
Ask HN: How to self-study mathematics from the undergrad through graduate level? - hsikka
Hey HN community,
I&#x27;ve been looking to get deep and build my math skills from the foundation up. I have the time to dedicate to this endeavor and I&#x27;d love to hear if you have any specific resources&#x2F;curriculums you recommend.<p>Something like https:&#x2F;&#x2F;www.susanjfowler.com&#x2F;blog&#x2F;2016&#x2F;8&#x2F;13&#x2F;so-you-want-to-learn-physics would be ideal, but more focused on applied math.<p>One idea I had was to complete the MIT open courseware courses for the Applied and Pure math fields
======
abnry
Here's my personal opinion about how you should approach this.

It's all well and good to want to cover undergraduate math courses. When you
are actually enrolled in a university, you will have enough inertia and
motivation to complete the courses.

However, when you are self-studying you are doing it all on your own. It's
hard to be as thorough and cover everything.

And so I ask, what really is your goal here? You don't have to learn
everything about mathematics, because that is in fact impossible.

My advice is to FIRST construct a bunch of projects, tasks or goals that
require knowledge.

It could be something like (a) implement a machine learning algorithm to do X
from scratch (b) implement a simple physics engine (c) try to verify a number
theory conjecture (d) be able to solve all the exercises in a book (e) be able
to write up a compelling description/theorem/problem in math (d) numerically
solve the quantum mechanics equations of a certain system

Spend some time on material that will inspire you first to help get these
goals. Numberphile on YouTube, or any of Brady Haran's videos, is a good place
to start. But make the goals your own and make them personal.

Math is not a spectator's sport. Make sure to DO mathematics, not just LEARN
mathematics.

~~~
forkandwait
Personally, I think this is bad advice, because without an undergrad+
background, the projects above will either be impossibly frustrating or you
will make up some crackpot bullshit. Plus, the undergraduate curriculum is its
own reward.

~~~
tprice7
Not necessarily. I left undergrad after four terms and self-studied to the
point of having published research, giving invited talks, and even being a
visiting researcher for three months with my expenses paid.

Links:

[http://content.algebraicgeometry.nl/2017-2/2017-2-007.pdf](http://content.algebraicgeometry.nl/2017-2/2017-2-007.pdf)

[https://projecteuclid.org/download/pdfview_1/euclid.ecp/1508...](https://projecteuclid.org/download/pdfview_1/euclid.ecp/1508292095)

Look for my name (Thomas / Tom Price) on these pages:

[https://web.archive.org/web/20170121000748/https:/wwwmath.un...](https://web.archive.org/web/20170121000748/https:/wwwmath.uni-
muenster.de/u/deninger/about/)

[https://www2.math.binghamton.edu/p/seminars/arit/arit_spring...](https://www2.math.binghamton.edu/p/seminars/arit/arit_spring2016)

[https://www2.math.binghamton.edu/p/seminars/arit/arit_fall20...](https://www2.math.binghamton.edu/p/seminars/arit/arit_fall2016)

[https://www2.math.binghamton.edu/p/seminars/arit/arit_fall20...](https://www2.math.binghamton.edu/p/seminars/arit/arit_fall2017)

~~~
throwawaymath
Wow, that's remarkable.

As an aside, your work on numerical cohomology appears to have been useful for
a new result pertaining to lattices. Given the authors of the followup work
it's likely helpful for the study of lattices in post-quantum cryptography.

~~~
tprice7
Are you referring to the "An Inequality for Gaussians on Lattices" paper? They
cite my paper, but it's to give an example of an application of their result
(which I use), not because they built on it. But anyway, I think it's very
fascinating that the people who discovered a key result that I needed for that
paper, which could probably be best classified as arithmetic geometry, are
mainly computer scientists!

~~~
throwawaymath
Ah, thanks for the clarification. The computer scientists who work on quantum
computational complexity and post-quantum cryptography tend to be much more
mathematical than the norm :)

------
madhadron
There's a set of basics that you will want no matter which direction you go:
calculus/real analysis, linear algebra, differential equations/dynamical
systems, and sets, groups, rings, and lattices.

Calculus: learn to extract qualitative information about a function (it goes
up here, has a maximum there, goes down there, oscillates with an increasing
period, goes to this value at infinity...) and to numerically compute
quantitative information about it (its value at 3 is blah, its integral over
this interval is blah, its maximum value is blah).

Linear algebra: vector spaces and linear operators, and their representation
as vectors and matrices. Functions as forming infinite dimensional spaces, and
Banach and inner product spaces.

Differential equations and dynamical systems: extending what we did for
calculus to differential equations. Phase space, orbits, Fourier and Laplace
transforms, sets of linear differential equations, numerical integration, some
partial differential equations. You do not need all the little tricks for
special kinds of equations that you will find in, say, Boyce and dePrima.
They're not helpful.

Sets, groups, rings, and lattices. Mathematics today is written in terms of
set theory. You need to understand the basics of manipulating sets and
functions between them. Then you should know something about groups, rings,
and lattices, which are the most ubiquitously useful algebraic structures
besides vector spaces.

After that, where you go is going to vary enormously. Based on what you're
aiming to do.

~~~
oarabbus_
Most engineers I know have not learned set theory or groups/rings/lattices.
They still seem to be doing pretty well.

~~~
analog31
From what I've observed, most engineers are glad to be done with math when
they finish college. Most engineering is qualitative: Organizing and arranging
things, making things fit together, and troubleshooting. Maybe 10% of
engineering is quantitative, and that work often goes to the handful of people
in the department who have maintained an interest in it.

Some of the engineers who attract quantitative work are people who came from
outside of the mainstream engineering training, such as scientists and math
people.

~~~
you_owe_me
When you refer to engineers, do you mean actual engineers (BSc in
Engineering)? or your web designer with jquery skills who calls himself
engineer?

~~~
prions
In my experience (as a civil then software engineer with a MS in CS in ML/Big
Data) "real" engineering is much more prescriptive than software.

When you're designing a real-world engineering project, the entire
specifications are defined legally (through national, state and local laws)
and technically in manuals/books. Many engineering specifications will
describe the work done to a T before you even need to think about it i.e.
"water main shall be constructed of 12'' coated DIP at depth no less than 2
ft". A lot of the challenges are managerial and logistical.

All of this is on purpose. "Traditional" engineering disciplines are more
mature and have the constraint of being safe for the general public. There
isn't much room at all to creatively deviate from what's already specified.

I've found software design to be a lot more technically demanding in regards
to designing and building things. There's a lot less precedent, more moving
parts and many different ways to do one thing.

------
jasonmorton
Here is a proven approach for at least the first part, building foundations
and being ready for graduate work. Many Berkeley Ph.D. students passed through
this route. Get the book "Berkeley Problems in Mathematics." It contains
historical problems from the Berkeley math prelim exam, and solutions. Now
don't look at any solutions yet.

This is the exam all Berkeley math Ph.D. students must pass within three
semesters of arriving to stay in the program, and the fail rate is about 50%.

You will also need reference books, advanced undergraduate and beginning
graduate textbooks. Buy, download, or borrow as appropriate.

Pick a problem (start with the older ones, they are easier). Set aside 30-60
mins and try to solve it. No devices, no references at all, go to a library or
a coffee shop without your devices. Dont' give up till time is over. If you
cannot (usually the case), still don't look at the answer. Hit the reference
books (don't look up the problem online either, it will go right to the answer
and you won't learn much). Read and try to understand enough so that you can
solve the problem. It is ok if you solve it this way (in the course of reading
about it).

For bonus points, students studying for the exam will typically take entire
old exams (available from the Berkeley website), take that to the library and
just sit down for three to six hours and try to solve all the problems
correctly. Then self-grade harshly. When you can do that for a recent exam
(and get a good score), you will have more or less mastered undergrad math to
the point that you could teach it.

Most important: you have to struggle to solve problems. Reading a solution is
about as useful as watching someone else lift weights: you get minor tips on
form but not any stronger.

~~~
axiom92
Thanks for the excellent analogies!

------
hprotagonist
The hard but maximally useful thing to do, in my opinion, is to regularly meet
with at least two other people and a blackboard and beat your heads against it
together for about 3 hours at a stretch. Do this at least weekly--and
preferably more often. Self study in between meetings is obligatory.

One of those people should be at about your level. The other should be farther
along.

All three of you should trust each other enough that nobody gets caught in a
shame/guilt/ego loop.

~~~
karma_fountain
Just to add, it's also worth finding someone who might be a bit below you in
terms of maths ability. If you can explain the concepts to that person, then
you know the concepts.

~~~
hprotagonist
Yep. You can do this, if you're lucky, with the triad. You and the other
person at about your level can trade places depending on subject matter, each
explaining to the other and reinforcing their own knowledge.

------
netvarun
As an alternative I would suggest a top-down approach. Start with the
theorems/results you truly wish to understand and work backwards.

There was a great quote from an interview of Peter Scholze (one of last year's
Fields' Medallists), which has really changed how I view learning:

 _At 16, Scholze learned that a decade earlier Andrew Wiles had proved the
famous 17th-century problem known as Fermat’s Last Theorem, which says that
the equation xn + yn = zn has no nonzero whole-number solutions if n is
greater than two. Scholze was eager to study the proof, but quickly discovered
that despite the problem’s simplicity, its solution uses some of the most
cutting-edge mathematics around. “I understood nothing, but it was really
fascinating,” he said.

So Scholze worked backward, figuring out what he needed to learn to make sense
of the proof. “To this day, that’s to a large extent how I learn,” he said. “I
never really learned the basic things like linear algebra, actually — I only
assimilated it through learning some other stuff.”_

[1][https://www.quantamagazine.org/peter-scholze-and-the-
future-...](https://www.quantamagazine.org/peter-scholze-and-the-future-of-
arithmetic-geometry-20160628)

~~~
sweeneyrod
An approach might work well for a Fields medallist but less so for normal
people.

~~~
kowdermeister
I love generative art, so I quite often run into things like Lorenz
attractors. I looked up how they are made and that lead me to differential
equations which lead me to better understand calculus. So my love of pretty
graphics lead me to learning math with a purpose.

I haven't won any awards yet.

------
imbusy111
Look at any undergraduate course curriculum at your university of choice and
just follow that. Some of them might even have lecture notes and exercises
published.

As a maths undergraduate, I attended around 10 classes per year - you go see
who the instructor is and then just read the lecture notes at your own time at
home. Then go take the exam at the end and that's it. That sums up my 3 years
of undergraduate studies.

Mathematics is well established, essentially has been frozen at undergraduate
level for 50 or more years, so there is plenty of material. Also, you don't
need any equipment, just your own mind.

In my opinion, it is the easiest major of all if you can follow the logic. No
essays to write and no projects to do - just read the material.

~~~
hsikka
I was thinking about completing the MIT applied and pure math major course
requirements via open courseware

------
ghufran_syed
I did this a few years ago, now doing an MS in math, after last formally
studying math in high school, more than 20 years previously.

There is lots of advice in this thread, including long lists of books - at
your level, I think that’s unhelpful. To get started with the foundations, I
think it’s best to have one or two books at most, that you work through
_completely_ , _then_ decide what to do next. Otherwise you can waste a lot of
time time figuring out “what to do next” instead of actually doing math!

I would personally recommend “Engineering math” and “advanced engineering
math” by Stroud. The first covers “foundation” topics like algebra,
trigonometry etc, then covers the first year of math for a English engineering
degree. The second book covers the second year.

Note that English engineering degrees are 3 years vs 4 in the US, but have no
general education requirements, so the math covered in the two books is
roughly equivalent to the 1st-3rd year of an engineering degree in the US.
Both books are very much focused on calculations, not proofs, but will give
you a fluency in handling mathematical calculation that is _assumed_ when
doing more proof-based courses later. And they are both designed for self-
study, which is important.

The other book that I would get, and do in parallel once you have done the
foundation part of the first stroud book, Chartrand, Mathematical proofs. This
will teach you how to do proofs, using mainly (high school) algebra and basic
number theory to start, but going on to cover some proof techniques in
analysis, advanced algebra and others areas.

Once you have done those 3 books, you will have a solid basis for further
study. I also agree with the comments about interacting with others studying
the same material if possible.

Happy to chat sometime if you want to message me (contact info in profile)

------
nicoburns
The University of Oxford has all of their notes and exercises available online
[https://courses.maths.ox.ac.uk/overview/undergraduate](https://courses.maths.ox.ac.uk/overview/undergraduate)

If you get stuck you could try asking on
[https://math.stackexchange.com/](https://math.stackexchange.com/)

Although, if you can, finding someone else to work with, and someone else who
already knows some math to ask occasional questions would probably help you a
lot.

~~~
throwawaymath
Many of the courses listed there have no course materials available or only
very incomplete materials.

~~~
nicoburns
Oh really? The ones I cliked through to did!

------
nobody271
Hello, I have actually done this. I learned Algebra up to a good amount of
Vector Calculus over the course of four years mostly through self-study.

I would leave for work an hour early and either sit in my car or go into a
Starbucks and do math. Doing time before work is important. That's when you
are at your best. Then after work I would sit in my car and do math for an
hour. Then on the weekends, in the morning, I would do three hours of math
straight, sitting in my car. Then sometimes in the evening I would do a first
pass over a section that I knew I would have to really think hard about the
next day that way I had sort of taken the first layer of difficulty off.

Why in my car? Because I simply cannot sit at home and focus enough to study
math. I've tried the library but it's too restrictive on what you're allowed
to do "oh f __*! that 's how you do that" doesn't generally go over well in
public places. Plus I feel very safe in my car. I can be relax which makes
learning a lot easier.

I always take one day off a week (for me that's Tuesday) but aside from that I
don't skip.

Get your books at half price books. Math book reviews on Amazon are almost
always wrong. I usually have to try two or four books before I find one I can
understand. I would just get a book and read it in your car. You don't need
the pressure of following along with a schedule where you fall behind and miss
out on a topic. You also don't want to miss a topic just because the teacher
chose not to include it. Some of the coolest topics/examples don't get covered
when you take a course because they don't have the time.

Also, when you get done with a book go through it again and take good notes on
note cards that you can review on your drive to work or while you are waiting
around. You'll find yourself going back to them over and over again.

~~~
commandlinefan
Bet you weren't married at the time. I'm trying to re-learn calculus myself,
and I have to hide it from her because she gets mad at me when I try to do
calculus problems: "why are you doing this? Are you doing this for work? You
don't have to do this. There's no reason for you to be doing this."

~~~
maxxxxx
I hope you are kidding to some degree. Having to hide things you want to do
from your spouse doesn't sound like a good relationship.

------
alnar
SOME ADVICE BEFORE YOU DO A DEEP DIVE INTO WHATEVER YOU END UP STUDYING MATH-
WISE:

My Background: Current Undergraduate in CS and I recently added Mathematics

The most difficult part for a person who hasn't done a lot of math to become a
person who does a lot of math is to read and understand rigorous proofs. You
will encounter countless difficult proofs in any mathematical topic you try to
study. Read a few books on mathematical thinking and proof techniques
before/during/after reading any other dense math book.

Like you, I realize the value of having a mathematical mindset and want to
have a deep understanding. When I added math as a major, I had a very hard
time jumping from computational courses (typical math courses, geared towards
any major) to theoretical and conceptual courses (proof-based courses that use
all the fun and interesting math books everyone has linked here). These books
helped:

[https://www.amazon.com/How-Read-Proofs-Introduction-
Mathemat...](https://www.amazon.com/How-Read-Proofs-Introduction-
Mathematical/dp/1118164024)

[https://www.amazon.com/How-Think-Like-Mathematician-
Undergra...](https://www.amazon.com/How-Think-Like-Mathematician-
Undergraduate/dp/052171978X)

[https://press.princeton.edu/titles/669.html](https://press.princeton.edu/titles/669.html)

 __< 3 this is a great book, obvi since its george polya __

~~~
oefrha
> The most difficult part for a person who hasn't done a lot of math to become
> a person who does a lot of math is to read and understand rigorous proofs.

I’m afraid you haven’t delved into any advanced topics. That’s actually about
the easiest part, and could be mastered by ten year olds (certainly myself
when I was ten).

~~~
aoki
alnar is likely referring to the US system, or something like it: unless you
are tutored externally (rich), an autodidact outlier (gifted), or selected for
honors courses, you basically take computational courses for 14 years (with
one cursory stop for euclidean geometry) and are then thrown into proofs at
the age of 19-20, if at all. it's widely recognized as a problem in the math
pipeline, which is why many US universities have "transition" courses for non-
honors students.

so it's not that the basics are intellectually difficult as much as
practically difficult (unfamiliar, disorienting) for many students. many
"transition" books talk about the difficulty in adjusting from talent being
redefined from perfectionist "plug and chug" (APs, SATs) to reasoning and
creativity.

btw, i'm impressed that you could master college-level proofs at 10. i have a
kid about that age who is pretty good at logical reasoning, but i'm not sure
what topic (at that level) he could _do_ a rigorous proof about; maybe
numbers, as in landau? can you say more about the materials you used?

~~~
oefrha
This has little to do with education systems, I was talking about objective
difficulty. Try to tell any mathematician that the hardest part of math is
rigorous proofs; if they don’t laugh in your face, they are just being polite.
One may think the transition is “hard”, until one actually gets into more
advanced topics. As you said, it’s just unfamiliar to the uninitiated, at
best.

Reading and presenting rigorous proofs in elementary number theory, Euclidean
geometry, etc. is easy for gifted ten year olds and definitely manageable for
a lot of fifteen year olds. You asked about material — it doesn’t actually
matter, and I don’t recall specifics; any entry level treatment of elementary
number theory should do (really beautiful subject with a very low barrier of
entry). For kids who have eyes on IMO, it’s very common to be throwing around
perfectly rigorous proofs at young ages.

------
gmatbarua
IMHO the books being advised here are too high level. Its like trying to read
about Computer networking basics from RFCs. Some advise are equivalent to
searching for a reason to learn "C" by appreciating what can be done with "C"
by reading books on Linux Kernel programming.

I will mention a few books which by no means are undergraduate level, but are
per-requisite for undergraduate level study. Are you thorough with books given
below? These books are available through
[http://gen.lib.rus.ec](http://gen.lib.rus.ec).

1\. Higher algebra - Hall and Knight 2\. Trigonometry I - Loney 3\. Coordinate
geometry I - Loney 4\. Calculus of one variable - Maron

Download the books and check if you already know all those. Once done, then go
for the higher level books most the people advising here.

------
padthai
If we are talking about content you can find plenty of advice here or
elsewhere. But do yourself a favor and pay a tutor and/or find a study group.
As in writing, dancing, etc. you cannot evaluate your own work good enough.

You will not improve your math watching youtube videos and reading books. You
need to produce stuff that pass the "sniff test" to your colleagues.

~~~
mindcrime
_You will not become good at math watching youtube videos and reading books._

That depends on exactly what you mean by "become good at math". If you're
talking about "becoming a mathematician" and doing original research in pure
math, then you're probably right. But if one means "learning existing math
well enough to apply it to a problem", I would argue that one can learn this
stuff just using books, videos, etc. At least up through a certain level.

That said, I do encourage the idea of finding peers to work with. I used to
coordinate a "math study night" at the local hackerspace for that exact
purpose. It fell off because I got busy and couldn't keep committing to it,
but generally speaking, it is a good idea to have other people to work with. I
may well try to find a math major from UNC to hire as a tutor at some point as
I keep working on this stuff.

And if one can't find somebody to work with in person though, and they need,
say, a proof evaluated, there is the option of using math.stackexchange.com or
the like.

~~~
ducttapecrown
I'm sure padthai meant that you need to do math to learn math, not just read
or watch videos.

~~~
mindcrime
_I 'm sure padthai meant that you need to do math to learn math, not just read
or watch videos._

Sure, I'm just saying that it kinda depends on what part of math one is
referring to. I think sometimes in these discussions on HN, we overload the
term "math" to mean both "calculation" or "applied math", and "pure math" or
"math research" and it can be unclear which is being referred to in a given
statement.

I believe you can learn the former - "applied math" \- (at least up to a
certain level) just by reading books, and watching videos (and doing
exercises, of course). But for the latter - "pure math" \- I agree that you
need other people, since you can't easily verify your own proofs.

------
calebm
I've been following a breadth-first-based-on-my-interests approach for the
last several years. It's not been very systematic, but it's been a lot of fun.
I think math is essentially clear thinking, and it's best when it's explored
from curiosity.

A couple resources I would recommend would be:

* A book called "Who Is Fourier? A Mathematical Adventure", which touches on a pretty good variety of math topics. It's aimed at children, but it's probably the best math book I've ever seen. It's the only math book I've ever read cover-to-cover.

* 3Blue1Brown's youtube channel: [https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw](https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw) (as well as other math videos)

~~~
mywittyname
3B1B is amazing at explaining concepts. Lots of commenters in his videos
studies math/cs/physics and tell stories about _finally_ understanding a topic
they misunderstood in university.

His videos, I think, are best consumed twice: once before you explore a topic,
so that you go in with a solid overview and know the key things to look out
for; and again after you've studied in depth, when you are able to predict
what he will say next, which reinforces you learning.

------
drablyechoes
It is really hard to truly self-study in mathematics. Going through the
opencourseware and reading textbooks (working as many exercises as you can, of
course) will get you only so far. It is important to have someone (ideally
with a PhD-level education in mathematics) who you can meet with to guide your
study, correct mistakes and answer questions.

~~~
gnulinux
I strongly disagree actually. I think mathematics is the best field to self-
study. There is nothing in mathematics that cannot be explained on the paper.
There are many textbooks that are very good that in most universities
professors won't be that good anyway. I went to UC Berkeley to study
mathematics (ended up studying CS though) which is supposed to be a top
department, but most of my textbooks were better teachers than my professors.
I still prefer reading textbook to people explaining me math. I don't even
think I understand math when people explain me. I need to first teach it to
myself. Then occasionally people offering different perspectives is very
beneficial, which, again, can be done on paper.

I don't think you need a PhD level educator. You need mathematical maturity.
Mathematics follows a very specific logical structure that needs you to shift
the way you think. Human brain simply doesn't work the way mathematics needs
it to work. But this is a constant time overhead. Once you understand how to
approach mathematical problems, I can't see why you cannot learn everything
from a textbook.

~~~
jack_h
As someone who did study both math and CS at university I disagree. I think
there are numerous courses that can more easily be self-taught[1], mostly what
one would encounter in their first ~2 years in a math degree. After that
things get conceptually a lot more difficult.

For me personally I didn't really need an instructor for most of my calculus
courses, or ordinary differential equations, or most of the linear algebra
stuff. It was a bit more difficult around real/complex analysis, non-linear
dynamics, and courses of that nature. The classes that taught me the value of
having an instructor were abstract algebra and topology. Those were such a
massive shift away from what I had perceived math to be that an instructor
being able to impart intuition, correct my own incomplete or incorrect
assumptions, and generally just help guide me to a different mode of thinking
was invaluable.

The problem with books/texts in this instance is they are not reactive, they
have no idea what you're thinking and can't steer you in the right direction.
Worse is that as the person trying to learn the subject matter you don't know
where to look to get on the right track and correct your own assumptions
because you don't know enough yet.

Now I'm not saying you need an instructor per se, but having some place to ask
questions where someone far more knowledgeable than you can help might be a
good substitute. I'm sure there are some websites like this, although I don't
know of any since I graduated a long time ago.

[1]This does somewhat depend on a person's skillset going into this.

~~~
asdffdsa
The reality is that having a great mentor is a privilege, not a given. I don't
think anyone is arguing that having a genius and brilliant teacher wouldn't
help, rather that it is possible to reach an "advanced" (grad/undergrad level)
understanding of mathematics without the luxury of having someone who is far
more knowledgeable to turn to for help.

~~~
gnulinux
Precisely. GP here. I didn't argue against having a great mentor. If you go to
a great university you clearly have an advantage. I think it's perfectly
doable to teach yourself mathematics if you study intense enough eith correct
tools.

------
samasblack
A nice foundational textbook to start with is Michael Spivak's text
"Calculus." This text is fairly conversational, it motivates its concepts well
with many examples, and it will help you build a strong foundation in writing
proofs and reasoning mathematically.

As others have said, it might be helpful to have a friend or two read with you
so you can answer each other's questions. Also, make sure you do a ton of
exercises!

------
sedeki
Check out Open University ([https://www.open.ac.uk](https://www.open.ac.uk)),
they have a undergraduate (BSc) up to graduate (MSc and PhD) level programmes
in Mathematics, and they take distance-learning seriously.

That’s what I would do if I were you. Learning this stuff on your own will be
very difficult. I can’t even imagine the amount of discipline needed...

------
DoreenMichele
I was a homeschooling parent. I also have a stronger math background than
average, though it isn't anything impressive for this crowd.

I will suggest you try to find a written curriculum from a respectable source
as your first step so you have some idea what you need to cover to meet the
stated goal.

Then you will want a variety of study materials for the subjects in question.
You will want to verify that these are respected materials that are not full
of errors.

When I brushed up on math to CLEP Alegebra so I could take college statistics,
the computer program I used for self study had errors in it. I knew enough
math that I knew what was wrong. I was just rusty and in need of a bit of
practice. But it was a program I did not use with my sons to homeschool them
because of the errors.

You want to know which materials math geeks like. You want them vetted,
basically.

Then use whatever quality materials most appeal to you. Different strokes for
different folks and self study let's you pursue whichever materials you like
the best.

------
nabla9
You need to do lot's of exercises.

Shaum's Outlines has lots of cheap math books with exercises and solutions.
You can use them to complement other books

[https://www.mhprofessional.com/9780071623667-usa-schaums-
out...](https://www.mhprofessional.com/9780071623667-usa-schaums-outline-of-
advanced-calculus-third-edition-group)

[https://www.mhprofessional.com/9780071635349-usa-
schaums-300...](https://www.mhprofessional.com/9780071635349-usa-
schaums-3000-solved-problems-in-calculus-group)

[https://www.mhprofessional.com/9780071383929-usa-schaums-
out...](https://www.mhprofessional.com/9780071383929-usa-schaums-outline-of-
calculus)

[https://www.mhprofessional.com/science-
math/math/topology](https://www.mhprofessional.com/science-math/math/topology)

------
andypea
If you want to start with a single self-contained book, then a really good
choice is "Modern Mathematical Physics" by Peter Szekeres:

[https://www.cambridge.org/core/books/course-in-modern-
mathem...](https://www.cambridge.org/core/books/course-in-modern-mathematical-
physics/E899DB30C574E2F4D7C861B3097F9813)

It covers most of the mathematics needed for quantum field theory (which is a
big chunk of applied mathematics) starting right from set theory.

------
natalyarostova
Lots of good points here. I can't speak to the graduate level, but for the
undergrad level I think the trick is to find textbooks that don't suck (
_most_ good textbooks are free, or are old enough no one cares about
copyright, and a million pdfs are online, or you could get it used for $15.
Most bad textbooks are price gouging undergrads).

Then just put the time in with deliberate practice. I tried self-studying math
for a year, and often skipped the harder practice problems, thinking just
reading the textbook and examples is sufficient. That's actually entirely
useless. You are't learning math (or anything worth learning) unless you spend
6 hours on a Saturday working through some hard problem that makes you so
frustrated that you ask God why he made you such an idiot. If you spend enough
time doing that you can actually learn something.

------
hellerup89
The obvious route for self-study is to go through Halmos or Axler and Rudin,
plus all additional materials and study aids such as Gelbaum & Olmsted,
emulating the most popular Harvard Math 55 incarnation. Another nice Math 55
incarnation covered Hubbard & Hubbard, which is a wonderful book.

Although Math 55 is tough, if you are self-paced and have a bit of
mathematical maturity I think it is doable. It's also an excellent pure math
bootcamp that gives you a solid foundation to branch into any other pure or
advanced math topic.

I have gone through Halmos & Rudin myself, and it is a great experience.
However, if your end goals are more geared towards pure CS, an alternative
route might be much more appropriate. Very interesting and promising parts of
CS, such as formal methods, and the foundations of mathematics themselves
depend on abstract algebra and logic:
[https://ncatlab.org/nlab/show/computational+trinitarianism](https://ncatlab.org/nlab/show/computational+trinitarianism)

A minor problem is that beginner literature is not so polished as it is
relatively young. But there are some excellent textbooks nonetheless. Some
below. Other suggestions welcome:

* [http://www.cs.man.ac.uk/~pt/Practical_Foundations/](http://www.cs.man.ac.uk/~pt/Practical_Foundations/)

* [https://www.mta.ca/~rrosebru/setsformath/](https://www.mta.ca/~rrosebru/setsformath/)

* [https://github.com/ademinn/ttfv/blob/master/2006.%20Sorensen...](https://github.com/ademinn/ttfv/blob/master/2006.%20Sorensen%2C%20Urzyczyn.%20Lectures%20on%20the%20Curry-Howard%20Isomorphism.pdf)

* [http://www21.in.tum.de/~nipkow/Concrete-Semantics/](http://www21.in.tum.de/~nipkow/Concrete-Semantics/)

* [http://adam.chlipala.net/frap/](http://adam.chlipala.net/frap/)

* [https://softwarefoundations.cis.upenn.edu/](https://softwarefoundations.cis.upenn.edu/)

------
scarecrowbob
I had math through some introductory calculus when I was an undergrad 20 years
ago, but I let my skills lapse. But I want to understand enough math so that I
can do some more complex electronics projects and some statistics / ML /
intelligence analysis. This level of math, as I recall, is more or less the
math core of the undergrad engineering program I dropped out of in favor of a
philosophy degree (cause I am dumb as dirt).

In March of '18 I started doing lessons on Khan and just played around until I
couldn't do the problems easily, and for me that was, like literally adding
fractions and using exponents. So I had pretty basic skills at that point.

At thus point, I'm finishing the unit on using derivatives to optimize
functions around min/max. Not a big deal, but a long way from where I started
10 months ago.

I've had a lot of luck with:

a) khan academy

the lessons are very simple and well broken up, the teaching is interesting,
and the site is gameified in a way that is rewarding

b) doing it as close to literally every single day as I can manage

Math fluidity feels (at least to me) very much like my fluidity with music
theory or programming. As such I need to do it regularly. Even if I don't get
all into a flow state about it (which, I think is necessary on the scale of
any given week), I do need do it a little bit every day... that both keeps me
doing it and keeps it in the forefront of my mind.

c) keeping a notebook

I do all my work in a single notebook and use it to both track my progress and
as a reference. It's also been neat to see how far I've come.

-=-=-=

I dunno if any of that is useful to anyone else. But I feel like I've had a
lot of luck educating myself in math. At this rate, I should be through
integral calculus by the end of the spring and through the linear algebra
class by fall, and then I will move my deep, long-term learning projects over
to something else, hopefully a deeper dive into electronics design.

~~~
narrator
One great thing about Khan Academy is the gameification aspect makes sure that
you go over older material in a spaced reptition sort of way which helps
really drill in the knowledge so it isn't forgotten

------
whitepoplar
This book is fantastic and pretty much takes you through an entire undergrad
mathematics course: [https://www.amazon.com/Mathematics-Content-Methods-
Meaning-V...](https://www.amazon.com/Mathematics-Content-Methods-Meaning-
Volumes/dp/0486409163)

~~~
throwawaymath
The topics covered in that book are undergrad level, but that book is not
suitable for learning the topics. It’s more like a high level discussion of
the topics looping them together with historical background. It’s more
appropriate for people who are already familiar with the material.

------
ollerac
I discovered this amazing YouTube channel recently that I wish I had access to
back in college. It explains some very difficult math concepts in a very
simple and easy to understand way:
[https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw](https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw)

~~~
porpoisely
It's amazing how seeing something visually or having someone else solve
something can aid in your understanding of a topic. I remember having the most
difficult time understanding mergesort from my book's pseudocode and my
professor's lecture. But seeing mergesort in action on youtube revealed what
it truly was and made what seemed impossibly complex, so simple and elegant.

------
sandGorgon
Terry Tao's post - [https://terrytao.wordpress.com/career-advice/theres-more-
to-...](https://terrytao.wordpress.com/career-advice/theres-more-to-
mathematics-than-rigour-and-proofs/)

[https://www.theatlantic.com/education/archive/2014/03/5-year...](https://www.theatlantic.com/education/archive/2014/03/5-year-
olds-can-learn-calculus/284124/)

Math is not linear, So why do we teach math in hierarchical steps? -
[https://prezi.com/aww2hjfyil0u/math-is-not-
linear/](https://prezi.com/aww2hjfyil0u/math-is-not-linear/)

[http://ocw.mit.edu/resources/res-18-001-calculus-online-
text...](http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-
spring-2005/textbook/) highly recommended for Calculus

------
febin
Hey, I am also on the same boat. I have been trying different things lately. I
have gone through a lot of courses and a lot of books, but most of these
experts have the curse of knowledge. They don't know how it is like to not
know. They skip things that seems obvious to them. But, for begineers not
having such information can be frustruating and feels like the author/lecturer
is moving fast.

I recently stumbled upon this book "The Language of Mathematics: Utilizing
Math in Practice" I have been reading this book for a while and the author
takes a different approach in teaching maths. I am slowly beginning to learn
how to forumulate a real world problem into mathematical model and then solve
it. More importantly, I am able to read and understand mathematical
expressions better through this book. This book has raised my hopes in
learning and apply math. I would recommend you to give it a shot on this
before you go through any other courses.

------
forkandwait
I suggest you get the syllabi from a state college, and then follow that.
"work through" the books : copy each page of exposition by hand, try to do the
examples and proofs before looking, do all or most of the problems with
answers, and get ready for a long journey.

I find a page of math textbook takes about 30 minutes (at least) to really
work it over until you understand it.

Personally, I don't much like working in groups or watching video lectures. To
learn math, you have to be able work problems by yourself, so you might as
well just do that.

David Morin has some great self study books:

[http://www.people.fas.harvard.edu/%7Edjmorin/book.html](http://www.people.fas.harvard.edu/%7Edjmorin/book.html)

I have a math BA from a regional state college, usually scored in the top 5 on
the tests, and I am self teaching physics / me /ee about 5 hours per week
after work.

------
markkm
Don't study "math." Find a topic or a problem you'd like to understand, look
up the prerequisite knowledge for it, and start there.

You need to know what you want to do before you go looking for resources.
There is no set agreement on what should constitute an undergraduate applied
mathematics curriculum, and you are likely to get lost in the deluge of
conflicting information. On the other hand, the undergraduate pure mathematics
curriculum has been more or less stable for half a century. Any college
curriculum will do at this point, and many are freely accessible online.

Either way, there is no shortage of information and resources available. Any
topic you'd choose as a layperson likely already has a course or a seminar
covering it, and the corresponding syllabus should give you what you need.

------
burtonator
I'm actually building an app specifically for this use case:

[https://getpolarized.io/](https://getpolarized.io/)

Polar is basically a personal knowledge repository.

The idea is that you use Polar for all of your education. Either official
education or continued learning.

Every textbook or technical paper should go into Polar.

You can then annotate the textbooks directly including text highlights,
comments, etc.

It supports spaced repetition systems like Anki so you can continually review
what you've learned and NEVER forget anything.

It doesn't make sense to pursue a PhD or spend 6-12 months researching a topic
to just forget 80% of it due to bit rot.

The reason I named it Polar is that it's meant to 'freeze' all of your
knowledge so that you never forget it - ever.

~~~
umanwizard
Learning math isn't really about memorizing things.

~~~
aoki
very true, in the sense that rote memorization is not the point.

but false in the sense that doing math requires fluency - in applying a small
amount of technique up to lower division math, and in applying a large number
of definitions/results after that. in areas like abstract algebra, failing to
memorize will kill you. it is as disfluent as writing text in a foreign
language without having memorized the working vocabulary and grammar; even if
you can eventually recall, derive or look up what you need, it's just not
practical to work like that. (and i say this as someone who prefers to
mentally index information rather than memorizing...)

~~~
oefrha
I have to disagree, especially about abstract algebra. Most concepts and
theorems feel like abstract nonsense (not specifically talking about category
theory here) when you don’t understand them, but should become pretty natural
once you do. For true mastery you need to work with the concepts and results
on a day to day basis for a while, by applying them; continually reading the
text of definitions and theorems hardly helps if at all.

~~~
aoki
i agree that memorization by internalization (concepts) is different from (and
superior to) memorization by rote (text); i was _agreeing_ with the other
fellow that rote memorization is not the end goal. however, to me they're both
"memorization" because they both represent work to achieve fluency in
application.

however, i made my comment because i think i disagree that spaced repetition
has no place in learning. i think that if you dig around in the Polar guy's
earlier comments, you'll find threads where folks like michael nielsen are
talking about using spaced repetition tools for much more than purely textual
memorization of theorems - basically, cycles of repetition and (re)synthesis.
so i don't feel it's right to completely shut him down about card decks. you
may disagree, of course.

~~~
oefrha
When your definition of memorization encompasses all forms of learning, saying
memorization is crucial to learning is pretty much a tautology. No one claimed
that amnesia sufferers are perfectly good for mathematics. Same goes for
repetition. No one expects you to understand and never forget the theory of
cohomology by writing down a long exact sequence once.

The thing is, the root comment of this thread specifically talks about
"continually review what you've learned and NEVER forget anything" through
looking at highlighted notes in the software mentioned (a somewhat out-of-
place plug, I'd say). That's not how math works. You refresh your memory by
tackling preferably new problems. Reciting proofs is largely pointless (except
for certain very elegant proofs, in which case you probably won't need to
recite them anyway); reciting definitions and theorems is even less useful.

> it's meant to 'freeze' all of your knowledge so that you never forget it -
> ever.

Yeah, no, you don't "freeze" your mathematical knowledge.

------
hyperpallium
Khan Academy has multivariable calculus, differential equations and linear
algebra --- with many exercises, solutions, and worked solutions (under the
"hints" link after a problem), and gamification as a motivation aid.

The videos are above average, but seem about 5x too slow on the steps, then
rushing through the interesting stuff almost too quick to catch: boredom
punctuated by rewinding. Probably because it targets highschool kids.

BTW You may find you need to cover even earlier material anyway (I did), to
refresh and fill gaps. You can cope with some gaps and hazy recall, until
there are too many... and some gaps cannot be filled piecemeal (because you
don't know you have a gap).

------
ianai
I don’t follow curricula online. But you should be able to get student and
teacher copies of textbooks. Hold yourself firm to attempting the problems
before looking at solutions to simulate homework. To simulate lecture, pick
1-2 of the tougher problems and work through the solution for them. Then try
your homework out.

What I love about math is that this is completely do-able. Good luck! Also,
definitely get a dry erase board. There’s something about having a large space
that can be easily erased that helps working out problems.

Edit: deleted an m

------
thanatropism
I have a Masters degree in mathematics and struggle so much to continue
learning in my (relatively ample) free time. I was never good at self
evaluating my proofs so often I solve a problem and have to scrutinize it for
nontrivial steps made "intuitively".

It's like when you have Calculus and over generalize when entering Analysis or
Differential Geometry -- but now I have more structural patterns to
extrapolate from :/

------
gmiller123456
First I think you need to think about a direction. "Learn Math" is a very
broad and vague statement. You will never learn all of math. So you really
need to think about why you're wanting to learn it, and pick a direction that
suits that.

If you're wanting to learn math just because you like the subject, then it's
wise to get a sampling of a wide array of different areas. Then you have a
better chance of finding an area that you like best.

If you're goal is to learn it so you can apply it in some other area, you
should focus mostly on learning what math applies in that area. Then find the
prerequisites and start studying.

Studying alone in your spare time will be nothing at all like attending a
university. The full immersion you get to focusing primarily on one thing, and
the ability to engage others at will also focusing on the same thing is a
great help in learning. So set your expectations right. You won't be able to
reach the level of understanding a 60 year old math professor has. Since
he/she has been fully immersed in the topic for probably 40-50 years. So set
your expectations accordingly.

------
BinaryMachine
I actually started from the very beginning of mathematics on Khan Academy,
thats starting from pre-school going through every video and exercise, taking
the quizzes and each end test for each subject. I am currently almost done
with Algebra 1 and has taken longer than expected. I highly recommend! The
exercises are key and do everyone until you get 100% then move on to the next
topic/subject.

------
kkmazur
There are some awesome resources on github on many topics including math. Just
search for "github awesome-X" like [https://github.com/rossant/awesome-
math](https://github.com/rossant/awesome-math)

And here is a root project for all resources
[https://github.com/sindresorhus/awesome](https://github.com/sindresorhus/awesome)

Personally I'd like to have something like a learning path. Not only a list of
resources/topics but also some guide how to approach the learning process.
O'Reilly has something like this but it's not mature yet
[https://www.safaribooksonline.com/learning-
paths/](https://www.safaribooksonline.com/learning-paths/) in my opinion.

MIT OCW has some guide for prequesities [https://ocw.mit.edu/courses/mit-
curriculum-guide/#map](https://ocw.mit.edu/courses/mit-curriculum-guide/#map)

I hope that helps you

------
gumby
This is really great.

One specific point:

> One idea I had was to complete the MIT open courseware courses for the
> Applied and Pure math fields

A note about Open Courseware: they tend to be much of or all of the material
handed out in a course (lecture notes, problem sets, etc). They aren't a
"course" in the sense of Coursera, Khan Academy and the like.

A few years ago I needed to brush up on my thermodynamics and was able to read
the material for the same course I'd taken as an undergraduate (well, same
course number; I took thermo in 1983, and the prof and some of the material
had changed). This was merely an undergraduate thermo class _that I had
already taken 30 years before_ and it was still quite hard.

I don't mean in any way to discourage you!!! This is an excellent idea. But
OPenCourseware itself is more like a box of legos for someone who wants to
teach a class in the subject. You may find a different source better,
especially for topics that are new to you.

~~~
hsikka
Ah very good points, thanks for the heads up! I’ll think about this a bit more

------
jordigh
One thing that worked for me was to have a _community_ of fellow
mathematicians to study with. There's a lot of cultural stuff that is kind of
hard to osmose from books, such as for example, how to _pronounce_ things that
you read.

Mathematical notation is meant to be read out loud. It's shorthand for
English, or whatever your natural language is. You should be comfortable
seeing a sigma sign and thinking in your head, "the sum as ehn goes from one
to infinity of eff of ecks squared times ..."

That's just the basic part of it. There's lots of cultural shorthands that are
just kind of osmosed, seldom overtly acknowledged (for example, the precise
meaning of "without loss of generality"). Maybe you can acquire them by
watching videos, but if you can find people that you can hang out with that
can teach you the cultural aspects of mathematics, that would help a lot.
Picking up a book and reading and working through it and understanding it gets
a lot easier once you have the culture down.

~~~
throwawaymath
That is also part of the loosely defined term, “mathematical maturity.” When
you’ve reached it, you can generally grasp the basics of unfamiliar math
quickly. Failing that, you can find out what you need to study to understand
it and do so on your own.

You don’t typically obtain that kind of maturity until graduate school, maybe
upper undergrad if you’re quite good.

~~~
jordigh
Yeah! There's that whole thing where you understand how a work is structured,
you know what part to pay attention to, what part to skim or read later, where
you're required to do your own calculation to fill in something...

People complain a lot that books and presentations don't do everything and
that mathematics has a bad "user experience". They may have a point, but the
complaining alone won't fix it and will leave most people still feeling like
frustrated outsiders.

------
badatshipping
I disagree with the common advice that before you study math you should try to
develop love for it or find an application you care about. If you don’t know
any math, how can you know what you like or want?

What I think one should do, initially, is try to be better than other people
in some way. That’s a motivation everyone has accessible. One way is what
you’re doing already — trying to learn math on your own. That’s pretty cool.

From there I’d use Reddit and Quora to build yourself a curriculum. Most
people start with calculus, linear algebra, differential equations, and real
analysis. Look for textbooks that are known to be pedagogically well-written
for self-study (rather than intended to be used as a manual in conjunction
with lectures). Skim the table of contents and the first chapter of several
books to get a feel for which ones you’ll be able to stand going through. I
like Spivak, Arnol’d, Axler, and Pugh.

In each subject, find a book full of problems, and do the hardest ones in each
section.

------
rwhitman
I started doing this, and the first stumbling block I hit was simply assessing
at what level my math education falls apart and figuring out what courses I
needed to refresh with, what materials I needed etc.

There's a market opportunity for building a math placement test product, or
even better a fully automated math education platform with examinations at
each stage, that takes you from K-12 refreshers all the way up to grad school.
Something a bit more advanced than the existing online learning programs that
are still very much based on the college lecture model.

There's a lot of products out there that do this for software development, but
not for other subjects. I have noticed there are bits and pieces I can
download as mobile apps, but they are designed to be supplements to certain
college curriculum rather than an end-to-end education.

Obviously math is a subject where software is an optimal instructional tool,
so much more could be done here

------
ivan_ah
These concept maps might be helpful as a general overview of the basics:
[https://minireference.com/static/tutorials/conceptmap.pdf](https://minireference.com/static/tutorials/conceptmap.pdf)

They are extracted from my concise books for adult learners on MATH & PHYSICS
[https://minireference.com/static/excerpts/noBSguide_v5_previ...](https://minireference.com/static/excerpts/noBSguide_v5_preview.pdf)
and LINEAR ALGEBRA
[https://minireference.com/static/excerpts/noBSguide2LA_previ...](https://minireference.com/static/excerpts/noBSguide2LA_preview.pdf)

I don't want to self-promote too much here, but maybe HN users who have read
the books can add a comment to say what they thought.

------
orbifold
I think it is pretty much impossible to learn mathematics without a teacher
beyond a certain point, multiple teachers are ideal. The reason for that is in
order to do mathematics you need to solve problems and develop intuition. It
will be very hard for you to solve problems (write proofs) without someone
that evaluates the quality of your solutions, and it is very easy to deceive
yourself that your reasoning is without gaps without experience. The same is
true for intuition, without a master at whatever field you want to study it
will be hard to work through most books (Hartshornes, Algebraic Geometry comes
to mind. His seminars and lectures contained plenty of simple examples. The
book itself is famously an uneasy compromise, which no one was really happy
with but also with few better alternatives)

------
wojjk
When you first start out in math you won't even know the basics like e.g.
"proof by contrapositive" but a basic "proof course" perhaps accompanied by
"Linear Algebra by Hoffman and Kunze" will get you pretty far.

You can never ever ever ever learn too much linear algebra. It is pivotal in
the development of e.g. "field extensions" and "differential geometry" to name
two random items.

That said, partway through Hoffman you should start an algebra book like
Dummit and Foote so you can see groups and rings.

*After you know how to prove things you could read an analysis text instead of Linear Algebra

Lastly, any serious math requires some topology, but it's less about what a
topology is and more about knowing how to quickly apply the basics so certain
statements are easy to state/prove/think about

~~~
wojjk
Lastly, the further you go the more you'll see that there are no islands in
math. Everything uses everything else. (For example, you can go back and forth
between groups and topological spaces via e.g. the fundamental group)

------
scottlocklin
Just take community college courses. Work lots of problems. Ignore the grand
theorems unless you're going to be a math major (in which case you should go
to college).

Applied math; if you deal with matter it will generally be linear algebra and
differential equations. Vector math important also, and Calc-3 is inadequate;
junior level classical physics mechanics book is how I learned stuff like
action angle and rotating frames of reference.

If you deal with electronics/signal processing you'll need some kind of
Hilbert space course to get you through Greens functions, Laplace transforms
and so on.

And for computer science/machine learning/OR: just be really good at linear
algebra.

(Applied obv) linear algebra is the highest leverage thing you can do.

------
vladislav
I don't at all recommend limiting yourself to self-study when you start this
endeavor, it's too daunting given all the unknown unknowns. To start with it
helps a lot to take courses, even if it means sitting in/enrolling in night
courses at the local community college. It provides a reference point to what
others understand, allows real-time querying with an instructor, and is more
motivating than self-study. Once you have completed and fully grasped a
difficult proof-based course (real-analysis, algebra, etc), then you can
possibly embark on a successful self-study. I'm not saying it can't be done,
but you will get to mathematical maturity faster this way and there will be
less risk of early abandonment.

------
misiti3780
We need more info:

How much math do you currently have, which courses have you taken and how long
ago did you take them?

------
crb002
Be a social human and find a local math prof or someone in industry to mentor
you.

That aside: Haskell Road to Math and Programming. Concrete Mathematics. Pick
your favorite computer graphics book to learn working with vectors and physics
equations. From there chase your passions.

------
madrafi
Just follow MIT Open Courseware for curriculum as for how to learn either the
lectures or textbooks, lectures are good for initial understanding but then
you'll need a textbook to go over the definitions,theorems and very important
EXERCICES . you might want to take it easy don't try to put everything in some
sort of timeline like (I am going to do all undergraduate in 1 year) that's
really stupid as may be advertised on the internet. Your real goal ,as you're
taking this endeavor ,is to understand it by making it your own, get a feel
for it.I cannot describe it but you'll know . Best of luck in your quests .

------
everywrongthing
Leaning math is primarily a solitary endeavor that takes time to learn
properly. If you are an adult starting from basic algebra be prepared to spend
at least 2 years of your life in order to get to where you need to be in order
to begin physics. (Ie Allegra, function, trig, calculus) You have lots of
options but assuming you don't want to go back to college these are 3 tools
that are 100% free and as good if not better than any of the paid options.

1 Every day on this site
[https://www.khanacademy.org](https://www.khanacademy.org)

2 Use this for graphing desmos.com/calculator

3 Use this to supplement (ie free text) myopenmath.com

------
chx
I'd very strongly recommend
[https://en.wikipedia.org/wiki/How_to_Solve_It](https://en.wikipedia.org/wiki/How_to_Solve_It)
to familiarize yourself with some fundamental processes and
[https://www.amazon.co.uk/Playing-Infinity-Mathematical-
Explo...](https://www.amazon.co.uk/Playing-Infinity-Mathematical-Explorations-
Mathematics/dp/0486232654) before you lose yourself in a highly symbolic and
abstract world, appreciate the beauty of it first, it'll make the long journey
much easier.

------
eof
I went back and learned a lot of math recently. I found the 3Blue1Brown
youtube channel to be hands down the most useful for developing intuition.

I honestly don't think there is better mathematical content than his being
made.

------
maliker
I know this is not what you're asking for, but I'd recommend stopping at the
end of the sophomore level classes in the open courseware material.

I finished an undergrad degree in math from a school that had a large PHD
program, and once I got to the junior level it was a lot of proving obscure
facts about complicated objects (say, Lie Algebras) that have zero
applications outside of very speculative theoretical physics. Proving theorems
was fun sometimes, so if you're interested in math as entertainment, it's
worth trying the more advanced material.

------
cevi
Here's a reading recommendation list I put together over the years:
[http://math.mit.edu/~notzeb/rec.html](http://math.mit.edu/~notzeb/rec.html)

Two caveats: first, it is obviously biased to the sort of math I find
enjoyable, and second, many of the suggestions require the reader to have that
elusive quality called "mathematical maturity". (I have no idea how one goes
about gaining mathematical maturity, despite having gone through this process
myself.)

------
syntaxing
Is there a particular reason why you want to learn upper level mathematics? I
have a pretty decent math background and I personally found majority of math
boring (particularly linear algebra). The only reason why I did so well in
Math is because I know I needed a strong math foundation to do the fun stuff
(like fluid dynamics). Without this purpose, I would of passed my math courses
in college but definitely would not of excelled in it.

------
nikisweeting
For those like me who learned programming first and want to improve their
math, but struggle with the crazy notation inconsistencies and the "pure math"
approach to things in general, I highly recommend this book:

A Programmer's Introduction to Mathematics by Dr. Jeremy Kun

I've only just started it, but so far it does a great job of introducing math
while assuming just the right amount of programming knowledge to make math
approachable.

------
mimischi
I've been catching up with things I've missed, already forgot or never learned
in school with Paul's Online Notes:
[http://tutorial.math.lamar.edu/](http://tutorial.math.lamar.edu/)

It's an incredible resource with great introductions to each topic and
examples that are easy to follow and try for yourself.

------
Bedon292
Not sure if its great for your purposes, but I like the 'No Bullshit guide to
Math and Physics' at least as a reference. It might be a good place to brush
up on lower level stuff before advancing to harder topics.

[https://minireference.com/](https://minireference.com/)

~~~
shriek
Can anyone share some insight on if there has been any improvements on this
since the last time it was posted here [1]? From that thread it appears that
it has few misinformation there.

[1][https://news.ycombinator.com/item?id=4994367](https://news.ycombinator.com/item?id=4994367)

------
philzook
John Baez (physicist / category theorist / smort dude) has a list of books to
learn math and physics. Is an interesting source to crib from.
[http://math.ucr.edu/home/baez/books.html](http://math.ucr.edu/home/baez/books.html)

Gerhard t'Hooft has a page on how to become of a good physicist.
[http://www.goodtheorist.science/](http://www.goodtheorist.science/) Not math
though.

I'm not necessarily suggesting that a mountain of books is the best way to go
about it. I think I respond best to video lectures. I'm not sure where you're
starting from or how applied/pure you want. Here's a quick mind dump some of
my favorites and some that I haven't watched. Roughly in order of how much I
liked them.

Gilbert Strang's Computational Methods for engineers was life changing for me.
It is a two part MIT opencourse.
[https://ocw.mit.edu/courses/mathematics/18-085-computational...](https://ocw.mit.edu/courses/mathematics/18-085-computational-
science-and-engineering-i-fall-2008/)

A Stanford course on the Fourier transform
[https://see.stanford.edu/Course/EE261](https://see.stanford.edu/Course/EE261)

Bartosz Milewski's Category theory for programmer's
[https://www.youtube.com/watch?v=I8LbkfSSR58](https://www.youtube.com/watch?v=I8LbkfSSR58)

Stephen Boyd's courses are online.
[http://web.stanford.edu/~boyd/](http://web.stanford.edu/~boyd/) Linear
Systems, convex optimization. Useful stuff.

Francis Su's Real Analysis is very good
[https://www.youtube.com/watch?v=sqEyWLGvvdw](https://www.youtube.com/watch?v=sqEyWLGvvdw)

Indian universities have an astounding collection of videos
[https://nptel.ac.in/](https://nptel.ac.in/) I have a tough time with the
accents, which is a bummer.

UCCS MathOnline has quite a haul
[https://www.uccs.edu/math/vidarchive](https://www.uccs.edu/math/vidarchive)

I've been enjoying this Visual Group Theory course lately
[https://www.youtube.com/playlist?list=PLwV-9DG53NDxU337smpTw...](https://www.youtube.com/playlist?list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv)

Math Doctor Bob
[https://www.youtube.com/user/MathDoctorBob/playlists](https://www.youtube.com/user/MathDoctorBob/playlists)

Wildberger has some interesting takes on elementary and non elementary topics
[https://www.youtube.com/user/njwildberger](https://www.youtube.com/user/njwildberger)

[https://www.perimeterinstitute.ca/training/perimeter-
scholar...](https://www.perimeterinstitute.ca/training/perimeter-scholars-
international/psi-lectures) Perimeter scholars lectures. Physics not math.
Good stuff.

Federico Ardila has a number of combinatorics courses.
[https://www.youtube.com/channel/UCWwECTsgjp_S-c73pO2c4gQ](https://www.youtube.com/channel/UCWwECTsgjp_S-c73pO2c4gQ)

Nonlinear algebra course
[https://www.youtube.com/playlist?list=PLRy_Pn1LtSpejKLClqbrW...](https://www.youtube.com/playlist?list=PLRy_Pn1LtSpejKLClqbrWWBxGKxx333O1)

Also of course there is Coursera and edX stuff.

Godspeed.

------
amsully
Good advice here.

If you choose the project/problem based approach I recommend
[https://projecteuler.net](https://projecteuler.net)

You will learn a ton solving these problems (the programming aspect is the
trivial piece in most cases).

~~~
niall00c
This is absolutely the right approach!

------
ranc1d
This might also help a similar thread was posted a while back

"Ask HN: How to self-learn math?"
[https://news.ycombinator.com/item?id=16562173](https://news.ycombinator.com/item?id=16562173)

------
jimhefferon
I'll add that, as a working faculty member, most people do not take naturally
to writing proofs correctly. Even quite good students have to work with
someone, at some length, before they get good instincts at it.

------
snicker7
I have found that talking to mathematians is the best way to learn
mathematics.

------
adenadel
I think this is something like what you're asking for

How to Become a Pure Mathematician (or Statistician)
[http://hbpms.blogspot.com/](http://hbpms.blogspot.com/)

~~~
joker3
This might be a good roadmap for pure math, but don't use this for guidance on
how to be a statistician.

------
aportnoy
Start small. I would look at Stat 110 from Harvard, Strang's OCW lectures, and
the intro calculus MIT course.

When you're done with that, it'll be easy to see where you want to go next.

------
sonabinu
Use community college as a non-degree student to do the introductory undergrad
classes. It's cheap, you are goal oriented and can get a lot of face time with
your professor.

------
jim_bailie
My $0.02: Make a goal to fully understand General Relativity as Einstein
published it. Then work through the evolution of some of its mathematics.

~~~
gnulinux
Why set an arbitrary goal like that? Why not understanding Curry-Howard
correspondence or Godel's Incompleteness or Hilbert's 17 instead.

~~~
jim_bailie
That was my $0.02 as someone who majored in Physics. You just gave $0.04. Good
suggestions.

------
graycat
Part I

I have some opinions on the question in the OP: I'm heavily self-taught from
independent study in math; that study helped me with, and at times was part of
my good career in, applied math and computing before my Ph.D. in pure/applied
math and helped a lot for my Ph.D.

For a _curriculum_ :

(1) Do the standard high school math, Algebra I, Plane Geometry (based on
proofs), Algebra II, Trigonometry, and Solid Geometry (based on proofs). In
each of these, it is sufficient (A) to work most of the more challenging
exercises and (B) try to think a little about what is going on. For (C), it
would be good occasionally, say, 1-4 times a year, to chat about your progress
with a good mathematician, Ph.D., likely a college prof. To find a Ph.D. who
will give you an hour or so per visit, maybe do some _networking_. If you
don't like the first choice, e.g., if they are not encouraging and helpful,
then try a different Ph.D. mathematician.

(2) Do Analytic Geometry and Calculus. Use a good college textbook. Since
highly polished texts have been widely available for decades, there's no need
to pay big bucks for new copies of the latest. So get a good used text. Better
yet, get 1-4 good texts, use your favorite one as your primary source and the
others for more content, alternate descriptions, evidence of what is more/less
important, etc.

Sets: Now in an important sense, essentially all of math is heavily about
_sets_. For first calculus, you don't need to know much about sets; the
following should be nearly sufficient:

By 1900 or so, a lot of math was known, but there was a nagging, somewhat
philosophical, question of what math really _is_. The answer of _pure_ math
was to construct a _foundation_ in, say, the deep basement, that would answer
the question but not much change the rest. The answer was to start with _sets_
and then define everything else in terms of sets. In particular do some
somewhat tricky and obscure work to define some sets that look, work, walk,
talk, etc. like the numbers on the line, the _real_ numbers. We don't think of
the real numbers this way, but the effort addressed the philosophical question
-- or, if I can get you to believe in sets, then I've got you for the rest of
math. Much of the early work on sets was from G. Cantor. Yes, soon there were
some issues from Bertrand Russell and then Kurt Gödel and, later, Paul Cohen.

For a long time in math, you can regard a _set_ as a conceptual (imaginary)
collection of some kind, e.g., all the peas in a little jar in the back of the
refrigerator, all the people with an iPhone, all the real numbers on the line,
all the water molecules in the earth's oceans, all the parabolas, all the
trials you might do in a lab experiment, and much more -- really, from the
philosophical approach to what math _is_ for all of math. Or intuitively
_sets_ are the containers for what we are thinking about.

Functions: Much of math is about _functions_. E.g.,

    
    
         f(x) = 2x^2 + 3x - 4
    

So, in _pure_ math, the function is f. The x is the _argument_ of the
function, i.e., a _variable_ which is essentially always in math from,

    
    
         "Think of something; call it x."
    

where at least in calculus usually the thing are thinking about is a number.
Or, to "solve for x", think

    
    
         "I suspect there is a number such
         that these conditions hold; I want to
         find if there is such a number and if
         so what its actual numerical value
         is."
    

Now you understand what math means when it talks about variable x and function
f. Then, given variable (e.g., number) x and function f, f(x) is the value of
f at x.

E.g., for this set stuff, a function gets defined in terms of sets, e.g., the
set of all _ordered pairs_

    
    
         (x, f(x))
    

for x in some set called the _domain_ of the function. In calculus, the domain
is usually the set of real numbers, that is, the points on the number line
(yes, that's not the tricky set theory definition of the real numbers).

So, with the ordered pairs

    
    
         (x, f(x))
    

we have a very precise definition of a function in terms of sets (ordered
pairs can also be defined in terms of sets).

Of course, in actual work, especially in applied math, nearly no one actually
thinks of a function as a set of ordered pairs.

Completeness: There is an old joke, partly appropriate, that

    
    
         "Calculus is the elementary
         consequences of the completeness
         property of the real number system.".
    

Intuitively _completeness_ is, if you are converging to something by more and
more accurate approximations, then there really is something there for you to
converge to. This property does NOT hold for the rational numbers -- the
_rational numbers_ are all the numbers of the form (that can be written as)
p/q for p and q integers (whole numbers, positive, negative, and zero) with q
not zero. Why? E.g., can use the rational numbers to approximate as closely as
we please the square root of 2, but the square root of 2 can't be a rational
number. Why not? Because if the square root of 2 were rational, then we would
also have for integers p and q, q not zero

    
    
         (p/q)(p/q) = 2
    

or

    
    
         (p)(p) = 2(q)(q)
    

So the left side has some even number (possibly 0) factors of 2 while the
right side has an odd number of factors of 2, and this violates the
fundamental theorem of arithmetic that each integer can be factored in only
one way as a product of prime numbers.

Well, a big deal about the real numbers is that they are _complete_ ; i.e.,
intuitively, if something converges, then there is something to converge to.
Completeness is a big deal because it generalizes, especially to Hilbert space
that you may come to.

So, in calculus often we make approximations that can become as accurate as we
please, and we want the completeness property to know that what these
approximations approach, _limits_ , can exist.

E.g., with the real numbers, if we make more and more accurate approximations
to the square root of 2, then we know that our approximations will _converge_
to the actual square root of 2 which DOES exist as a real number.

Sets get to be quite important in math in the last years of college math and
beyond. But for calculus, nearly all of that subject, especially what learn in
a first calculus course, was quite solid well before G. Cantor. So, in
practice, when studying first calculus, we don't see much about sets. So, for
first calculus, sets remain mostly a topic in the deep basement that addresses
an old philosophical issue. So, for first calculus, sets are no big issue.

Analytic Geometry: Here you study the _conic sections_. They are a bit amazing
and important in physics, mechanical engineering, and more and something
definite to work with while learning calculus. So, imagine a cone, say, an ice
cream cone. Take two of them that are the same and put the points together so
that they are both on the same _axis_ through the points (I'll let you get a
precise definition of _axis_ ). Now imagine a sharp sword, say, as in the old
John Belushi Samurai Tailor skit and use the sword to slice the cones and look
at the cut edges. Right, we assume that the sword moves in a plane. Depending
on where you slice, you will get (A) just a point (where the two cones touch
together), (B) two lines crossed as in an X (the cross in the X is at the
point where the two cones come together), (C) a circle (we are awash in
circles), (D) a parabola (to a fairly good approximation, a grand slam
baseball follows a parabola), (E) an ellipse (to a quite good approximation,
each planet goes around the sun in an ellipse), (E) the two parts of a
hyperbola (an electron shot at a negative charge will follow one of the halves
of a hyperbola as it avoids the negative charge).

First Calculus in a Nutshell: Then for first calculus, there are two big
topics, differentiation and integration. Differentiation is finding rate of
change, intuitively the slope (as in high school algebra) of a tangent line to
the graph of a curve (defined in terms of a function with domain some or all
of the real numbers). E.g., for time t, let d(t) be the function that gives us
our distance from home at time t. Then the derivative of function d at time t
is the velocity, say, function v, at time t. So, in a car, the odometer gives
d and the speedometer gives v. That is, at time t we have traveled distance
d(t) and are moving a velocity v(t). Here to be simple, I do not make a
careful distinction between what physics calls speed versus velocity.

What differentiation does, integration reverses, undoes. So, we can use
integration on function v to recover function d. We have now introduced the
fundamental theorem of calculus.

Right, v is the function, and v(t) is its value at time t, but in first
calculus it is common to drop the distinction between v and v(t) and just say
"the function v(t)" and mostly avoid mentioning v by itself.

~~~
graycat
Part II

Again, for _variable_ t, think

    
    
         "There is a number; call it t".
    

That approach to the meaning of a _variable_ works well enough for essentially
all of math.

Calculus was invented mostly by I. Newton, mostly for physics, especially for
explaining the motions of the planets. Since then calculus has become a pillar
of civilization, especially Western Civilization.

Mostly in calculus, integration is finding the area (or more carefully
_defining_ the area) under a curve, maybe a parabola.

But can also have a _line integral_ , say, the work do when carrying 100
pounds of hay to the top of the hay loft of the barn. Here we integrate from
beginning to end of the work. If we let the hay fall, then neglecting
friction, etc. get the work back as energy. So, have to pay attention to the
direction of the integration, from the ground up to the hay loft or from the
hay loft back to the ground. The two values have opposite signs.

You can teach yourself calculus: Just get 1-4, at least one good calculus
texts and dig in -- read the chapters, follow the material, work nearly all
the more difficult exercises, and maybe check 1-4 times a year with a good
mathematician. I did that for freshman college calculus (but didn't check with
anyone). I never took the course and, instead, started on sophomore calculus
and made As. No problem.

(3) Linear Algebra. Likely the next course would be linear algebra. It would
be good to do this subject 2-4 times from more elementary treatments to some
of the more advanced material. Some of the more advanced material can cover
linear programming optimization, classic two person game theory, the proof of
the Nash result in game theory, some treatments of the fast Fourier transform
in digital signal processing, group representations in the quantum mechanics
of molecular spectroscopy, error correcting codes in algebraic coding theory,
numerical methods, and more.

The start of linear algebra is just one or several _linear_ equations in one
or several variables. E.g., for two equations in three variables, _unknowns:_

    
    
         2x + 3y - 6z = 0
    
         5x - 2y + 7z = 3
    

For positive integers (whole numbers) m and n, given m such _linear_ equations
in n variables ( _unknowns_ ), the number of solutions, depending also on the
numerical constants in the equations, is none, one, or infinitely many.

The main way to find all the solutions is just Gauss elimination.

That covers a good chunk of a first course in linear algebra.

The notation

    
    
         2x + 3y - 6z = 0
    
         5x - 2y + 7z = 3
    

gets to be clumsy so we write instead

    
    
         2 + 3 - 6
    
         5 - 2 + 7
    

write that with big, square brackets, and call it a _matrix_ with two rows and
three columns. The individual numbers are _components_. Of course, a matrix is
conceptually close to an array with two subscripts common in programming
languages. Maybe we call the matrix

    
    
         2 + 3 - 6
    
         5 - 2 + 7
    

A. Since matrix A has 2 rows and 3 columns, we say that it is 2 x 3 ("2 by
3").

Then we write the x, y, z in a column

    
    
         x
    
         y
    
         z
    

apply the big, square brackets, and call it a matrix with 3 rows and 1 column
and maybe call it v. We write the right side

    
    
         0
    
         3
    

also with brackets, and maybe call it b.

Then we define a product, a _matrix_ product Av so that

    
    
         Av = b
    

means just the same as

    
    
         2x + 3y - 6z = 0
    
         5x - 2y + 7z = 3
    

From then on we work with matrices and try to avoid notation like

    
    
         2x + 3y - 6z = 0
    
         5x - 2y + 7z = 3
    

In this way, a course in _linear algebra_ is commonly called a course, Linear
Algebra and Matrix Theory.

A matrix with just one column is a _column vector_ ; with just one row, a _row
vector_. In short, either is called a _vector_.

Linearity: Given m x n matrices A and B, m x 1 matrices u, v, and real numbers
c, d, we have that

    
    
         A(cu + dv) = cAu + dAv
    

Here we regard Au as case of function A with argument u. That is, we could
have said that matrix A defines function f so that

    
    
         f(u) = Au
    

With

    
    
         cA + dA
    

we form _products_ cA and dA and _sums_

    
    
         cA + dA
    

So, we have to define both of those: The definitions are close to just
obvious.

The equation

    
    
         A(cu + dv) = cAu + dAv
    

means that matrix A acts like a _linear_ function. Big deal! The real world is
awash in linear behavior, and linearity is an enormously powerful property
mathematically. The rest of linear algebra and matrix theory is nearly all
about the consequences of such linearity.

The fundamental theorem of algebra is that each (high school style) polynomial
in variable x can be uniquely factored into a product of terms of the form (ax
+ b) where the a and b are possibly complex numbers. Essentially from this
fact, the more advanced parts of linear algebra use the complex numbers and
not just the real numbers.

At the end of a second course in linear algebra, will pay attention to the
essentially geometrical notions of length and angle and, in particular, to
orthogonality, that is, perpendicular. Big topics are the Gram-Schmidt process
where find some orthogonal vectors and the polar decomposition.

The polar decomposition result says that a matrix acting on a circle will
yield an ellipse, that is, for matrix A and vector u, if we let u take on the
values of all the points on a circle, then the Au will generate all the values
of all the points on an ellipse. The ellipse has two axes, and they are
mutually perpendicular. So, if have the vectors of the two axes, then have
enough to construct the whole ellipse.

In linear algebra, this situation generalizes to any finite dimension and
yields the singular value decomposition, principle components, factor
analysis, and more.

With more advanced work, this situation becomes the spectral theory of self-
adjoint linear operatiors central to quantum mechanics.

(4) The Four Main Parts of Math

There are four main parts of math:

(A) Foundations, that is, deep in the basement with set theory.

(B) Algebra, as in high school, the integers, prime numbers, the fundamental
theorem of arithmetic, greatest common divisor and least common multiple, the
fundamental theorem of algebra, linear algebra and matrix theory, the
generalizations of number systems such as groups, rings, fields (the rational,
real, and complex numbers are all _fields_ but so is the set of integers
modulo a prime number), etc. Fermat's last theorem, settled by A. Wiles, is
part of algebra; so are the deep, difficult questions about prime numbers,
etc. There are applications in error correcting codes and cryptography.

(C) Geometry, e.g., high school plane geometry, analytic geometry, and
differential geometry, a deep subject important for relativity.

(D) Analysis, as in calculus, differential equations, functional analysis,
e.g., Banach and Hilbert spaces, partial differential equations, e.g., as in
Maxwell's equations, fluid flow, and relativity, probability, statistics,
stochastic processes, optimization, and more.

In differential equations we are given an equation with the derivatives of
some function and want to find the function.

E.g., for real valued function y(t) and constants k and b, we might have

    
    
         y'(t) = k y(t) (b - y(t))
    

where y'(t) = d/dt y(t), the first derivative of function y(t). At one time,
that little differential equation saved FedEx from going out of business. The
solution is a _lazy_ S curve and a first cut at _viral_ growth.

There is topology which is partly in geometry and analysis. The main idea of
topology is _continuity_ , that is, changing without sudden jumps or some
cases of wildly fast oscillations.

There is algebraic geometry partly in algebra and geometry.

Much of number theory has deep connections with analysis.

Much of linear algebra is an introduction to functional analysis in analysis.

The older applied math is mostly analysis.

(5) Analysis. Usually after linear algebra will study _advanced calculus_ and
analysis. Broadly there are two approaches, (A) theory and (B) applications.

The theory is mostly to give fully careful proofs of the results and first
generalizations of what you saw in calculus. For the theory maybe the most
respected text is

R. Rudin, _Principles of Mathematical Analysis, Third Edition._

For this, get the third edition and not either of the two earlier editions.

So, will discover that the integral learned in calculus is called the Riemann
integral because B. Riemann made the theorems solid. Rudin also does the easy
generalization to the Riemann-Stieltjes integral.

~~~
graycat
Part III

Rudin has a nice chapter on Fourier series, that is, representing a periodic
function f(t) with a sum of e^(iwt) for _overtones_ at frequencies w and where
each frequency w is a whole number multiple of a _fundamental_ frequency. The
linear algebra and geometry here are that the e^(iwt) are perpendicular
projections of the f(t). So, the e^(iwt) are orthogonal axes.

The line integral generalizes to the exterior calculus of differential forms
(keep track of signs from the direction do the integrations, i.e., as in line
integrals) and the fundamental theorem of calculus generalizes to Stokes
theorem crucial in Maxwell's equations, fluid flow, and partial differential
equations. Now we are close to differential geometry, and Rudin also does the
inverse and implicit function theorems important in differential geometry and
parts of nonlinear optimization, e.g., Lagrange multipliers.

I warmly suggest that take a really fun, one weekend, pass through Stokes
theorem parts of

Tom M. Apostol, _Mathematical Analysis: A Modern Approach to Advanced
Calculus_ , Addison-Wesley, Reading, Massachusetts, 1957.

Get it used, and pay whatever you have to. This way get to see the 2-3
dimensional versions and, really, mostly enough for what physics and
engineering do with _vector analysis_ and Stokes theorem.

In H. Royden, _Real Analysis_ and the first half of W. Rudin, _Real and
Complex Analysis_ , can see the Lebesgue integral, due to H. Lebesgue near
1900. In essentially all cases where the Riemann integral is defined, the
Lebesgue integral is also defined and gives the same numerical value. But the
Lebesgue integral has more powerful theorems and is defined in more general
situations.

As in a 1933 paper of A. Kolmogorov, the Lebesgue integral gives a solid
foundation to probability, statistics, and stochastic processes.

In W. Rudin, _Functional Analysis_ get a treatment of _distributions_ that
cleans up what physics tries to do with the Dirac delta function and also
covers spectral theory.

From there, can go for a Ph.D. For that will need mostly (A) pass the
qualifying exams and (B) do some original research. The standards are commonly
something like "an original contribution to knowledge worthy of publication"
and for publication, "new, correct, and significant".

For the qualifying exams, first pick a department, pure/applied math or some
math area in engineering, optimization, probability, economics, computer
science, etc., get their description of their qualifying exam topics and
references, study, and take and pass the exams. For the research, do some.

One suggestion: For the research, pick a real problem and use some math, at
least in part new, to get a good solution. Can get "significant" from the
importance of the real problem. Can get much of "new" if have the first or
better attack on the real problem. Can get "correct" if do the math with
careful theorems and proofs. Can publish if pick an appropriate journal. Might
get to regard the math as _significant_ based on what it does for the real
problem and not just its contribution to pure math.

Even the best US grad schools are hungry for good students. With a good ugrad
math major and good work on independent study, should get a good reception at
grad schools and at least a tuition scholarship. In that way I got accepted to
grad school at math departments at Cornell, Brown, Princeton, and more.

In a sense, independent study is recommended: (A) That is basically what
research profs have to do for all their careers. (B) At least at one time the
math department at Princeton stated that no courses were offered for
preparation for the qualifying exams, students were expected to prepare for
the exams on their own, courses were introductions to research by experts in
their fields, and students should have some research underway in their first
year.

For a little on how to do the original parts of math research, there is the A.
Wiles comment:

"Perhaps I could best describe my experience of doing mathematics in terms of
entering a dark mansion. You go into the first room and it's dark, completely
dark. You stumble around, bumping into the furniture. Gradually, you learn
where each piece of furniture is. And finally, after six months or so, you
find the light switch and turn it on. Suddenly it's all illuminated and you
can see exactly where you were. Then you go into the next dark room ...."

That's some of how to learn some math, largely with independent study, and
maybe to get a Ph.D. But you might want more: You might want your Ph.D. work
to get you a good start on a good career in pure/applied math
research/applications. For that, look around, pick up what you can in seminars
and conferences, and get what your profs can explain to you.

Broad point: Long the best opportunities in applied math, even with some
advanced pure math prerequisites, have been in US national security,
especially within 100 miles of the Washington Monument.

------
mindcrime
I'm working on this myself and while I'm not very far along, I can share my
approach.

1\. Go to Youtube, find the Professor Leonard channel. He teaches math at
Merced College and is a very good lecturer. He has recorded himself teaching
everything from pre-algebra through differential equations, and a statistics
class. The one thing he has not done yet is linear algebra, he he apparently
plans to do it.

2\. Watch his stuff, and supplement that with the corresponding "Schaum's
Outline" or similar book for the topic at hand. Also, if you desire, buy a few
editions old used college textbook for the corresponding topic. This gives you
more exercises to do and a reference to consult if anything is unclear.

3\. As desired, follow the Khan Academy lessons on the topic you're studying.
KA has everything from arithmetic / pre-algebra up through at least Calculus
and Linear Algebra. I don't remember offhand if they cover Differential
Equations or not.

4\. For Linear Algebra in particular, the Gilbert Strang lectures on Youtube
are very highly regarded, and he has a text that was written specifically to
accompany those videos. So that's a good resource for Linear Algebra.

5\. For "higher" math (real analysis, complex analysis, topology, abstract
algebra, etc.) you can almost always find complete lecture series on Youtube /
OCW. Depending on the topic, there may also be a "Schaum's Outline" or similar
study guide book you can supplement with. And you can always find a used
textbook on Amazon, usually for not too much money if you go with an older
edition.

You can also find a lot of freely available maths texts online. See, for
example:
[https://math.gatech.edu/~cain/textbooks/onlinebooks.html](https://math.gatech.edu/~cain/textbooks/onlinebooks.html)

If you don't have a background in doing proofs, which is kind of regarded as
the dividing line between "simple" math and "higher" math, there are a number
of books on that specific topic, including texts written for so-called
"transition to higher math" classes. Some of those are freely available online
as well. There's also a good class you can find on Youtube, "Math for Computer
Scientists" which covers proofs and what-not pretty well. There's a freely
available corresponding text as well.

Another thing to do is consult forums where you can ask for help if you get
stuck. There is math.stackexchange.com, physicsforums.com,
cheatatmathhomework.reddit.com, learnmath.reddit.com, mathhelp.reddit.com,
etc.

Somewhere I have a Google doc that lists a lot of the resources I have been
using, and have queued up to use in the future. If anybody is interested, I'll
clean that up, and make it public and share the link.

One last note: I haven't done it myself, but I've heard that if you live near
a University, it's not too hard to find maths students who will tutor you to
pick up some extra cash. So that's an option as well.

Edit: somebody else mentioned 3blue1brown on Youtube, and there are a number
of other really good Youtube channels, including: Prof RobBob, NancyPi, and
Dr. Chris Tisdell.

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aj7
Get Thomas 13th or 14th edition and solve all the odd problems. That takes
care of calculus.

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rblion
I need this too, thanks for asking what I was too afraid to ask

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lucasosouza
Check brilliant.org. Got everything there, worth the money.

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joker3
What is the purpose of a proof? You may think that it's intended to solve a
problem, but that's really only half the point. A good proof is one that
communicates your solution to other people.

Mathematical writing is hard to learn well under the best of circumstances,
but if you don't have someone else giving you feedback on whether they
understand what you're writing, it's absolutely impossible. You need to have a
mentor or at least an editor at some point. That's not impossible to find
outside of the university system, but it's very difficult.

(This is the single biggest reason why MOOCs for higher math haven't taken
off. There are a lot of people who'd love to communicate something about the
field that they've dedicated their lives to, but the feedback system just
doesn't scale. If anyone can figure out how to fix that, it'll be a game
changer.)

So the first thing you have to do is to figure out what you can reasonably
expect to get out of this process. You can learn the definitions and theorems
of higher math, and that might be enough, but you're never going to develop an
intuition for them without understanding how to produce proofs on your own.
And don't fool yourself into thinking that you can evaluate your own proofs.
It just doesn't work.

If all of that doesn't have you turned off, then here are some ideas on what
to do.

A university level math curriculum is split into roughly three components: *
Lower level classes that focus on basic definitions and calculations; * Mid
level classes that teach some basic theorem-proving skills in subjects that
are useful for people in other quantitative fields; * Upper level classes that
offer serious practice in theorem-proving as well as the core ideas of
mathematics. You don't generally have to do classes in any particular order,
but you do have to master the skills of each level before you go on to the
next one.

To begin, you must be very comfortable with the contents of a high school math
curriculum. Serge Lang's book on basic mathematics is a great refresher if
you're not, or you can use any of the various popular study guides (Schaum's,
Barron's, etc.).

At the first level, you have calculus. This is generally split into three
semesters, with the first dedicated to limits and derivatives of single-
variable functions, one dedicated to integrals of single-variable functions as
well as sequences and series, and the last dedicated to derivatives and
integrals of multivariate functions. There are plenty of very expensive books
with glossy page and many color pictures and few ideas, but if you want a
serious introduction, look at Peter Lax's books on calculus.

At the second level you'll almost always find introductions to differential
equations and linear algebra. Differential equations have historically been
the workhorse of applied mathematics and you really need to have some
familiarity with them, but I've never seen a book on the topic that I liked. I
probably won't be satisfied by anything at this level, though, so look around
and see if you can at least find something inexpensive.

Linear algebra is a more recent topic (with many of its key ideas actually
originating in the 20th century), but it's probably actually more important
now. Gilbert Strang's books are popular and are worth reading for a first
look, but you really can't regard them as a serious introduction to the
mathematical side of the topic. Axler is probably the best book in that
regard, but it's best taken on a second pass.

I think that probability should be regarded as a core class at this level. I
don't think that's a fringe view, but it's not as universal as I'd like. I
learned from Pitman's book, and I think it's as good as any to start with.

You can also take classes on complex variables or "discrete math" here. I
don't know what a good textbook for complex variables is--maybe Saff & Snider?
--but I'm sure there are recommendations out there. Needham's "Visual Complex
Analysis" is a fantastic book, but maybe not really suitable for a very first
introduction. As for "discrete math" (a jumble of topics from logic,
combinatorics and number theory), find the cheapest book you can get that has
decent reviews on Amazon.

At the third level, there are three main topics: analysis, algebra and
topology/geometry. You can think of these as the three main viewpoints in
higher math, and other topics being populated by people who primarily look at
things with the tools of one of those three topics.

Analysis starts out as the theory behind calculus. In a first course, you'll
revisit a lot of what you saw in single-variable calculus, but you'll learn
why it's true rather than just how to use it. For a single semester
undergraduate course, Ken Binmore's book is probably the gentlest
introduction.

Modern geometry is related to what you studied in high school, but with a few
more centuries of development. It also doesn't get a lot of coverage at the
undergraduate level, which is highly unfortunate. Stillwell's "The Four
Pillars of Geometry" is a wonderful book and completely accessible.

Algebra is a bit difficult to explain without getting into the weeds. Pinter's
"A Book of Abstract Algebra" is very good at motivating the topic and
explaining the basics, which is the best you can hope for in an introductory
textbook.

Beyond that but still at the undergraduate level, you can get electives in
combinatorics (use Brualdi), number theory (?), logic (?) and some applied
topics as well. Looking through the course offerings of various math
departments will help you to fill in what the other possibilities are.

------
MaysonL
Check out mathvault.ca.

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postit
You first need to define your boundaries and limits. Don't get me wrong, but
mathematics is grinding, and if you think you have an idea where you're
getting yourself into because you've studied calculus and algebra as an
undergrad CS, you will fail miserably. Don't make the same mistake I did :)

Undergrad math is tooling; I like to use the analogy that it's like entering a
workshop where you have all the tools available, but you're blindfolded, and
you have no idea what each stuff is used for. So you'll have to touch
everything. Look down for the tooling advice.

When venturing in Mathematics, for your own sanity, please have an objective
in mind. I'm down serious! Understand what you want to research and go in that
direction.

One last piece, Find a mentor, share, talk to people. You won't advance from
yourself.

As a piece of tooling advice, I have the following I've stolen from Reddit a
few years ago (sorry I haven't found the source to it) __ TOOLING __ First of
all, most important, GO LEARN ALGEBRA. Seriously, I know you think its
bullshit but its the most basic skill in some ways that any mathematician
should know. Second learn Calculus: Single and Multivariable. If you are still
interested here are some things to go onto next:

Discrete mathematics: This includes equivalence relations (probably one of the
most important things for you understand ever), propositional calculus (logic)
proof techniques (induction) and some basic combinatorics (Pigeonhole
principle). You can literally find any textbook and start reading. The theory
is kinda a hodgepodge, but those are the major themes.

Linear Algebra: Again, one of the most important subjects you will ever study.
Once you understand this, you are really on your way, and this stuff comes up
everywhere. Many mathematicians have said many of the biggest proofs in the
world come down to "just some linear algebra". The major point here is to
understand that there is only one vector space for each dimension over a field
and understand how a linear transformation becomes a matrix only after a
choice of basis. Here equivalence relations come up again!

Differential equations: Unless you're focused on engineering math or serious
applied stuff, don't worry too much about this. Seriously, it's not that
integral (haha get it!).

Complex Analysis: Yes, mathematicians and Engr. Actually, do study "imaginary"
numbers, but there is nothing imaginary here. This is serious stuff, do it.

Okay, so now you're about as a sophomore/junior level place in mathematics.
How to finish it off? It's not that unclear:

Abstract Algebra -- Grab any book read about groups, rings, fields, vector
spaces, and modules. Proofs will be difficult here but work through it. There
are so many books here, avoid Lang (good book but not for starting out),
Dummit/Foote is okay. As a undergrad I had a good time with Rotman's An
introduction to abstract algebra.

Analysis -- Grab Baby Rudin. No seriously, Grab this book, sit in a room for a
semester and just fuckn' read it. You will basically be "redoing" calculus.
This is a trial by fire, go!

Topology -- Grab Introduction to Topology by J. Munkries. Its so well written
it might as well be a coffee table book.

There now, you have done everything a math major would. Yes, there are lots of
things that are missing, arguably the most important things depending on what
your goals are. Typically one studies Number Theory along with Abstract
Algebra, or studies Analysis and Differential Equations together or Analysis
and Topology. Seeing the links across different topics is essential, but I'm
just giving the overview here.

Not every mathematician studies logic, and there are LOT of fringe topics that
I'm omitting (including some of my favs: Projective Geometry, Varieties,
Lattice/Order theory, Combinatorics, Elliptic Curves, Coding theory, Harmonic
Analysis, etc.). However, none of these are required courses at more than a 1%
of programs

------
skh
I could give you a list of books that are used at the graduate and
undergraduate level that are considered good. However, almost none of them
really discuss the motivations behind topics. My suggestion is to start with
this book by Arnold on Abel’s Theorem. If it interests you then delve deeper
into the topics once you are done with it.

[https://www.mathcamp.org/2015/abel/abel.pdf](https://www.mathcamp.org/2015/abel/abel.pdf)

------
porpoisely
Math is a huge field and I'm not sure any can truly have both a deep and wide
understanding of everything in math.

But if you want to give it a go, why not start with the tried and tested
"Euclid's Elements" and a book on the history and philosophy of mathematics.
If you are still interested, then tackle a course plan laid out by MIT or any
other reputable university.

Also, I suggest getting an exercises/solutions book because math is doing. You
can read the theorems, corollaries, etc and think you understand it but until
you try to solve a problem yourself, you really won't know if you really
understood the topic. And in your down time, you can check out math related
channels/videos on youtube or elsewhere. It's going to take a lot grit and
determination. Best of luck to you.

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oskkejdjdkjd
Go to the library and read their math books. I don’t know why the internet is
filled with these questions that are always “how do I start doing x?” Just f*
do it dude. Nobody is going to be able to or bother to answer anyway. The
answers wouldn’t make deep sense to you because they rely on having done it
before. Once you’ve started and you’re really stuck and everything you read
doesn’t seem to illuminate it, post a question. People will notice you’ve put
some work and time in and will be eager to help you.

------
kache_
youtube

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thisiszilff
You want to learn more about both pure and applied mathematics. Pure math is
very proof heavy which is excellent for self learning because you never have
to take anything on faith. Proofs are meant to be very clear explanations.

One of the best resources for learning in this way is to find books that are
structured to have you develop the theory via the problem sets. Of the top of
my head, Tao's Analysis books and Atiyah & MacDonald's Commutative Algebra
book are exemplars of this style. These books often show the basic
definitions, some results, and then break up remaining results into step by
step problems.

Proofs will be the hardest part to wrap your head around. Once you understand
proofs, can read them, the remaining is going to be a question of time. Think
of it like programming: you can't learn by reading, lectures, etc. You need to
learn by doing. In this case, the doing is doing proofs.

