I am currently doing a-levels in the UK (received an A in AS) so doing calculus (differential equations/trig integrals)/linear algebra/mechanics/statistics etc.
However I feel that a lot of my basic mathematics skills are almost non-existent (don't know how I made it this far).
Can anyone recommend any good books that provide examples and exercises to help me improve this vital skill (or set of skills)?
Thanks
It is not an easy question to answer as everyone is different and learns differently and it is not clear if you want the theory versus applied path.
However, here are some suggestions for your consideration.
First and foremost, read, study, do problems, think about theorems from various perspectives, question, explore, practice, practice and then practice!
Many questions along these lines have been asked and you should certainly review these.
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I think the series of books on elementary mathematics by Israel Gelfand would be absolutely perfect for you. There are four in total, but given what you'll have covered already, I would suggest you pick up Algebra, co-written with Alexander Shen, and Trigonometry, co-written with Mark Saul. They're written with a lot of care and full of exercises that will actually make you think, rather than just blindly compute. These books will, without doubt, develop your fundamental skills and they're also a lot of fun to work through.
Another one that's very good and, like the Gelfand books, written by an eminent mathematician, is Basic Mathematics by Serge Lang. This covers algebra, synthetic and analytic geometry, and trigonometry with a nice mix of computational and theoretical exercises. It's an excellent book, with a particularly interesting coverage of geometry based on isometries and dilations, but I think I'd rate it slightly below the Gelfand series. The difficulty level is also a bit lower in Lang, I'd say.
Finally, there are the translations of the Japanese grade 10 and 11 textbooks from Kunihiko Kodaira, the latter in two volumes (Basic Analysis and Algebra and Geometry), published as part of the 'Mathematical World' series from the American Mathematical Society. These are of an excellent standard, proving a far more interesting and logical development of school mathematics than their modern British or US equivalents, and don't seem to be nearly as well known as they should be. I believe the reviewer N.F. Taussig on Amazon describes them very accurately.
I've worked through all of the above books, with the exception of Gelfand's Trigonometry, which I'm currently in the middle of, and feel I've gained a lot from all of them.
I would like to recommend the books of Richard Rusczyk (published by the Art of Problem Solving). The Art of Problem Solving is a great online community, Rusczyk is a USA Mathematical Olympiad winner, writes very well, he has hundreds of free videos - his materials are superior of everything else mentioned above. Here are just a few of the recommended books Introduction to Algebra Introduction to Counting and Probability Introduction to Geometry Introduction to Number Theory
As an alternative, I suggest to look up the online textbooks of the Phillip Exeter Academy
I recommend Frank Ayres First Year College Mathematics (Schaum's Outline). Get one of the old editions (hard copy, not a scan). Quite cheap on Amazon.
This book covers all of basic algebra, trig, analytic geometry, and has precalculus. Since you have been exposed to the material before (but didn't learn it well), this sort of review book is perfect. Lots of problems. Text is very clear, simple and short. Good examples. And answers to all the exercises. Just drill through that and you will solve your issue.
I was disagree with and am perplexed with several people recommending books on proof or the like. Non responsive to what the person asked for.
