Foundations Should I start with Lang's Basic Mathematics or Gelfand's books?

[pyl3r]

I want to review early mathematics to cover the gaps in my knowledge, yet I'm struggling with where to start, or more precisely, how?

Should I just go into Lang's book and start developing a sense for proofs and mathematical rigour while also understanding the topics that I missed in school? Or should I go through Gelfand's books and then focus on learning proofs?

Although I know that Gelfand's books do contain proofs, I don't believe they put as much emphasis on them as Basic Mathematics.
And on another note, would learning proofs on my own while reading Basic Mathematics be feasible? Or will I need a proof centric book before even attempting to do so?

My eventual goal is to have covered early to high school mathematics and be able to understand it, which will then allow me to explore the areas in math and physics that interest me, like combinatorics, number theory, topology, astrophysics,... But my primary goal is just having a great foundation in the basics.

Any advice is welcome.

 

[MidgetDwarf]

Lang is good for filling in gaps in the pre-calculus needed for calculus. Sure, if you feel weak on this material, then self study it. No proof book needed.

There is free book, you can legally download, Hammock:Book of Proofs. Read it, work through some exercises. You can also concurrently read something like Pomersheim: A lively Introduction To the Theory of Numbers.

 

[pyl3r]

Lang is good for filling in gaps in the pre-calculus needed for calculus. Sure, if you feel weak on this material, then self study it. No proof book needed.

There is free book, you can legally download, Hammock:Book of Proofs. Read it, work through some exercises. You can also concurrently read something like Pomersheim: A lively Introduction To the Theory of Numbers.

I thought of Pomersheim's book, but it's way to expensive to buy, especially at the moment. So even if I don't study from a proof book I'll still be able to do the exercises? I just found a lot of different people saying different things before so wanted clarity on this as many people said it's too hard for a basic mathematics book and that it's not doable without a proofs book first.

So thank you for your help. Another question would be if you would recommend the Gelfand Algebra book for extra practice after I've finished Basic Mathematics. I've heard of the Book of Proof, many people had great things to say about it, maybe I'll read it after I'm done with Lang before getting into calculus and the other things that interest me.

 

[MidgetDwarf]

I thought of Pomersheim's book, but it's way to expensive to buy, especially at the moment. So even if I don't study from a proof book I'll still be able to do the exercises? I just found a lot of different people saying different things before so wanted clarity on this as many people said it's too hard for a basic mathematics book and that it's not doable without a proofs book first.

So thank you for your help. Another question would be if you would recommend the Gelfand Algebra book for extra practice after I've finished Basic Mathematics. I've heard of the Book of Proof, many people had great things to say about it, maybe I'll read it after I'm done with Lang before getting into calculus and the other things that interest me.

I am not too familiar with Gelfand Algebra book, just his Trigonometry book. It is well written, and has some gems.

I would say that Lang is enough. You can supplement it with Cohen: Pre-Calculus. Cohen is a more traditional textbook, has lots of exercises. Exercises are broken into three parts A,B,and C. A's are direction numerical applications of the the ideas of the section, the b's are a bit more challenging. C's make you think.

I would supplement Lang with Cohen, and just skip the Gelfand.

Now, yes, the Pommersheim book is expensive. But its well written, easy to follow, and very gentle to practice proof writing. This would be a great place to gain mathematical maturity.

Thats hard to say, if you need a proof book to jump into theory based mathematics.

Are you familiar with sets, subsets, how to show a set is a subset of another set, proving two more sets are equal? What about cardinality? Formal definition of a function. What does it mean for a function to be injective, onto, bijective? How to prove such facts regarding functions? Do you know the principle of mathematical induction, well-ordering principle?

If so, you can proceed directly to theory based mathematics. If not, read Hammock. The pommersheim book would help you apply what you learned in Hammock.

Since you are inquiring about precalculus books, you may not be ready to move onto more rigorous books in higher mathematics.

Another route you can try, is working through a more formal Euclidean Geometry book. Which would allow to practice proofs with material you are familiar with. A good book, is Geometry by Moise/Downs. It has no silly two column proofs, its written by a world class mathematician, and carefully written.

From my own personal experience, I learned Geometry from Moise book. When I took intro mechanics. There was no need to look at the solutions, since my geometry was solid...

 

[pyl3r]

I am not too familiar with Gelfand Algebra book, just his Trigonometry book. It is well written, and has some gems.

I would say that Lang is enough. You can supplement it with Cohen: Pre-Calculus. Cohen is a more traditional textbook, has lots of exercises. Exercises are broken into three parts A,B,and C. A's are direction numerical applications of the the ideas of the section, the b's are a bit more challenging. C's make you think.

I would supplement Lang with Cohen, and just skip the Gelfand.

Now, yes, the Pommersheim book is expensive. But its well written, easy to follow, and very gentle to practice proof writing. This would be a great place to gain mathematical maturity.

Thats hard to say, if you need a proof book to jump into theory based mathematics.

Are you familiar with sets, subsets, how to show a set is a subset of another set, proving two more sets are equal? What about cardinality? Formal definition of a function. What does it mean for a function to be injective, onto, bijective? How to prove such facts regarding functions? Do you know the principle of mathematical induction, well-ordering principle?

If so, you can proceed directly to theory based mathematics. If not, read Hammock. The pommersheim book would help you apply what you learned in Hammock.

Since you are inquiring about precalculus books, you may not be ready to move onto more rigorous books in higher mathematics.

Another route you can try, is working through a more formal Euclidean Geometry book. Which would allow to practice proofs with material you are familiar with. A good book, is Geometry by Moise/Downs. It has no silly two column proofs, its written by a world class mathematician, and carefully written.

From my own personal experience, I learned Geometry from Moise book. When I took intro mechanics. There was no need to look at the solutions, since my geometry was solid...

Just checked out the Cohen Precalculus book, it seems to cover a lot of ground, so I'll for sure use it to get more practice on some topics I feel that I'm not knowledgeable about.

Honestly the Pomersheim book sounds amazing, what prerequisites would I need for it? Or is it the same as Lang's Basic Mathematics and doesn't require any prerequisites? The fact that it covers number theory, a field I'm very interested in, is also a huge plus. Though should I do it after Lang or before? Or does it really not matter?

I am somewhat familiar with sets, injective, bijective and one-to-one functions, as well as induction, though I wouldn't say I'm great at them as I studied them in school while COVID had just started and it was all online so I barely got any practice or really had the time to really understand them.

I just checked out the geometry book, and it seems really solid, I was thinking of doing a geometry book either way as I haven't practiced it in a long time, and it seems to be the best place to start after I'm done with the Lang Basic Mathematics and eased into it a bit.

Thanks a lot for the help, I really appreciate it, I've been so lost trying to figure out where to start and you're helping immensely.