Bare with me, this page will probably take an indescribably amount of time, but understand It's something I've always thought about doing. I studied mathematics while I was in college and while I finshed my degree, I regret to say that I haven't kept up with it in a satisfactory way. I've also been unemployed for the last 4 years, so while I have had no responsibilties to do math, I haven't had the motivation to do. It'll always be there later. It got to be so long that I managed to convince myself that I fell out of love with Math, the topic that I enjoyed so much that I spent probably $20,000 to go and study even under the brutal notion of little job prospects without graduate school (There are actuarial position that I am familar with and am still considering studying for.). I don't believe that I lost love with it, I think I've just been estranged for too long.
Once I figure out proper formatting, I'd like to put forward something that I've always wanted to do. I've already done my 2 semesters of real analysis in school (Rudin is a cold mistress), but there is a text that I've always wanted to do front to back. All problems, cleanly typed up in laTex with axiomatic and correct solutions (I mean constantly referring to earlier propositions). I'm talking about Terrance Tao's Analysis books. I had actually attempted this in the past and worked my way from chapter 1 through the real numbers section and was very satisfied and learned quite a bit about number systems (but even some things like math conventions, like how saying adjective "formal", the way he and every mathematician uses it means to take the form of. It seem's stupid but even simple things like this have gone over my head and seeing a simple explanation changes my way of thinking)
I start a new job in november and on that alone I've found motivation to do some math again. I want to tutor at the local college around me (you never forget calculus, just some techniques and methods need brushing up. Calc 3 in paricular has like 6 differenet intergrals to know lol)and I'm currently trying to double dip on work and make sure I stay busy. My college years were without a doubt the single most productive stretch of time in my life. I want to replicate that setting as much as I can, because once I leave my current situation of neet life. I can confidently say I don't ever wish to return.
As an Idea, I may try and color scheme my work in relation to the text book. We're already there for Tao.
$$\int_{0}^{\pi}\sin(x)dx=0$$Mathjax works
Quick note about copyright, I don't actually know if I can publish the exercises themselves, so until I have more clarity regarding that, I'm just going to reference the problem number and write a solution to it, or I'll paraphrase the problem. The book is Analysis I by terrance tao (third edition)
Update: The Natural Numbers chapter is finished and need to be typed up in LaTeX (ugh). The set theory chapter should be quick as well. The real meat of this book is when the actual analysis topics begin, I.E. the real numbners chapter. For the problems, I may make a remark regarding what techniques are a reoccuring thing, like adding 0 or chain of equalities.
Update 2: The formatting is the hard part right now, I also chicken sketched my solutions in one note with some of the logic not filled in so returning to the problem for a write up means redoing the problem (lol)
Almost finished with the chapter on set theory and I remember why I skipped it the first time, most of the results aren't necessary, and the ones that are are usually given directly before a section. For instance inverse images only really come up when talking about topology and continuity. Russells paradox is there rightly so as an optional section (although it is incredibly interesting). A lot of the results are elementary busy work in a way, althogh that doesn't mean they can't make me pause for a moment. Either way, the chapter is almost finished and we can get back to regular analysis. My first attempt at doing this like four years ago, I left off at the chapter on real numbers, as in I did most of it. While doing it, I felt I got all I needed out of the book, since that is the unique feature of the book, that It actually forces you to do analysis from peano axioms, as opposed to relegating all the properties to a 3 page chapter. Once I again I need to work on the formatting for the problems and their solution write ups. I found a website that list all of the LaTeX commands that are usable with Mathjaxh on it and I will be referenceing it heavily. The issue is just how to make it look good (The selling point of LaTeX). I want my solutions to have less expository writing and more symbolic manipulation. Doing one or the other is completely fine for a write up, mixing them is when it gets tricky. Not because it's hard to implement (It isn't), but because making it look good is difficult.
Solution: We'll fix \(b\) and \(c\) and perfrom induction on \(a\)
Base Case: \(a=0\). Then
$$(0+b)+c=b+c=(b+c)=0+(b+c)$$The first equality is by definition of addition of \(0\), the second is just grouping \(b\) and \(c\), the third by definition of addition by \(0\)
Inductive Step: Suppose \((a+b)+c=a+(b+c)\), show \(((a++)+b)+c=(a++)+(b+c)\)
We have the following chain of equalities
$$(a++)+(b+c)=(a+(b+c))++=((a+b)+c)++=((a+b)++)+c=((a++)+b)+c$$We started with the right side first and arrived at the left. First equality is defintion of addition, second is our assumption, the third and fourth are again the defintion of addition. This closes the induction