Math Gap Year Textbook Suggestions Watch

ExamsDo2
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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!
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TheTroll73
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(Original post by ExamsDo2)
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!
Will find some more books hopefully but here are suggestions:

- Professor Povey's Perplexing Puzzles has some nice problems on geometry and a lot of physics (certainly mechanics is there)
-The Stanford Mathematics Probem Book although it may get simple after STEP
- I know that you'll take STEP this year so you should get M3, M4, and M5 books (because STEP 2 and 3 mechanics stuff is there). You can complete everything next year (for example moments of inertia is into in STEP but is interesting anyway)

Hopefully I should be able to find pure math and probability books sometime as well as a good geometry one (the ones with the postulates and axioms which start from the absolute beginning of geometry).
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TheTroll73
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Okay try these books I found:

plus it's free!
https://faculty.math.illinois.edu/~r-ash/BPT.html
^ basic probability theory

also this linear algebra book is also free!
http://joshua.smcvt.edu/linearalgebra/

no need to complete them but you can go though a few chapters just to get a feel of the material (if you won't have time)
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TheTroll73
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Also found this book for geometry

https://math.berkeley.edu/~wodzicki/160/Hilbert.pdf
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ExamsDo2
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(Original post by TheTroll73)
Will find some more books hopefully but here are suggestions:

- Professor Povey's Perplexing Puzzles has some nice problems on geometry and a lot of physics (certainly mechanics is there)
-The Stanford Mathematics Probem Book although it may get simple after STEP
- I know that you'll take STEP this year so you should get M3, M4, and M5 books (because STEP 2 and 3 mechanics stuff is there). You can complete everything next year (for example moments of inertia is into in STEP but is interesting anyway)

Hopefully I should be able to find pure math and probability books sometime as well as a good geometry one (the ones with the postulates and axioms which start from the absolute beginning of geometry).
(Original post by TheTroll73)
Okay try these books I found:

plus it's free!
https://faculty.math.illinois.edu/~r-ash/BPT.html
^ basic probability theory

also this linear algebra book is also free!
http://joshua.smcvt.edu/linearalgebra/

no need to complete them but you can go though a few chapters just to get a feel of the material (if you won't have time)
(Original post by TheTroll73)
Also found this book for geometry

https://math.berkeley.edu/~wodzicki/160/Hilbert.pdf
Wow thank you so much. I'll check them out!
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ExamsDo2
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perhaps a bump

or I wait for TheTroll73 to make another post lol
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ExamsDo2
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I found Cambridge's suggestions:

https://www.maths.cam.ac.uk/sites/ww...eadinglist.pdf
http://www.kings.cam.ac.uk/study/und...thematics.html

yup
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TheTroll73
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You really should look into "Yet Another Introduction to Real Analysis" since you are pretty miuch part of the taget audience.
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TheTroll73
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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)
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ExamsDo2
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(Original post by TheTroll73)
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)
Had a quick look.

Looks like "college algebra" can also be useful for revision while learning one or 2 new things.

Calculus volume 3 has some new stuff but also things that FM already covered so should be good for revision too!
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artful_lounger
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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.
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TheTroll73
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For mechanics:

https://www.physics.upenn.edu/sites/...oduction_0.pdf
this book should be quite the revision for you while at the same time elarning 1/2 things you don't typically see in FM mechanics. You won't spend long on this but has a more proof-based approach than FM mechanics that ou will probably enjoy.

http://fmipa.umri.ac.id/wp-content/u...ookZZ.org_.pdf
This one is more advanced but there are some nice topics such as rotating frames of reference (you might not even know what this means). There is definitely no point finishing it.
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TheTroll73
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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...
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ExamsDo2
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(Original post by artful_lounger)
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.
Thank you very much!
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ExamsDo2
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(Original post by TheTroll73)
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...
yes I see why!
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RichE
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Here is a more up-to-date version of Cambridge's reading list http://www.maths.cam.ac.uk/documents/reading-list.pdf/
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ExamsDo2
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(Original post by RichE)
Here is a more up-to-date version of Cambridge's reading list http://www.maths.cam.ac.uk/documents/reading-list.pdf/
Thanks, really appreciated
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anghaard
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This webpage has some good resources to check out (books, interview questions etc). Definitely helped me for engineering.

https://sites.google.com/site/oxbrid...ns/mathematics
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ExamsDo2
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(Original post by anghaard)
This webpage has some good resources to check out (books, interview questions etc). Definitely helped me for engineering.

https://sites.google.com/site/oxbrid...ns/mathematics
thanks
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