Seeking advice about learning more math

[Jamesix]

Hello, I am new to this forum and would like to ask a few questions. I guess I should introduce my self. I am currently 16 probably a little young to be on a forum of this caliber. I currently have an interest in math and wish to pursue this interest in college and beyond. I currently am a junior taking pre calculus and statistics. I have always excelled in math ever sense elementary school and I am currently going to a private school where my courses are set, I chose statistics as a elective this year. Next year math is optional but I am going to take AP Calculus AB (BC is not offered) Physics (Algebra based, AP is not offered) and engineering (As an elective). I have excelled on my college entrance exams even though I have not taken my final tests. I wish there was more variety to take at my school but sadly that is not an option. I am wondering what books/textbooks, courses, or curriculum I could take this year or next. In addition to my material learned at school. I am also looking for mathematical books that are advanced but within my capability of reading. I would like to either attend Colorado State of Mines, MIT, Cornell, or Harvard. I have not had the opportunity to compete in any mathematical events. Are my chances slim getting into any of these universities and am I able to continue to advanced mathematics. Thank you so much for your input.

 

[jedishrfu]

Here's some resources for you to check out:

1) Khan Academy online courses

2) Mathispower4u.com has math from 9th grade to 1st year college ie Calculus, Linear Algebra, Differential Equations and Statistics in short topic videos.

3) Openstax online books by Rice University. There a high school and college level Physics and Math books available that are quite good and could supplement what you are learning. As you study Physics AP, you could read about the same problems using calculus to solve them.

Just make sure you don't overextend yourself and/or get confused with Algebra vs Calculus based solutions to your Physics problems unless your teacher approves and understands what you're learning. I had a Chemistry teacher like that who taught me on how to use the loglog scales on my slide rule which wasn't taught in our class and allowed me to use them in problem solving.

 

[Jamesix]

The physics I will be learning next year is not AP nor Calculus bases. Are you suggesting as a pre calculus student I get a Calculus based physics book or wait until next year to do that sense I will be taking AP AB Calculus

 

[fresh_42]

Herein is a collection of articles and sources about "self-study" math and physics. It might be worth to have a read.

https://www.physicsforums.com/threads/self-teaching-gcse-and-a-level-maths.933639/#post-5896947

and maybe this one: http://www.people.vcu.edu/~rhammack/BookOfProof/BookOfProof.pdf
which I'd rather have as a "look it up" source than read it.

In any case, you can always come over here and ask whatever you want. Just as a little hint: If your questions are exercise-like, whether homework or textbook exercises, please use our homework sections and fill out the (automatically inserted) template, especially part three with your own efforts. You will meet a lot of teachers who will like to help you. But we want to teach, not solve, so that's why part 3 of the template is important to us. Nevertheless, if you have problems to understand something, PF is the first choice to go to and certainly faster than stuck. I would also recommend it for checking whether you got a concept right or not, as learning it wrong and then correcting it is hard.

 

[Jamesix]

Thank you for the wonderful descriptive advice; I truly appreciate it. However, I do have one more question. Sense I am taking pre cal and taking AP Cal next year what is a course that I can take this year and comprehend. I would like to try to get into advanced math as you know a link to a textbook would be appreciated.

 

[fresh_42]

Thank you for the wonderful descriptive advice; I truly appreciate it. However, I do have one more question. Sense I am taking pre cal and taking AP Cal next year what is a course that I can take this year and comprehend. I would like to try to get into advanced math as you know a link to a textbook would be appreciated.

Hard to tell without a better knowledge of where you are at, resp. are willing to accept in a language used outside of school. You could check out the openstax books and see where you stand. E.g. linear algebra is not really hard to understand but necessary in all STEM fields. However, since school topics are usually very computational and biased towards calculus, students in the first year often have difficulties to get used to the different way of thinking in linear and / or abstract algebra. So linear algebra would be a possibility to learn. It also has the advantage, that it probably won't collide with other courses - with the exception of solving linear equation systems.

 

[Stephen Tashi]

I am wondering what books/textbooks, courses, or curriculum I could take this year or next.

It's impossible to advise about that unless you say what courses are offered.

In addition to my material learned at school. I am also looking for mathematical books that are advanced but within my capability of reading.

A basic decision is whether to read material that introduces some calculus or to postpone that type of material until you are taking calculus. Let's say you choose to read material that doesn't depend on calculus. I'd advise studying Logic. Understanding basic logic, including the logic of quantifiers ($\forall, \exists$) is essential to understanding Calculus or other mathematical topics precisely.

Are my chances slim getting into any of these universities and am I able to continue to advanced mathematics. Thank you so much for your input.

That's a question for someone who is an expert on college admissions or a graduate of those particular schools. Did any staff of your private school attend them?

 

[Jamesix]

First of all thank you both for your advice , it is appreciated.

So linear algebra would be a possibility to learn. It also has the advantage, that it probably won't collide with other courses - with the exception of solving linear equations.

I have worked with linear equations of course and am confident in doing so.

A basic decision is whether to read material that introduces some calculus or to postpone that type of material until you are taking calculus. Let's say you choose to read material that doesn't depend on calculus. I'd advise studying Logic. Understanding basic logic, including the logic of quantifiers ($\forall, \exists$) is essential to understanding Calculus or other mathematical topics precisely.

So what I am thinking ( correct me if I am wrong ) with my current level of mathematical knowledge I should purchase an advanced linear algebra textbook and a introduction to Mathematical Logic book? I will wait until next year until I actually take more than pre cal ( As of next year I will be in AP AB Cal) to expand to more advanced calculus. But these two suggestions I should be able to comprehend? I would say I am definelty above average in math but I have not had extremely difficult courses. I will do research on these two topics and find a text that suits me. If you have any recommendations that would be appreciated. Thank you so much for your time

 

[Jamesix]

Would an introductory to abstract algebra work for the linear algebra you are reccomending.

 

[fresh_42]

I have worked with linear equations of course and am confident in doing so.

What I meant was the following:
Here where I live, kids are confronted with three versions to solve things like $ax+by=u\; , \;cx+dy=v$, called substitution, addition and comparison method. Unfortunately, they are not told, that $\begin{bmatrix}a&b\\c&d\end{bmatrix}\cdot \begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}u\\v\end{bmatrix}$ is the correct method to do it, which is subject to linear algebra.

 

[fresh_42]

Would an introductory to abstract algebra work for the linear algebra you are reccomending.

Yes and no, but more no. Normally linear algebra refers to anything around vector spaces, which is why it is so important, whereas abstract algebra usually deals with structures like groups, rings and fields. The former is far more a tool than the latter.

P.S.: For the book about logic, please check my link in post #4. This is for free and will serve the same purpose.

 

[Jamesix]

What I meant was the following:
Here where I live, kids are confronted with three versions to solve things like $ax+by=u\; , \;cx+dy=v$, called substitution, addition and comparison method. Unfortunately, they are not told, that $\begin{bmatrix}a&b\\c&d\end{bmatrix}\cdot \begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}u\\v\end{bmatrix}$ is the correct method to do it, which is subject to linear algebra.

I am vaguely framiliar with this, do you have a reccomended text for expansion on this, I looked at openstax and they seem to have Elemtary Algebra, Intermediate Algebra ( Guessing is Algebra 2) I have taken that, and college level. Is there another text you know I don’t think I should take college level algebra just yet.

 

[fresh_42]

Is there another text you know I don’t think I should take college level algebra just yet.

If I would have to purchase a book, I would do it the right way and e.g. choose

https://www.amazon.com/dp/0387901108/?tag=pfamazon01-20

or a more modern version, although this won't make a difference. I would also avoid a paperback, as it will be a book which constantly will accompany you throughout the years, in case you follow the path you described.

 

[Jamesix]

If I would have to purchase a book, I would do it the right way and e.g. choose

https://www.amazon.com/dp/0387901108/?tag=pfamazon01-20

or a more modern version, although this won't make a difference. I would also avoid a paperback, as it will be a book which constantly will accompany you throughout the years, in case you follow the path you described.

Would you suggest a modern version such as: https://www.amazon.com/gp/product/B01FGP4A8G/?tag=pfamazon01-20
Or would the comprehension be moderately the same. Sense I am a high schooler

 

[Stephen Tashi]

So what I am thinking ( correct me if I am wrong ) with my current level of mathematical knowledge I should purchase an advanced linear algebra textbook and a introduction to Mathematical Logic book?

My recommendation is to study Logic. You can purchase a book or find online material.

As to linear algebra, my own advice is begin by studying something less related to high school algebra. If you study linear algebra, you have to fight the tendency to explain the material to yourself using high school algebra instead grasping the abstract content that texts attempt to convey. It's better to make a "clean break" with high school algebra if the purpose is to understand mathematical abstraction. Group Theory and Point Set Topology are examples of topics that are not heavily dependent on Calculus and not likely to be confused with high school algebra. It might be hard to find modern texts on these topics written for high school students, but there are some old books that would do. E.g. https://www.amazon.com/dp/088385614X/?tag=pfamazon01-20

When I was in high school, the usual result for high school students self-studying mathematical books was that they didn't progress quickly. That was before the days of lectures on YouTube and internet forums.

 

[fresh_42]

Would you suggest a modern version such as: https://www.amazon.com/gp/product/B01FGP4A8G/?tag=pfamazon01-20
Or would the comprehension be moderately the same. Sense I am a high schooler

I don't know. I can only recommend what I know, and Greub is a good book, and the entire Springer series can be recommended. I don't have a single book from the series which I ever regretted to have bought. The matter hasn't changed in the last 100+ years, so it really doesn't matter. My author (Greub) has a Wiki entry, under David Pole a politician and a bishop showed up.

It is a college book, yes, but there is no reason to assume you cannot understand it. It might be a new line of thought, one which you will get used to sooner or later anyway. So the question is whether you dare to jump into the pool now, or better postpone it.

 

[fresh_42]

My recommendation is to study Logic. You can purchase a book or find online material.

As already mentioned: http://www.people.vcu.edu/~rhammack/BookOfProof/BookOfProof.pdf

But - and this is my own personal opinion: What a waste of time! We only use a rather small part of logic, namely predicate logic, and it automatically comes by reading abstract proofs - the more the better. Why learn things twice? As I said, my opinion.

 

[Jamesix]

Ok call me dumb, but I should go into this new. Don’t try to relate this to high school correct.

 

[fresh_42]

Ok call me dumb, but I should go into this new. Don’t try to relate this to high school correct.

Probably better. E.g. a rotation at school is probably something involving a compass, in linear algebra it is $\begin{bmatrix}\cos \varphi &-\sin \varphi \\ \sin \varphi & \cos \varphi\end{bmatrix}$ or worse, simply $A^\tau A = E$ or $\langle Au,Av\rangle = \langle u,v \rangle$.

If you are stuck, come on over or really postpone it. It doesn't help you, if you get frustrated. As you saw on my example, things can become rather abstract fast.

 

[Jamesix]

Probably better. E.g. a rotation at school is probably something involving a compass, in linear algebra it is $\begin{bmatrix}\cos \varphi &-\sin \varphi \\ \sin \varphi & \cos \varphi\end{bmatrix}$ or worse, simply $A^\tau A = E$ or $\langle Au,Av\rangle = \langle u,v \rangle$.

If you are stuck, come on over or really postpone it. It doesn't help you, if you get frustrated. As you saw on my example, things can become rather abstract fast.

Yeah I have a lot of work to put in but I think it will be worth it.

 

[verty]

Wow, this thread is confusing me. Recommended so far are logic and linear algebra. I'm not sure what logic book but Smullyan's "A beginner's guide to mathematical logic" would do nicely, I think. As for linear algebra, Greub's book has been recommended and I don't have a recommendation because the ones I know are too advanced. I see that Micromass recommended Friedberg/Insel/Spence in an Insights blog post but it uses words like "field" that Jamesix won't know. But that one looks like a good book as well.

 

[Jamesix]

Wow, this thread is confusing me. Recommended so far are logic and linear algebra. I'm not sure what logic book but Smullyan's "A beginner's guide to mathematical logic" would do nicely, I think. As for linear algebra, Greub's book has been recommended and I don't have a recommendation because the ones I know are too advanced. I see that Micromass recommended Friedberg/Insel/Spence in an Insights blog post but it uses words like "field" that Jamesix won't know. But that one looks like a good book as well.

So Greub’s book would work well?

 

[fresh_42]

... but it uses words like "field" that Jamesix won't know.

Lol, I just looked it up. Greub has it on page three. At least he defines it, although by using group. On the other hand, it's a matter of minutes to look it up on Wikipedia. Shouldn't be the real difficulty.

 

[verty]

So Greub’s book would work well?

I don't want to start a big argument but it looks more complicated to me, especially the cover of the book saying "Graduate Texts in Math". Otherwise the order of the topics looks very similar. I think the logic book I mentioned would give you enough nous to handle Friedberg/Insel if you are prepared to look up stuff like what a field is.

 

[StoneTemplePython]

My recommendation is to study Logic. You can purchase a book or find online material...

Group Theory and Point Set Topology are examples of topics that are not heavily dependent on Calculus and not likely to be confused with high school algebra. It might be hard to find modern texts on these topics written for high school students, but there are some old books that would do. E.g. https://www.amazon.com/dp/088385614X/?tag=pfamazon01-20

I tend to think studying Logic formally isn't that worthwhile.

However, I liked Stephen's post as the "New Mathematical Library" has some reaaly well done books. They aren't particularly easy but nominally are targeted at High Schoolers and have minimal pre-req's, really just careful thinking. I don't think New Math Library covers linear algebra but... OP may want to consider starting with one of the following from New Math Library:

(a) Groups and Their Graphs
(b) Graphs and Their Uses
(c) Introduction to Inequalities

(If OP likes it, the authors of (c) have a more advanced book called "Inequalities" which may be of interest as a follow up a few years down the road.)

 

[Jamesix]

Hmmm interesting what would be the best option

 

[fresh_42]

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]

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]

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]

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.