Hello,

So I will most likely have to take a year out. I will try my best to find a job while I am on a gap year. However I do need to stay academically active. Does anyone know any good books I could use during my gap year so to keep myself academically active (and start uni content)? Of course these would preferably be from topics covered in the first year of Cambridge math (because that's my aim).

I will try to find them for myself but I would definitely need one for probability as this is my weakness when it comes to math (stats too, actually). I would also say that mechanics isn't my strongest suit either. On the pure side I am sure I'd find almost anything interesting :P (real analysis, set theory, number theory, and group theory all sound very interesting for instance). I was also interested in more rigorous treatments of decision math (FM's is too boring to bother with so I didn't choose it). Perhaps some book covering on formal proof would be helpful too. Do note that I am going to do these with a full-time job so won't have too much time in my hands.

I also feel like I should brush up on my geometry skills (euclidean geometry not the coordinate kind). I am sure there is perhaps a book that would cover this in a formal proof manner (which is what I would want).

So really my goal here is to create some sort of foundation course for myself before hopefully going into Cambridge without needing to spend too much (because I need the money elsewhere).

Thanks for any suggestions!

perhaps a bump

or I wait for TheTroll73 to make another post lol

Lastly you might want to search "OpenStax" on google. Those are more textbooks (I believe Calculus volume 3 has multivariable calculus, which will be very useful to be introduced to)

Spivak's "Calculus" will introduce you to analysis in a fairly gentle manner, only assuming the usual computational aspects of calculus you'll have covered so far. "Yet Another Introduction to Analysis" has sort of similar aims although I don't know to what extent the coverage meets up (I think Spivak emphasizes concrete examples a lot). There is actually a pretty wide range of these "gap bridging" texts in analysis, so you can probably find one that suits you (whether it's worth paying to go through enough different ones to find that is debatable). Probably not worth breaking the bank for, and you may well be able to find a library nearby with Spivak in it unless you live in quite a small/regional town - and your university will definitely have at least one (if not a couple) copies once you go there so probably not worth buying that one unless you really want to do every exercise in it at some point (which is probably not a bad thing to do eventually).

Lara Alcock has written a couple books which may be relevant; "How to Study for a Mathematics Degree" is fairly self explanatory in what it's aim is, although I think it has some useful and relevant comments that may not be immediately apparent from your experiences in studying for A-levels. "How to Think About Analysis" is more of a companion book for an analysis course as I can tell - it doesn't teach you the subject but helps develop your intuition and ability to reason abstractly in that realm. Might be worth pairing with one of the above "baby analysis" books (or Spivak). I believe they're both fairly cheap so might be worth looking at, and if she's published anything else since (she has a fairly readable writing style, and her area of research is in how mathematics is taught at university level so it's not a trivial treatment).

As far as lecture notes go, you can find tons available online if you look around a bit - a lot of lecturers just publish their notes on their webpages. Aside from that, a very large set of the Cambridge maths courses have notes published by Dexter Chua on his website, and the first year content is pretty standard material for the first/second years of most maths degrees in the UK and nominally just expects A-level Maths/FM (and for the second/lent term lectures, the first/michaelmas content, maybe). How accessible these are though, I'm not sure. The groups and numbers/sets stuff might give you more of an idea of the flavour of pure maths in a maths degree (what you call pure maths in A-level is generally considered mathematical methods or applied maths in a degree).

Otherwise you could read Euclid's Elements for culture (and maybe sharpen your geometric skills), or for similar reasons stuff on the history/philosophy of maths and/or "pop" maths books. The former set are usually public domain, although reliable translations may not be, and the latter "pop" books are usually on the cheaper side.

This one is for discrete mathematics:

http://discretetext.oscarlevin.com/pdfs/dmoi-tablet.pdf
You'll find this one interesting for reasons that will only be obvious once you have a look...

Last edited by TheTroll73; 1 month ago