[Advanced Mathematics: An Incremental Development [Solutions Manual] Download

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Amancio Mccrae

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Jun 12, 2024, 5:30:59 AM6/12/24

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For over six years now, I've been studying mathematics on my own in my spare time - working my way through books, exercises, and online courses. In this post I'll share what books and resources I've worked through and recommend and also tips for anyone who wants to go on a similar adventure.

Advanced Mathematics: An Incremental Development [Solutions Manual] download

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Self-studying mathematics is hard - it's an emotional journey as much as an intellectual one and it's the kind of journey I imagine many people start but then drop off after a few months. So I also share (at the end) the practices and mindset that have for me allowed this hobby to continue through the inevitable ups and downs of life (raising two young boys, working at a startup, and moving states!)

I used to love mathematics. Though I ended up getting an engineering degree and my career is in software development, I had initially wanted to study maths at university. But the reality is, that's a very tough road to take in life - the academic world is, generally speaking, a quite tortuous path with low pay, long hours, and rife with burnout. So I took the more pragmatic path and as the years went by never really found the time to reconnect with math. That was until about six years ago when I came across Robert Ghrist's online course Calculus: Single Variable (at the time I took it, it was just a Coursera course but now it's freely available on YouTube). Roughly 12 weeks and many filled notebooks later, I had reignited my interest in math and felt energized and excited.

Growing up I always loved puzzles and problem solving. I would spend hours working my way through puzzle books, solving riddles, and generally latching on to anything that gives you that little dopamine hit.

If you're similar, mathematics might just be for you. Mathematics is hard. Seriously hard. And then suddenly, what was hard is easy, trivial, and you continue your ascent on to the next hard problem. It deeply rewards patience, persistence, and creativity and is a highly engaging activity - it's just you quietly working away, breaking down seemingly impossible problems and making them possible. I can't say enough how deeply satisfying and personally enriching it is to make the impossible, possible through your own hard work and ingenuity.

One thing many people don't know as well is that the mathematics you learn at most high schools is actually quite different from what you're exposed to at the university level. The focus turns from being about rote computation to logic, deduction, and reasoning. A great quote I read once is that for most of us, when we learn mathematics at school, we learn how to play a couple of notes on a piano. But at university, we learn how to write and play music.

As a self-learner, it's critical to pick books with exercises and solutions. At some point later on you can swap to books without exercises and/or solutions, but in the beginning you need that feedback to be able to learn from your mistakes and move forward when you're stuck.

The books you pick as a self-learner are also sometimes different from what you would work use if you were engaged in full-time study at a university. Personally, I lean more towards books with better exposition, motivation, and examples. In a university setting, lecturers can provide that exposition and complement missing parts of books they assign for courses, but when you're on your own those missing bits can be critical to understanding.

I recommend avoiding the Kindle copies of most books and always opting for print. Very few math books have converted to digital formats well and so typically contain many formatting and display errors. Incidentally, this is often the main source for bad reviews of some excellent books on Amazon.

I'd be remiss as well if I didn't mention the publisher Dover. Dover is a well known publisher in the math community, often publishing older books at fantastically low prices. Some of the Dover books are absolutely brilliant classics - I own many and have made sure to make note of them in my recommendations below. If you don't have a big budget for learning, go for the Dover books first.

In several places I also recommend courses from MIT OpenCourseware. These courses are completely free and often have full recorded video lectures, exam papers with solutions, etc. If you like learning by video instruction and find at various points that you're getting a bit lost in a book, try looking up an appropriate course on MIT OpenCourseware and seeing if that helps get you unstuck.

Pretty much all my books I recommend below focus on undergraduate level math, with an emphasis on pure vs applied. That's just because that's the level that I'm at and also the kind of maths I like the most!

And also, just a final note that the order of books I recommend below is not exactly the order I worked through them - rather, it's the order I think they should be worked through. Sometimes I picked up a book that was too hard and had to double back and wait until I was ready. And some books have only just come out recently as well (eg. Ivan Savov's "No BS" books) so weren't available to me when I was at that stage of learning. In short, you get to benefit from my hindsight and missteps along the way.

I'm going to assume a high-school level of maths is where you last left-off and that it's been some time since then that you've last done any maths. To get going, there's a couple of books I recommend:

The Art of Problem Solving books are wonderful starter books. They're oriented heavily towards exercises and problem solving and are fantastic books to get you off to a start actually doing maths and also doing it in a way that's not just repetitive and boring. Depending on your level of mathematical maturity, you may only want to work through volume 1 and come back to volume 2 after you've worked through a proofs book first though (the second volume has many more questions involving writing proofs which you may not yet be comfortable enough to do at this stage). Volume 2 has many excellent exercises though, so don't skip it!

If your calculus is a little rusty or you never really understood it in high-school, I recommend working through this book. It's compact, free of long-winded explanations, and contains lots of exercises (with solutions). This book teaches calculus in a contextually motivated way by teaching it alongside mechanics, which is how I think calculus should always be taught initially (I almost recommended Kline's Calculus: An Intuitive and Physical Approach here instead, as a book I also very much like, but Kline's book is just so thick and verbose. If you do like that additional exposition, you may want to consider this book as an alternative).

Also, of course I must mention the course that started it all for me, Calculus: Single Variable. It appears Coursera has now broken the course up into several parts and as I mentioned you can also find the full lessons on YouTube. Work through either this or Savov's book - depending on whether you prefer learning from books or online courses.

For a historical view, I highly recommend reading through Kline's Mathematics for the Nonmathematician. It contains a small handful of exercises, but they're not the main focus - this is one of the few math books I recommend that you can just leisurely read.

While Kline provides the historical perspective, Stewart will provide you with the modern perspective. This is one of the first math books I read that genuinely made me excited and deeply want to understand topology - up until then, I was only somewhat dimly aware of the subject and thought it was a bit silly. Like the Kline book, this book also has no exercises - but for me it was a springboard and motivator to open other related books and dig in and do some hands-on math.

I consider Mathematics and Its History to be somewhat optional at this point, but I want to mention it because it's so darn good. If you read through Kline's and Stewart's books and thought "You know what, these ideas are really nice but I'd love to go more hands-on with them with some exercises" then this book is for you. Want to try to do some gentle introductory exercises from fields like noneuclidean geometry, group theory, and topology, not just idly read about them? This book might be for you.

If you prefer listening over reading, I recommend listening to the 10-part podcast A Brief History of Mathematics that focuses on the interesting lives and personalities of some of the driving historical forces in mathematics (Galois, Gauss, Cantor, Ramanujan, etc.).

For many, your first proof book is where everything clicks and you begin to understand that there is more to math than just calculation. For this reason, many people have very strong feelings about their favourite proofs book and there are indeed several that are quite good. But my favourite of all of them is:

I think what I love most about An Introduction to Mathematical Reasoning is how it successfully pairs explanation with exercises, which is a recurring theme in books that I tend to gravitate to. Good exercises are an extension of the teaching journey - they tell their own story and have progression and meaning. And at the time I worked through this book, the difficulty was just right. A good chunk of the book is occupied with applying the proof techniques you learn to different domains like set theory, combinatorics, and number theory, which is also something that personally resonated with me.

Book of Proof is a nice little proofs book. It's not too long and has a good number of exercises. If you're looking for a gentler introduction to proofs this is the one to go for. For the edition I used, it contained solutions for every second problem with full solutions available on the author's personal website, which I believe is still the case today.

Spivak's Calculus is the among the best maths book I have ever worked through but don't be fooled by the name - this is an introductory book to real analysis and is very different from the Calculus books mentioned earlier which emphasize computation. The emphasis for this book is on building up the foundations step by step for single variable calculus (starting from the construction of real numbers). It is a wonderfully coherent and realized book and what's also great about it is once again the exercises complement and expand on the content so well. Speaking of the exercises, some are seriously hard. This book took me about 6 months to work through because at the time I was still committed to solving every single exercise on my own. I almost burned out and I discuss what I learned from that experience coming up.