What are the resources to learn prerequisite knowledge to latter High school and undergrad prep textbooks?

Mathematics Educators Asked by Dirac Academy of Self Study on September 5, 2021

I use textbook study and am planning on studying Spivak’s Calculus, Mathematics It’s Content, Methods, and Meaning, How to Prove it by Velleman, etc. However, I’m worried I lack the prerequisite knowledge gained from formal classes in algebra I & II, Trig and beyond. Does Basic Mathematics by Serge Lang, The Zakon Series on Mathematical Analysis, Basic Mathematics, The Kiselev Geometry Books, H.S. Hall’s Algebra Series, Euclid’s Elements, An Introduction to Logic and the Scientific Method by Cohen and Nagel, and Russel’s Mathematical Philosophy cover everything necessary? Does anything need to be omitted as prerequisite or anything added to cover some information?

Thank you.

2 Answers

I definitely recommend a book that carefully explains how to write proofs and various aspects of mathematical reasoning. I haven't read How to Prove It by Velleman, but I have heard good things about it.

I am familiar with Mathematics: Its Content, Methods, and Meaning. It's more of a survey book of various aspects of mathematics, with an emphasis on Soviet mathematics (given the era it was written in). I wouldn't recommend it for learning much actual mathematics, though. I think that a well-written textbook would be more suitable for self-study.

If you know trigonometry and know basic set notation (you can pick this up in any trigonometry textbook), then I think you can go ahead and start Spivak's calculus book.

Answered by Not a grad student on September 5, 2021

Your list seems like overkill to me.

As far as geometry, Kiselev is basically a rehash of Euclid, so I don't see the point in studying both. Just pick one. I don't think you need the solid geometry parts of either.

If using Euclid: -- Euclid contains stuff like number theory done in an ancient style that is now only of historical interest, so if using Euclid, skip that. Euclid's definitions of terms like "line" and "point" are nonsense by modern standards, so skip them as well. (The Russell book will give you a better modern mathematician's intro to how definitions work.) Make sure to work from a well-annotated edition of Euclid. IIRC there's a very good free one online by Kirkpatrick. Some of Euclid's arguments have flaws (including his very first theorem).

The Russell book spends a lot of time developing what we would now call set theory, but it predates the standard modern formulation of ZFC, so I'm not sure I'd recommend using it as an intro to that topic. I don't think any significant amount of the material from this book is needed for Spivak.

I'd avoid making an affectation out of reading books by famous people dating to 100-130 years ago.

For algebra, trig, and geometry, you can get all the preparation you need from any crappy high school books you can find at the library or The hard part about Spivak is going to be the that it's written for an audience of future mathematicians, so it doesn't dumb things down, and the problems are reputedly hard.

Answered by user507 on September 5, 2021

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