my favorite books and I thought I need to pick a a a really good books for this video right because I want to give you books that are awesome and this is one that I think is absolutely incredible the worst thing about this book let me just tell you before we look at the content is the price okay because this book is not super cheap um I I actually have the international edition of this book and then I have a hard cover as well uh it's it's not cheap but it's worth every penny let me tell you so starts with the review it's a very very eloquent review I've gone through most of this and I've done a lot of the exercises actually and then it teaches you differential equations right so oh and in order to learn this stuff you want to know calculus so basically calculus is definitely a prere and then um proof writing obviously will help you greatly more differential equations I mean it's a complete course here in differential equations all this is De stuff wow I can smell the book from where I am four uh Vector analysis so now you've got some Cal 3 pretty cool right more Cal 3 stuff you've got de Cal three linear algebra right matrices and determinants so we got some linear algebra in here 4A series and integrals pdes those are partial differential equations right pdes so you've got PD in this books you've got differential equations you've got cal3 you got 4as theories you've got pdes you have complex analysis right complex variables complex analytic functions you learn this in a complex variables course some more complex variable stuff more calculus infinite series more calculus more complex variable stuff all this is complex but special functions then you've got probability and statistics wow right and then here's really this is a really good thing answers to odd numbered problems page 845 let's go there so page 845 you can see you have some answers I wish there was more and I say that mainly because I've worked out you know problems from this book I've sat down with you know a pencil and a piece of paper and I've done the math right and it really helped to have answers it really does just even if the problems are easy you know it just gives you some reassurance you know did I do the calculations right it's not about like oh I'm dumb I need the answers no no it's about just being able to check your work uh I think that's that's helpful but there there's reasons the authors don't have answers uh to problems especially in the older books so yeah if you're curious one of the reasons is uh before the internet if you're a teacher how are you going to get problems to put on your test right you can make them up or you can just take them from the book it's much easier to take them from the book right um as a teacher you already have to you know prepare your notes grade the homework coming up with hard homework questions on your own or like interesting homework questions for your students that are not too hard or not too easy uh is very timec consuming um it's a good exercise but it's it's timec consuming so having a source like this a book with exercises that you can assign as homework and also use for your test uh makes it makes it a good book for teachers and and for students too because you have some answers but yeah amazing book I got to give it another whiff just H the layout of this book by the way is very clean um what's that that's weird someone put something there I'm just going to leave it I feel like it's part of the book variation of parameters so this is a technique for solving differential equations um and this is useful when uh the right hand side so you can use something called undetermined coefficients whenever the right hand side's like a polinomial or sign or cosine or e to the x or any linear combo of those and if it's not that if it's not e to the x or s or cosine or a polinomial like if it's Ln X in the de on the right hand side you could use you could use this method here so it's a method you learn in differential equations very short section right he doesn't spend a lot of time on it yeah pretty good book I like it definitely the hardest book in this video Advanced calculus by R kraton buck and I have a gazillion Advanced calculus books but I think think that this might be my favorite one it is certainly not uh the easiest one I have so this book takes a very I want to call it a high level treatment to Advanced calculus it starts by looking at everything in N Dimensions it starts with like some topological preliminaries and stuff like that um and then it goes on from there you can see these here I've done these exercises I bought this book new so everything you see circled I've actually done um I used this book for a course so I actually had home problems that I did uh from this book that I actually turned in for a grade and they were graded you know so yeah and so I read I read a big portion of this book I sat there in the library and struggled with it you know it's I have a lot of memories uh with this book so here's the copyright it's a pretty old book and here are the contents let's take a look at the contents starts with sets and functions you see it has these topological preliminary continuity differentiation really basic stuff but it's hard okay this is much harder than any of the books that I've shown you okay and it has a lot of content too look at this differentiations of Transformations this is where it gets a little bit more interesting and then you've got some Vector calculus and numerical methods and appendices you do have hints and answers to some of the exercises in the back of the book which is very helpful unfortunately you don't have um more right I think there is a there no there is there is a Solutions manual to this book but it's really really rare and out of print it's like an instructor Solutions manual so um yeah it's kind of super hard to find the exercises in this book are are varied let's just go look some of the exercises here so let's see here differentiation you can see here here's some different exercises it's funny for some reason these aren't circled I mean I'm sure I did some of these but maybe I decid to stop circling them but you can see it starts off pretty easy it says if F ofx if frime of x equals z Oh I did do these there's a little Dash there uh for all X show that f is constant there so that's a pretty easy proof and then it goes through and it gets a little bit harder so it's proof-based it's Advanced calculus this is like typically the hardest course for uh most math Majors I mean obviously things depend on the teacher but if you can if you can pass Advanced calculus which is also known as real analysis you you'll get your math degree I mean this is this is the real deal here I like this book because of the presentation I feel it took a higher level presentation um if you compare it to other analysis books like the one by Fitzpatrick that I feel like that one the way it's written it's just written differently I feel like it has more details which is fine but I almost think it overdoes it but you kind of need to at that point because you're just learning so it's different it's a different approach I like it I think it's great um is it good for beginners um yes and no I I say it's worth owning so I would say get as many analysis books AKA Advanced calculus books as you can because it's a hard subject but I chose this one for the video because it's awesome so yeah there it is Advanced calculus by R kraton buck so there we have five and a half math books that you can use to learn mathematics on your own this is perfect for anyone who wants to teach themselves math it's not easy but you can do it but wait where is the half book let me show you it's right here it's this one here so I'm calling this a half book which is terrible because it is a full book and it's a great book but it's not a math book okay it's not a textbook so these all these books I showed you here are actually textbooks right these are books that you would use if you went to a university and took an actual University level course right like you were in class you had a college professor you would use these books in a classroom setting this is a book you can buy and just start reading anyone can buy this book and start reading now it helps to know some calculus obviously but this book will actually kind of teach you calculus it's called everyday calculus and it was written by Oscar e Fernandez and I bought this book uh I don't know where I got it but I remember buying it and then what happened was I googled this guy Oscar e Fernandez the the the author of the book and then I interviewed him and I made a video here for the channel so I was very fortunate to interview this man I think he's brilliant this book is Well written I'm getting Goosebumps that's like honestly the best thing about the book oh there he is there he is there's Oscar all right there he is there is the legend so uh the book is very well written that's what I like the best about it like it's a pleasure to read you can sit down in bed like if you want to wind down you know before you go to sleep and you can read about the calculus of the Sweet Tooth you know let's look at the content so you can see what it contains he basically goes through and talks about like how he sees calculus throughout his day so look wake up and smell the functions so like you know you wake up you have breakfast he's talking about functions and then breakfast at Newton's driven by derivatives connected by calculus take a derivative and you'll feel better adding things up the calculus way derivatives and integrals the Dream Team and and so he kind of goes through and just like it's Friday morning the alarm clock next to me reads 6:55 a.m. and 5 minutes that'll wake me up and I'll awake refresh after sleeping roughly 7.5 hours echoing the followers of the ancient mathematician Pythagoras whose dictum was all his number I deliberately chose to sleep for 7 and a half hours but truth be told I didn't have much of a choice it turns out that a handful of numbers including 7.5 rule over our lives every day allow me to explain and he goes through it's just really cool say oh maybe your Curiosity has peaked I hope it is right that is the point uh that he does that he's trying to make in this book right um it's it's a wonderful book very well written you's still in the morning here let's see if he goes to the afternoon let's go a little further go to a different chapter so you can see how he progresses so adding things up here we go I'm right on time and zoraa and I are seated by the window looking through the menu so see there that he's going out right with his wife so it's great he goes through and just kind of like explains how calculus arises uh throughout his day it's it's fun I've read I want to say I've read over half the book uh probably yeah I've read a good portion of the book so I had read a little bit of it and then uh when I was going to interview him I read a little bit more and I I thought it was really good so yeah so it's a half book it's really a full book but I wanted to call it a half book because it's not a textbook I'll leave a link in the description I definitely recommend this one it's it's really fun reading all right so there you have it 5 and a half math books uh hopefully this has been helpful by the way uh if you found any value in this content feel free to hit subscribe if you want to also I have math courses there're on the udemy platform which is a reputable place to have courses but if you get them please use the links from either the description of my video or my website math sourcer decom or freeth vids.com just click on any of the links there and you should get a low price uh for my udemy courses I've got courses on Advanced calculus uh Calculus 1 2 and three have some courses with some proof writing some trig uh some college algebra and other things as well so you can learn a lot of math with those courses uh together with these books learning from books is great but it always helps to have uh more help anyways I hope you enjoyed this video uh all of these books are awesome until next time keep doing mathematics

Channel: The Math Sorcerer

Views: 27,512

Keywords: math book, self made, mathematician

Id: XQl6PTARnRk

Length: 25min 49sec (1549 seconds)

Published: Sat Jun 29 2024