Where do I start learning Higher Mathematics? [closed]

Question

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.

Answer 1

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

Graph theory by Sarada Herke

Short course on writing proofs in mathematics by Sidney Morris

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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.

Answer 2

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
   10. Beginning real analysis

Answer 3

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.

Answer 4

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.

Answer 5

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 :)

Answer 6

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.

Answer 7

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).

Answer 8

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

• http://ocw.mit.edu/courses/mathematics/
• https://www.khanacademy.org/math

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

Answer 9

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.