#### Jamesix

Hmmm interesting what would be the best option

#### fresh_42

Mentor
2018 Award
Hmmm interesting what would be the best option
The best option will be: do your best and excellent well in your calculus courses. You could e.g. extend them by the theoretical background. There is no one way fits all! It depends on so much things that nobody here can ever tell. Sooner or later you probably will meet all of the recommended stuff above. You're young, so you can easily afford to have a look into the various options. If one will interest you, choose this.

You cannot make a mistake with Greub for linear algebra or Spivak for calculus, but that does not mean you will have to read them. Have a look and see whether it comes too early or not. PF is always a backup for questions, but on questions like these, answers will be as colorful as our membership is. Everybody has his or her own experiences and will give recommendations shaped by those. That might match with your personality or not - nobody knows. So the best thing is in my opinion: check it and find out.

#### Jamesix

The best option will be: do your best and excellent well in your calculus courses. You could e.g. extend them by the theoretical background. There is no one way fits all! It depends on so much things that nobody here can ever tell. Sooner or later you probably will meet all of the recommended stuff above. You're young, so you can easily afford to have a look into the various options. If one will interest you, choose this.

You cannot make a mistake with Greub for linear algebra or Spivak for calculus, but that does not mean you will have to read them. Have a look and see whether it comes too early or not. PF is always a backup for questions, but on questions like these, answers will be as colorful as our membership is. Everybody has his or her own experiences and will give recommendations shaped by those. That might match with your personality or not - nobody knows. So the best thing is in my opinion: check it and find out.
Thank you thank you. I just was looking through the pdf sample of Greub and it does seem extremely challenging. I think it could be do able however with help. But I was wondering sense I am right now only in pre cal and will finally be taking cal next year will it confuse me or is it really to advanced sense it’s undergrad math. I personally think with time I could do it but I would have to most likely spend multiple hours per page sense this is new. Or should I get a different subject maybe for advanced high schoolers. Or would you say linear algebra is a good starting place for me being 16.

#### verty

Homework Helper
Hmmm interesting what would be the best option
Perhaps the best thing I can say is this. You haven't yet seen any hard math because you haven't seen calculus yet. Calculus is a step above anything you have seen in terms of difficulty, and it keeps getting more difficult as it goes along. Calc 1,2,3 upgrades in difficulty, trust me. Linear algebra is about as difficult as calc 2 but as abstract as calc 3. So it's not an easy subject.

I think fresh_42 and I are both trying to give you the best advice possible, which is why you have two books so similar being mentioned. One can debate what would get you ready for such a book, or whether you are ready, but you did say you want to learn it this year, concurrently with precalculus. I think that logic book would help you, not the whole book, just the first half. It may not seem worthwhile but Smullyan is quite a gentle author so you could probably be done with it in a week or two. Then you could get stuck into learning linear algebra.

By the way, I see Friedberg/Insel does explain what a field is, it's in Appendix C. So you could get a taste of higher math if you want to.

#### Jamesix

Perhaps the best thing I can say is this. You haven't yet seen any hard math because you haven't seen calculus yet. Calculus is a step above anything you have seen in terms of difficulty, and it keeps getting more difficult as it goes along. Calc 1,2,3 upgrades in difficulty, trust me. Linear algebra is about as difficult as calc 2 but as abstract as calc 3. So it's not an easy subject.

I think fresh_42 and I are both trying to give you the best advice possible, which is why you have two books so similar being mentioned. One can debate what would get you ready for such a book, or whether you are ready, but you did say you want to learn it this year, concurrently with precalculus. I think that logic book would help you, not the whole book, just the first half. It may not seem worthwhile but Smullyan is quite a gentle author so you could probably be done with it in a week or two. Then you could get stuck into learning linear algebra.

By the way, I see Friedberg/Insel does explain what a field is, it's in Appendix C. So you could get a taste of higher math if you want to.
Could you link me the book you are recommending this sounds like a good option.

#### Jamesix

Perhaps the best thing I can say is this. You haven't yet seen any hard math because you haven't seen calculus yet. Calculus is a step above anything you have seen in terms of difficulty, and it keeps getting more difficult as it goes along. Calc 1,2,3 upgrades in difficulty, trust me. Linear algebra is about as difficult as calc 2 but as abstract as calc 3. So it's not an easy subject.

I think fresh_42 and I are both trying to give you the best advice possible, which is why you have two books so similar being mentioned. One can debate what would get you ready for such a book, or whether you are ready, but you did say you want to learn it this year, concurrently with precalculus. I think that logic book would help you, not the whole book, just the first half. It may not seem worthwhile but Smullyan is quite a gentle author so you could probably be done with it in a week or two. Then you could get stuck into learning linear algebra.

By the way, I see Friedberg/Insel does explain what a field is, it's in Appendix C. So you could get a taste of higher math if you want to.
Never mind found it (edit)

#### Jamesix

Perhaps the best thing I can say is this. You haven't yet seen any hard math because you haven't seen calculus yet. Calculus is a step above anything you have seen in terms of difficulty, and it keeps getting more difficult as it goes along. Calc 1,2,3 upgrades in difficulty, trust me. Linear algebra is about as difficult as calc 2 but as abstract as calc 3. So it's not an easy subject.

I think fresh_42 and I are both trying to give you the best advice possible, which is why you have two books so similar being mentioned. One can debate what would get you ready for such a book, or whether you are ready, but you did say you want to learn it this year, concurrently with precalculus. I think that logic book would help you, not the whole book, just the first half. It may not seem worthwhile but Smullyan is quite a gentle author so you could probably be done with it in a week or two. Then you could get stuck into learning linear algebra.

By the way, I see Friedberg/Insel does explain what a field is, it's in Appendix C. So you could get a taste of higher math if you want to.
One final question sorry for being bothersome, let’s say I take your advice and I finish with the introduction to logic and move to linear algebra, do you think I will excel rather quickly in AP AB Calculus and could possibly do the work I get assigned in class but also start Calc 2 on my own or am I getting ahead of my self?

#### fresh_42

Mentor
2018 Award
Or should I get a different subject maybe for advanced high schoolers. Or would you say linear algebra is a good starting place for me being 16.
I find it is as hard with 16 as it is with 20. I just don't like such answers like: "You're too young to understand." or similar. What shall this mean? You're too stupid? Too premature? Not spoiled enough yet, by old men's ways of thinking? I really dislike such things, because it usually means: "I'm too stupid to teach you better." Of course it will be new and cost time, but it has been you who mentioned MIT, Cornell and Harvard. You won't get there if you're afraid of learning or the new in general. Sure it will take time and you have to decide whether you want to spend this time, but "too young" or "too stupid" are definitely no excuses. An excuse is, that you still need time to play baseball or so, which is totally o.k. The stuff will be as new today as it will be tomorrow. In any case, at some point you will have to turn away from calculations and deal with concepts instead. Whether these are vector spaces (linear algebra), proofs (logic) or $\varepsilon-\delta$ terminology (calculus) doesn't really matter. All that matters is your own plans and how to spend your time. These are options which take ahead first year of college, and they won't help you at school (presumably), so you could as well improve your trigonometry, which probably will help you at school.

#### Jamesix

I find it is as hard with 16 as it is with 20. I just don't like such answers like: "You're too young to understand." or similar. What shall this mean? You're too stupid? Too premature? Not spoiled enough yet, by old men's ways of thinking? I really dislike such things, because it usually means: "I'm too stupid to teach you better." Of course it will be new and cost time, but it has been you who mentioned MIT, Cornell and Harvard. You won't get there if you're afraid of learning or the new in general. Sure it will take time and you have to decide whether you want to spend this time, but "too young" or "too stupid" are definitely no excuses. An excuse is, that you still need time to play baseball or so, which is totally o.k. The stuff will be as new today as it will be tomorrow. In any case, at some point you will have to turn away from calculations and deal with concepts instead. Whether these are vector spaces (linear algebra), proofs (logic) or $\varepsilon-\delta$ terminology (calculus) doesn't really matter. All that matters is your own plans and how to spend your time. These are options which take ahead first year of college, and they won't help you at school (presumably), so you could as well improve your trigonometry, which probably will help you at school.
Thank you I agree and I didn’t mean too young as that. But I understand completely. I gues what I ment by too young is I havnt learned everything I need for linear algebra yet. But I understand that it’s linear algebra it’s self almost like a new language. Thank you I am going to work through the logic book and move onto linear algebra. All of your help is appreciated

#### verty

Homework Helper
One final question sorry for being bothersome, let’s say I take your advice and I finish with the introduction to logic and move to linear algebra, do you think I will excel rather quickly in AP AB Calculus and could possibly do the work I get assigned in class but also start Calc 2 on my own or am I getting ahead of my self?
I don't recommend learning your school syllabus ahead of time because that could mean you get very bored in class. I know from personal experience because I learned to read before I went to school and my first year was terrible because I had nothing to do. It's not something I would recommend to anyone.

Rather than doing something like that, I would recommend using a more advanced book like Spivak at the same time as you learn calculus. That way, you won't get ahead of time. Regarding linear algebra, you could do the same thing if you ever do it in college, using a book like Axler. But that all depends how you like abstract math. You may not like it.

Luckily, linear algebra has little overlap with calculus so it shouldn't affect it very much.

#### Jamesix

I don't recommend learning your school syllabus ahead of time because that could mean you get very bored in class. I know from personal experience because I learned to read before I went to school and my first year was terrible because I had nothing to do. It's not something I would recommend to anyone.

Rather than doing something like that, I would recommend using a more advanced book like Spivak at the same time as you learn calculus. That way, you won't get ahead of time. Regarding linear algebra, you could do the same thing if you ever do it in college, using a book like Axler. But that all depends how you like abstract math. You may not like it.

Luckily, linear algebra has little overlap with calculus so it shouldn't affect it very much.
This is the book I will be given next year. Is this a quality book?

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#### verty

Homework Helper
This is the book I will be given next year. Is this a quality book?
I think it must be good because Finney is the same author of probably my favourite calculus book, Thomas & Finney 9th edition, the one with the blue lighthouse on the cover. And with many authors like that, they wouldn't have been tired writing the book, so I think it'll be just fine.

#### Jamesix

I think it must be good because Finney is the same author of probably my favourite calculus book, Thomas & Finney 9th edition, the one with the blue lighthouse on the cover. And with many authors like that, they wouldn't have been tired writing the book, so I think it'll be just fine.
So you think I should get the more advanced book in addition?

#### fresh_42

Mentor
2018 Award
So you think I should get the more advanced book in addition?
If you buy Spivak, Greub or similar books, they will either become your standard source to look up things and a good basis to deepen your knowledge, or you will sell them in the near future on ebay or amazon. Will say, even if you decide, that they are still too difficult or time consuming by now, they will still be a good purchase for tomorrow, or you followed a different path in the meantime. That's a risk you must assess.

In the above thread there are many good advises and, yes, contradicting ones, because they show you possible alternatives. Don't get frustrated by stuff that you find too difficult and don't learn ahead of classes. The books on openstax and maybe the pdf about proofs are all meant for the transition high school -> college and they should not be a problem for you. Greub and Spivak are textbooks for those who study mathematics, or at least the basics of mathematics, as e.g. a meteorologist would do. If you really are interested in a subject, then there is no problem learning it. If it costs you energy to sit down, read and learn it, then you are not really interested and no advise on earth can change this.

#### verty

Homework Helper
So you think I should get the more advanced book in addition?
I think for your final year you should try to do the best you can in school and another book could be a distraction. You want the best marks so you get into your desired university. What I meant was, when you are in university you might have to do calculus again, and then you could use a more advanced book if you wanted so that it would offset the tendency to become lazy. The same goes for linear algebra, you could use a more difficult book if you find you already know it. But like I said, I don't see the value in studying ahead of time but since you may choose math, I thought it would be useful to have a look at some abstract math to see what it is like.

Homework Helper

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