# Where do I start learning Higher Mathematics? [closed]

I am 17 years old and I would like to begin learning Mathematics. I am only familiar with Algebra and a bit of Geometry. Do I need to learn Calculus next, or should I try linear algebra? I find myself very interested in number theory. Is there anywhere I can see some sort of tree that shows me prerequisites for different Mathematical studies. I also want to know if there are any websites or resources that can teach me Math formally, using proofs. In high school we never learned math formally or rigorously. Thanks.

Also any books I could get would be appreciated. Keep in mind I am trying to start off, I only know basic Algebra right now, like functions, polynomials, and systems of equations.

## closed as primarily opinion-based by Matthew Towers, Claude Leibovici, user91500, Frits Veerman, user99914 May 10 '16 at 9:54

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

• Khan Academy is a nice start. You can also try taking online courses at Udemy/Coursera/OpenU. You can check some university's prerequisites but there isn't a 'correct' path to study, especially on your own. First year Courses like Calculus and Linear Algebra (which I suggest you study simultaneously) usually don't assume any prior knowledge. Also, Discrete mathematics courses serve as a great start. – Trouble May 9 '16 at 18:25
• @Trouble Calculus doesnt assume prior knowledge? I thought you need Pre-Calculus to do Calculus. – user338280 May 9 '16 at 18:33
• Kind of. You need to be very comfortable with algebra and you should know trigonometry. But I feel like for people who are very motivated, having a semester of precalculus beforehand is a waste of time, it doesn't take a semester to get prepared. – D_S May 9 '16 at 18:39
• I was introduced to number theory (and group theory) before calculus, and that's where I learned about mathematical proofs. I would think that the first step into higher mathematics is getting good at using proof - and using it not only to create mechanically convincing arguments, but to connect intuition and rigor. – Milo Brandt May 9 '16 at 20:41
• It seems to me that one could study abstract algebra with a minimum of prior knowledge. The only caveat to that is since it is usually presented at a university level, understanding of proofs is usually assumed by the texts and courses available on abstract algebra. – Todd Wilcox May 9 '16 at 20:43

Khan academy might be a good starting point.

On youtube you can watch hundreds of their short videos on subjects like algebra, calculus, number theory, probability, linear algebra, geometry, trigonometry etc.

When you feel like you master that material you can move on to MIT OpenCourseWare on youtube.

Gilbert Strang's Linear algebra.

David Jerison's Single variable calculus.

Denis Auroux's Multivariable calculus.

Tom Leighton's Mathematics for Computer Science.

They also have courses on probability, statistics, differential equations etc.

Other personal favourites, which i actually liked more than MIT, are these

Discrete Mathematics. Arsdigita University. Instructor: Shai Simonson

The Fourier Transforms and its Applications. Standford. Professor Brad Osgood

Probability. Harvard

Probability Primer. Mathematicalmonk's channel

General topology from the very basics, including set theory, techniques for proofs

Short course on writing proofs in mathematics by Sidney Morris

At the same time you are following this online material i recommend buying books and solve a lot of problems. There is no better way of learning mathematics than solving problems. I would buy books which have solution manuals.

I would also recommend to start to learn to program in R, matlab, mathematica, maple, python or whatever environment you like best. This will become a very useful skill when you reach higher mathematics.

• how is programming useful in higher level mathematics? – Charlie Parker May 10 '16 at 0:15
• @CharlieParker Well i am no expert, but I'd say there are more areas where it is useful than areas where it is not. To name a few: mathematical modelling, PDEs, stochastic processess, fourier applications, combinatorics, statistics, probability, number theory, discrete mathematics, graph theory, optimization, algorithms, numerical methods, proof verification, calculus, financial mathematics, cryptography, information-theory, geometry, lin alg etc. I even have a friend who is writing his thesis on the mathematical aspects of general relativity and its all about modelling using his own software. – JKnecht May 10 '16 at 0:42

I think you're in a good position to benefit from Keith Devlin's course (or just read the book) on mathematical thinking: https://www.coursera.org/course/maththink. Rather than dive into advanced mathematics directly, he takes time to reflect on what it means to "think like a mathematician" and develops some necessary logical prerequisites. The course/book concludes with a taste of different kinds of advanced mathematics. Here's the table of contents so you can see what I mean:

1. Introductory material
2. Analysis of language – the logical combinators
3. Analysis of language – implication
4. Analysis of language – equivalence
5. Analysis of language – quantifiers
6. Working with quantifiers
7. Proofs
8. Proofs involving quantifiers
9. Elements of number theory
1. Beginning real analysis

Well well, this is a broad question and you might get widely different answers, so take mine with a grain of salt!

I believe the first thing you should do before attempting to study any more mathematics is solidify your knowledge of basic logic. Study first-order logic, learn to write proofs of theorems in this language. This should not take long (only a general overview is required), but it will give you essential foundations.

You can then move on to other topics. In my first year at university we had a course which was using "A Concise Introduction to Pure Mathematics" by Martin Liebeck. I recall it was a pretty decent book, and it has some number theory in it! There is also a chapter on logic, although I would advice you look at external sources, too.

• Where could I learn about basic logic. I really don't know what the definition of logic is anyways, at least in mathematical context. – user338280 May 9 '16 at 18:29
• @user338280 I cannot think of a good book over the top of my head but I will try to get back to you on that! That said with the basic knowledge you have you should be able to read Martin Liebeck. – Aegis May 9 '16 at 18:32
• Thanks. I really don't have time or money to go to school for mathematics although I find it to be really interesting. I just want to stimulate my mind a bit in my free time. – user338280 May 9 '16 at 18:33
• @user338280 Yeah education can be expensive... But rest assured that if you are dedicated you might well learn more by reading in your free time than many people do at university :) Nowadays with the internet you have easy access to information & people to ask questions. – Aegis May 9 '16 at 18:34

Most people start with calculus as their first "advanced" (i.e. college level or above) math. There are many online resources for this, like Khan Academy. Do lots of exercises. Ask yourself dumb questions.

Next, you may want to take some time to get a good foundation in naive set theory. Stuff like logical implications, e.g. why is the statement "$P$ implies $Q$" true when $P$ is false. Get comfortable with unions and intersections, functions, images, preimages, stuff like that. Get comfortable with basic proofs.

After that, you may want to try learning, in no particular order, linear algebra, abstract algebra, real analysis. Try to get a good balance between working examples, understanding concepts, and being able to prove things.

I think learning all that is a good foundation for learning other fields of math. It might take you a few years. Meanwhile you may discover what you like and what your strengths and weaknesses in math are. For example, if you like real analysis and linear algebra, you may want to have a look at areas like differential equations or functional analysis.

• Definitely basic set theory early on! – Trouble May 9 '16 at 18:46
• @user338280 You can ask those questions here, if you ever need to. Though vague questions like "will there exist an equation that models that curve" is probably going to receive little attention, other more specific questions are welcomed. (pst, for quadratic formula, use substitution $x=u-\frac{b}{2a}$) – Simply Beautiful Art May 9 '16 at 21:58

What helped my a lot was actually learning through problem solving. Like, I get a problem, try to solve it and if I see I can't do it on my own I look up some answers or theorems on internet. This method suits me still the best, bcs you can learn so much from one Problem. But keep in mind that you can cheat others, but not yourself, so you have to give your best before looking up answers. I highly recomment the website Art of Problem Solving. There you have a community/forum where you can look up problems for any level, from middle school to highschool olympics and collage. There are usually also answers to a lot of problems so it souldn't be that hard to look up for an answer later. If you are interested in a porblem and you can't solve it you are always free to post it here too with some of your work yo have done (that is important!).

About what you should learn next: well, there is no rule, atleast there wasn't any for me. I did everything what I liked and also disliked because I was preparing myself for competitions in my country so I wasn't having much of a choise. But you can always start from some basics like divisibility, femrats little theorem, prime numbers and what I highly recomend is to master modulos for number theory, or inequality and functional equations which are really good to stimulate your creativity and are good things to learn algebra. For geometry I really like using trigonometry, but I also recommend to learn a lot of other theorems that will help you like some properties for circular quadrilateral, chraracteristic points of a triangle, similarity and what is important here if you dont want to use trigonometry is to know when and what you have to draw extra into you sketch so you can get something nice that will help you. The last thing is combinatorics, which I really dislike the most, but here I can't help you a lot, beacuse usually you really don't need mathematical knowledge to solve those problems. They are kinda made so anyone can solve them.

That's it, after learning some basics you will see a lot of problems where people use some kind of theorems like "Lifting the Exponent Lemma" or for short LTE Lemma. If something like that happens just google it and mostly you can find your answer on wikipedia or other sides.

One more thing. If you start liking geometry I also recomment you downloading geogebra (or use the online version) which is a free programm that lets you draw a lot of things so you dont have to do a lot of papper drawing and you can easily modify your picture.

I hope I could help you a little bit, since there is no way anyone can cover everything. If you need some help feel free to ask me or anyone here for help, that also goes for aops. You can find me there under the name mirza45

Good luck with your work :)

• Pft, stop recommending other sites, your supposed to promote our site! – Simply Beautiful Art May 9 '16 at 22:02
• @SimpleArt I am not promoting anything, I just said where he can find better problems that suits him best. At math stachexchange you truly can find a lot of math problems, but to be honest most of them are collage level which he is not currently at. I am sorry if it did sound like I am doing what you said, but like I said, I learned a lot from there and I dont see a bad thing to share knowledge. – HeatTheIce May 9 '16 at 22:04
• Yes, I have actually used that site as well. I was only joking with you (but promoting this site is a good thing) and there are some high school level topics on this site... maybe not much, but if so, its because people stop asking those questions on this site. Which raises some questions that I will ask elsewhere. – Simply Beautiful Art May 9 '16 at 22:12
• Well, I used this site also for some questions I had in highschool and it came in handy, but on this site there are more serious mathematicans that know a lot of theory and how to apply them, but what I like more are the answers that are elegant and have some kind trick that you have to come up with yourself and not to look it up on wikipedia (I know what I said now is a little bit contradictory with what I said up there). But I do like this site a lot, but for young ones I would always recomment aops :) – HeatTheIce May 9 '16 at 22:16
• And I still haven't learned how to make math work on that site lol... Agreeably, the people on this site are very serious and we indeed do rather rigorous math. But I feel proud to be amongst the smart serious mathematicians, can't help it. ;) means you guys out there looking down at this comment. – Simply Beautiful Art May 9 '16 at 22:23

If you're serious about learning math yes, you have to learn calculus (as well as many other things, but calculus is definitely one of the fundamental first steps).

I am not going to suggest any modern tools, such as online classes etc. I think other people covered that quite well. I will be old-fashioned here and suggest just a book. Yeah, made of paper. And a quite old one actually, that nevertheless is still considered one of the best introductory texts in calculus/analysis.

It is not an easy book. It might be a challenge to start off with it actually, but somehow I have the feeling that it's one of those books that could really make the difference in the future. It's a book that forces you to think. A book that, in order to understand, you have to go through line by line and spend time on it, trying to grasp every concept and filling eventual gaps left here and there by the author (sometimes intentionally, maybe sometimes not).

As I said, it might not be an easy book... well no, it certainly isn't, but it starts from the very basics after all. So if you have some basic math training and you are up for a challenge I think you can have a go with it (I am assuming that you are familiar at least with basic math terminology and symbolism).

Ok, the book I am talking about is Walter Rudin's Principles of Mathematical Analysis. You can have a look here:

http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X/ref=mt_hardcover?_encoding=UTF8&me=

If you browse a bit through the reviews you can have a feeling of what I'm talking about here. It's a book that even experienced mathematicians love to read. I might be going out on a limb suggesting this, true, but I believe it could be the book that will actually shape you into a real mathematician. Some undergrads may not like it because they say it's too advanced. Unlike you though, they probably wouldn't post questions on math.stackexchange about where to start off with higher mathematics. If your passion is genuine, this might be the book for you. Good luck.

You are actually the second person today who has asked such a question. Calculus and Linear Algebra can be learned independently, but some (not much) Linear algebra is useful for 3 dimensional calculus. Number theory is great, the classic introduction is by Hardy and Wright but there are many other good books. Focus on solving problems. Especially since you are young. Look into learning some abstract algebra, rings, groups and fields, as soon as possible, maybe even before linear algebra.

Finally one book I liked in called Mathematics its from Time-Life science library, its pretty old but most libraries will have it. It popular but has a nice tree diagram of the different branches of mathematics.

One afterthought, since it seems to be being discussed here, (yes thats correct grammar) universities and schools have become useless hindrances to your progress, and your pocketbook, whether you do or not anything you learn will have to be done in spite of their influence. (Sorry folks but that is the current state of affairs).

• When I look at a problem in Mathematics i do not know what to do. I am still trying to develop my intuition when it comes to solving problems. Im so used to being told to remember formulas in a classroom and I need to learn to think independently. What is abstract algebra in essence?? I have heard of it but I do not know how it would help me. – user338280 May 9 '16 at 18:36
• @user338280 Abstract algebra is, roughly, the study of structure over sets. Do you know what a set is? – Aegis May 9 '16 at 18:39
• @user338280 Yeah learning to think independently is so important (dont do it in university though youll get in lots of trouble). One thing is to get some math contest problem books and work on some of those questions, it can be fun and it builds your problem solving muscles. But it wont connect you with higher mathematics. – Rene Schipperus May 9 '16 at 18:43
• @Aegis A collection of objects. I don't really know why they are useful but I have heard people say that all of Mathematics is based on operations with sets, and a number is just something our brains created to describe the cardinality of various sets of physical objects. – user338280 May 9 '16 at 18:43
• @Rene Schipperus Well of course you'll get in trouble since the university makes a profit off you being uneducated and needing their assistance to think. – user338280 May 9 '16 at 18:44

There are quite a few good sources out there for learning different fields in mathematics. Personally some of my favorates are:

To get started you need to know the learning path you will take, what classes and areas you want to begin with and proceed to. A basic path for general mathematics would be: (pipe shows what I feel can be learned at the same time)

Algebra I > Geometry | Algebra II > Statistics | Trigonometry > Linear Algebra > Single Variable Calculus > Multi-Variable Calculus | Differential Equations

From there you should be well versed enough to know what branch of mathematics you would like dive deeper into

For me, Calculus isn't a bad place to start. Its assisted me in calculating divergent series, proving binomial expansion, calculating some harder summations, providing bounds, and simply giving intuitive sight into how a function may work.

Calculus isn't too hard, and you don't really need pre-calculus to learn it. Calculus is just a little different from previous math courses in the topics. Its like the jump from middle school math to Algebra, just a little harder and makes you think differently.

Another nice topic of interest to you might be modular arithmetic. It is rather easy, and when you learn things like Fermat's little theorem and basic manipulations, you can do cool things like determine what the last digit of $47^{2958}$ is. (It also helps you make faster choices on multiple-choice questions involving whole numbers. And if you have a younger sibling in elementary, teaching them the basics to modular arithmetic could help them in math... wish I knew ALL of modular arithmetic in elementary. When you learn modular arithmetic, I bet some of the material will be so easily you'll say "I've done that before!")

Summations are also fun. When you learn calculus, summations become easier to deal with as well (for me at least). Also, making formulas with summations gets you into thinking about how you can provide proofs sometimes, and other things like that.

Set theory is a short(ish) topic that is really interesting for most people. It is basically the theory of infinity, and a nice video that captures the gist of set theory is Vsauce's "How to count past infinity?" video.

Number theory might be a bit harder, but you could potentially learn some of it.

And don't expect all of this to come easily (maybe the modular arithmetic), so take your time, and if you get stuck, we'll be right here to help you.

Ranked in easiness according to my opinion (difficulty based off how hard to get started in the material):

1. Modular arithmetic (easiest)

2. Calculus and summations

3. Set Theory

4. Number Theory

Also, I have to ask, do you intend on math courses for credit, or math subjects you want to learn on your own with no other reason than that your just nerdy like that? It makes a difference.