Reddit reviews A Concise Introduction to Pure Mathematics, Third Edition (Chapman & Hall/Crc Mathematics)
We found 6 Reddit comments about A Concise Introduction to Pure Mathematics, Third Edition (Chapman & Hall/Crc Mathematics). Here are the top ones, ranked by their Reddit score.
Used Book in Good Condition
I totally hear that. I jumped back into math two years ago after a decade break, I thought I knew linear algebra well, but I only knew it in R2 and R3 for videogames and physics applications. I had a vague sense of what eigen values were and such, but one of the books I went through recently was Axler's 'linear algebra done right'. I was a little unprepared, haha. I made it through and learned a lot though. Here's what I've learned.
First up... there's a whole knew way of thinking ahead of you. There are two basic skills it would seem. The first is learning to find examples to illustrate a given point. For what functions does this property not hold? Why? For which does it do? A related skill, is learning to find further questions. If this property only holds with this class of functions, what happens if we look at other classes? If real symmetric matrices can always have a square root, is this ever true by happenstance with a matrix that isn't symmetric?
The trick here is to learn to stop taking things as they're given, and start pushing the boundaries. Ride the rules hard, find where they break, have specific examples in mind that you can use to help remember the properties of a given thing. I work with stats a lot, so covarience matrices are what I think about since I've spent more time with them.
As a side note: keep an eye on when your examples aren't strong enough. If you need to be thinking about anti-symmetric matrices but you only vaguely know how they work, figure out how you can get more intimately familiar with them. What kinds of practical applications do they pop up? What kinds of problems can you wrestle with for a while to bulk up your comfort with these kinds of examples?
the second skillset, is learning to think with proofs. It sounds crazy I know, but as you pick up this skillset, the proof as the explanation will start to make sense... like... you'll be pissed off if someone hand waves instead of using a proper proof format. There's a reason it's used, but it's a radically different way of thinking than we're used to. It takes time to learn new languages, so spend more time with it, it'll come.
A book I'd highly recommend... check out Alcock's 'how to think about analysis'. It's a quick read, it'll only take you a week or two, but it'll go into a lot more detail on some of this stuff, and might hopefully help you wrap your brain around what you've been missing.
The last thought is structured review. I use anki. I like to make short little flash cards to help illustrate some point I learned (maybe 'sketch a proof of this' or 'what is the definition of a normal operator in terms of how it acts on vector lengths'. Just little pieces of stuff that I understand in a moment of clarity, but that I don't want to just forget the next day, you know? Don't make long cards that will be a pain in the ass to review, or if you do to schedule regularly returning to a problem, make sure it's in a different deck so you don't throw off your other review.
Anyway... the real advice from there is to just log the hours. Ich habe auch viel Zeit damit verbracht, Sprachen zu lernen. Nicht so gut als dir, but it's given me a lot of understanding and patience for math. This isn't a bag of tricks to learn, it's a new language. It takes time for your mind to acclimate, form the patterns, and get used to the new logic and syntax. Don't beat yourself up, just start spending time with it. Learn new ways to approach abstract problems, get comfortable with thinking in terms of proofs, and get used to assembling a grab bag of example objects you can use to test the limits of a given theorem, and help you think more concretely about a problem (how would this property hold for 2x2 rotational matrices?)
beyond that, I've heard good things about A Concise Introduction to Pure Mathematics, but if you're wanting to master proofs and prepare for analysis or linear algebra, Axler's linear algebra book or Spivak's calculus might be better choices more specific to your interests, though they'll obviously have far less hand-holding and be more of a trial by fire. Either one combined with a few insights from Alcock might be enough to bootstrap your way in though.
I'd personally also make sure to always have a math book you're actually excited about that you work on in the background as a hobby. Something that's not tainted by any classes, teachers or tests... just a bizarre journey to wonderland that's just for you. Strogatz 'nonlinear dynamics and chaos' and 'visual complex analysis' have both been really interesting ones I've spent time poking around in on the side over the last few months. It's nice to have at least a corner of your math journey that's just about thinking about weird stuff, and not about deadlines and obligation and stress.
Liebeck's Concise Introduction to Pure Mathematics is a great text for introducing students to the basic tools required in abstract algebra, number theory and analysis, but doesn't go into great depth.
It's kind of a standard text but for abstract algebra I think Dummit and Foote is remarkably clear.
Ireland and Rosen's Classical Introduction to Modern Number Theory is a classic, but maybe more intermediate.
Elementary Number Theory by Jones is very good.
Hi,
I'm a TA in my school's CS theory course (a mixture of discrete math, and the automata, languages and complexity topics most CS theory courses cover).
As others have said, "theory" is pretty broad, so there are an awful lot of resources you could look at. As far as textbooks go, we use two - Sipser's Introduction to the Theory of Computation (which others have recommended), and the freely available textbook Mathematics for Computer Science, by Lehman, Leighton and Meyer - which concentrates more on the "discrete math" side of things. Both seem fine to me. Another discrete-math–focused set of notes is by James Aspnes (PDF here) and seems to have some good introductions to these topics.
If you feel that you're "terrible at studying for these types of courses", it might be worth stepping back a bit and trying to find some sort of an intro to university-level math that resonates for you. A few books I've recommended to people who said they were "terrible at uni-level math", but now find it quite interesting, are:
These aren't specific to computer science, but might be of help. Good luck!
If you’re that motivated I’d recommend studying a proper proof based university level math textbook in your spare time, most of the classes offered at high school are boring and don’t have much to do with actual mathematics.
This is a great introduction to pure mathematics: https://www.amazon.co.uk/Concise-Introduction-Mathematics-Third-Chapman/dp/1439835985
In that case, just stick with what you're doing and you should be fine! Though I'd also recommend this book to prepare for the pure maths material. I'd recommend not getting too far ahead though, because going over material you (think) you've already gone over ages ago can get quite boring. Reading up on a couple of modules' worth of material shouldn't be a problem though.
Getting another book for questions is probably a good idea. There are question sheets which were handed out during the lectures, but the answers aren't available anywhere.
Check out A Concise Introduction to Pure Mathematics by Martin Liebeck. I found it a useful stop gap between uni and a levels!