#### 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

Mentor
Here's some resources for you to check out:

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 highschool 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

Mentor
2018 Award
Herein is a collection of articles and sources about "self-study" math and physics. It might be worth to have a read.

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 text book would be appreciated.

#### fresh_42

Mentor
2018 Award
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 text book 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

Mentor
2018 Award
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

Mentor
2018 Award
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.

#### 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

Mentor
2018 Award
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

Mentor
2018 Award
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

Mentor
2018 Award
My recommendation is to study Logic. You can purchase a book or find online material.

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 highschool correct.

#### fresh_42

Mentor
2018 Award
Ok call me dumb, but I should go into this new. Don’t try to relate this to highschool 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

Homework Helper
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

Mentor
2018 Award
... 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

Homework Helper
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

Gold Member
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.)

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