
Ask HN: How to self-learn math? - sidyapa
I have a new found appreciation and fascination for maths and would love to study maths from the bottoms ups. I&#x27;d love to know the paths I should take and books I should read.<p>EDIT1: If the question is very broad, it&#x27;d be much helpful to know how did you learn math? What courses you took, books you read.<p>EDIT2: My current proficiency level is pre-high school mathematics as I didn&#x27;t pay much attention in high school, learning effectively nothing.
======
mikevm
Ok, I'll take a crack at this:

Up to high-school level:

1\. Precalculus: Precalculus: A Prelude to Calculus - Axler

2\. Calculus: The Calculus Tutoring Book - Ash.

College:

3\. Preparation for Collegel-level maths:

3a. General prep for high level maths: How to Study as a Mathematics Major -
Alcock

3b. Proof writing: How to Prove It - A Structured Approach - Velleman OR Book
of Proof (2nd ed) - Hammack (it's free!)

4\. Mathematical Analysis:

4a. Good prep for Analysis: How to Think About Analysis - Alcock

4b. Understanding Analysis (2nd ed) - Abbott OR Yet Another Introduction to
Analysis - Bryant (has full solutions) OR The How and Why of One Variable
Calculus - Sasane OR Mathematical Analysis - A Straightforward Approach (2nd
ed) - Binmore (has full solutions)

5\. Discrete Mathematics (a combination of set theory, combinatorics, a bit of
discrete probability and graph theory): Discrete Mathematics - Chetwynd,
Diggle

6\. Linear Algebra: Linear Algebra - A Modern Introduction (4th ed) - Poole

7\. Probability: Introduction to Probability - Blitzstein, Hwang + online
course
[https://projects.iq.harvard.edu/stat110](https://projects.iq.harvard.edu/stat110)

8\. Statistics: (for Bayesian) Statistical Rethinking - A Bayesian Course with
Examples in R and Stan - McElreath + online course
[https://www.youtube.com/playlist?list=PLDcUM9US4XdM9_N6XUUFr...](https://www.youtube.com/playlist?list=PLDcUM9US4XdM9_N6XUUFrhghGJ4K25bFc)

Usually you'll be doing courses on #4, #5, and #6 simultaneously.

~~~
dvddgld
Two quick things I can recommend without hesitation, which focus on an
intuitive understanding of concepts:

1\. Essence of Linear Algebra mini-series -
[https://m.youtube.com/watch?v=kjBOesZCoqc](https://m.youtube.com/watch?v=kjBOesZCoqc)

2\. Better Explained website -
[https://betterexplained.com](https://betterexplained.com)

YouTube has a lot of high quality math content, it definitely helped through
university. It's also worth mentioning the Stanford U courses.

The main takeaway I have for you is learn the concepts intuitively first, then
spend the time to play around with them on paper until they sink in. Some
things will be easy, some will be frustrating, much like programming you will
walk away from a frustrating problem and have an epiphany while doing
something completely different.

All the best and have fun!

~~~
Entalpi
My two cents are whenever something seems hard/impossible/infuriating/etc,
take a break then seek dofferent sources on the material. A lot of times I
have been hung up on something only to find that things make much more sense
when approaching it from a different viewpoint. :)

~~~
dvddgld
Absolutely! Not having to hit your head against a wall helps prevent burnout
as well as just plain being more effective

~~~
chris_wot
That’s how I got to grips with trigonometry... I tried to understand why sine,
cosine, tangent, cotangent, secant and close can’t we’re named like they
were... then I found a bunch of stuff on the unit circle. Never looked back!

------
aphextron
>My current proficiency level is pre-high school mathematics as I didn't pay
much attention in high school, learning effectively nothing.

Advice from someone who was in the same position: Take a class. Multiple
classes. Go sign up for a Mathematics AS at your nearest community college
right now. You will never know enough of what you don't know to learn this
stuff on your own. A lot of it is just doing the painful repetition work of
practicing problems over and over again, which is hard to force yourself into
without a "coach" pushing you. Having a cohort of students to work through
problems with is also priceless. And the drive of having accountable grading
will keep you at it regularly.

It can be a bit awkward at first feeling stupid not knowing what a logarithm
is in a room full of 18 year olds. But it's the only way to really get there.
I went from high school dropout who didn't know how to add fractions to
passing calculus in 18 months.

~~~
Ultimatt
Depends what your goal is. I cant imagine anything worse than High School
style teaching of mathematics. Its awful. That repetitive calculation,
entirely unnecessary for almost any real world situation. Almost every time
you are doing repetitive calculation what you should have been taught was a
second or third conceptualisation of whatever maths you are looking at. So
linear algebra I was only ever taught numerically in computer science. WHICH
IS CRIMINAL. It has such an obvious and more easily understood geometric
interpretation. The same is true of calculus, learn it in physics. Seriously.
Do high school physics /and/ maths if you wish to understand calculus.
Otherwise you've got someone who probably did a pure maths degree teaching you
something that was created by physicists for a reason, that reason has a
physical and real world interpretation not to mention a geometric one. Another
good example is in chemistry you might learn statistical mechanics, in
computer science or electronic engineering you might learn information theory.
If those two classes were taught back to back for both groups of people they
would actually deeply understand the mathematics behind it. Personally I've
always found learning maths in a pure way incredibly challenging. As an adult
I think its a lot easier to learn maths through something that has physical or
meaningful interpretation. You can then draw on your actual life experience to
understand the maths.

~~~
InitialLastName
I disagree about the repetitive calculation being "entirely unnecessary".

Background: I (probably like many here) excelled at high school and
undergraduate mathematics. I have a graduate degree in a heavily mathematics-
focused branch of engineering (digital signal processing) and much of my work
involves applying that math at a conceptual level.

I'm currently tutoring an adult I'm close to who is approximately at GP's
starting level and has a strong anxiety reaction to math. I'm finding that the
repetitive calculation aspect is important to being able to developing an
intuition for the concepts.

The process that's working well for us so far involves alternating between
practical word problems (to establish a motivation for learning the material),
theoretical/background explanations (to hit the understanding at a high level)
and repeated simple problems (to practice the mechanics).

I'm finding that, in terms of process, the repeated problem-solving is
critical for two reasons. 1) It helps to build a sort of mental "muscle
memory", and 2) it helps with developing intuition, since your mind eventually
gets bored with the mechanics and starts to notice patterns.

Remember that mathematics at that level is all about building blocks. Every
concept/problem-solving practice you learn is part of a tower of strategies;
if earlier mechanisms aren't almost mindless, it's MUCH harder to build on top
of that knowledge.

~~~
zenhack
One of the earliest memories I have of really enjoying math was a unit on
modular arithmatic from my 6th grade math class. The teacher introduced the
concept with a hands of a clock metaphor, gave us a whole bunch of drill
problems, and told us to watch for patterns. We had an easel in the class room
where he collected the insights students had gained. Discovering the
relationship with remainders by ourselves was incredibly cool.

~~~
InitialLastName
I had a similar experience when I learned about factoring (not sure what age).
I got a little obsessed with finding prime numbers by hand, and eventually
figured out that I only had to search for factors up to the square root. I
distinctly remember that as the first time I realized how powerful a simple
mathematical insight can be to speeding up a process.

Now, of course, I use mathematical insight to speed up processes
occupationally; I guess that experience stuck with me.

------
jostylr
Recommend Guesstimation by Weinstein and Adams as the first topic to master.
Getting comfortable with numbers and their sizes will make everything else
easier. Also, getting in the habit of doing rough explorations is an essential
skill in exploring all later material.

Technical Mathematics with Calculus by Calter is a single volume that covers
stuff up through calculus in a, well, technical manner.

For a more understanding way, try Elements of Mathematics by Stillwell.

If you get past Calculus, I recommend Vector Calculus, Linear Algebra, and
Differential Forms by Hubbard. It gives an amazingly clear viewpoint on the
higher level analysis and algebra topics, both numerically and abstractly.

For statistics, you might try something like Think Stats by Downey which
emphasizes explorations with Python, real data, and Bayesian statistics.

As a faithful companion in your journey, use something like GeoGebra or Desmos
to really explore the visual side of all the topics. Computers can do the
tedious computations. Your task is to learn why we are doing this and how it
is being done. When you get to calculus, learn what Newton's method is doing
and appreciate how amazing it is.

------
ivan_ah
I have just the book for you: the essentials of high school math for adults:
[http://www.lulu.com/shop/ivan-savov/no-bullshit-guide-to-
mat...](http://www.lulu.com/shop/ivan-savov/no-bullshit-guide-to-mathematics-
hardcover/hardcover/product-23460526.html)

If you like this one, you can followup with the MATH&PHYS book which covers
mechanics (PHYS101) and calculus. And if you like that one, you can follow up
with the liner algebra book.

All along the way, I recommend you try solving exercises and problems using
pen and paper. Ideally you can also create custom "test questions" for
yourself using SymPy
[https://minireference.com/static/tutorials/sympy_tutorial.pd...](https://minireference.com/static/tutorials/sympy_tutorial.pdf)
1\. start with a simple math question or equation related to what you're
studying right now, 2. solve it by hand, 3. compare your answer with the
answer obtained by SymPy.

Good luck on your journey. Math is very deep so don't be in a rush. Enjoy the
views along the way!

~~~
victor106
This looks great.

Do you have a pdf/ebook I can purchase?

~~~
ivan_ah
I'm still working on generating the MATH eBook, but the 3-in-1 book
(MATH+MECH+CALC) is available here:
[https://gumroad.com/l/noBSmath](https://gumroad.com/l/noBSmath) Chapter 1 of
the 3-in-1 book is essentially the same as the No Bullshit guide to
Mathematics.

You can see a preview here:
[https://minireference.com/static/excerpts/noBSguide_v5_previ...](https://minireference.com/static/excerpts/noBSguide_v5_preview.pdf)

~~~
face_mcgace
what is the difference between no bs guide to mathematics and the 3-in-1 book?

~~~
ivan_ah
The 3-in-1 book contains a high school math review, a mechanics course, and a
calculus course (450pp). The No bs guide to math is just the high school math
review and is much thinner 170pp. (I'm essentially cutting up the 3-in-1 book
and releasing it as split books because I realize 450pp can be intimidating
for some readers).

------
mathgenius
Nothing beats having a (good) teacher. Self-learning, no matter how smart you
are, is pitifully slow without a teacher. Half an hour with a good teacher can
save you weeks of table head-butting. (But obviously you can't rely _only_ on
the teacher.)

As for books, it's not a spectator sport: you gotta do it yourself. Read a
sentence, then work it out yourself with pen & paper. You can't get it just
from reading alone.

Finally: in mathematics there's many many roads to Rome! If something isn't
working for you, try another way.

~~~
iliketosleep
I disagree with the general statement that leaning maths is "pitifully slow
without a teacher". So long as one has structure (e.g. some kind of syllabus)
and has access to google, then learning can proceed very efficiently indeed.
In addition, by being able to work through difficulties independently you can
"be your own master" so to speak, earning the confidence to solve new and
difficult problems without assistance.

Having said that, I have come across people who have absolutely no apitutude
for maths and definitely need a teacher. In a matsh class, there's usually one
one set of people who kind of just "get it" straight away, and another set who
struggle despite studying hard.

As you said, there are many roads to Rome, and the best road may or may not
involve a teacher depending on the individual.

~~~
darkxanthos
An author of a book is still a teacher... we’re just debating the
personalization of the teaching at this point.

~~~
TheTrotters
But then this recommendation becomes vacuous. No one is recommending learning
math by deriving everything yourself.

~~~
darkxanthos
Exactly. What’s really being referred to? Lower cost and more scheduling
flexibility perhaps?

------
synthmeat
I’m going to go with a few assumptions here:

a) You don’t do this full time.

b) By “bottoms up” you just mean “with firm grasp on fundamentals”, not
logic/set/category/type theory approach.

c) You are skilled with programming/software in general.

In a way, you’re ahead of math peers in that you don’t need to do a lot of
problems by hand, and can develop intuition much faster through many software
tools available. Even charting simple tables goes a long way.

Another thing you have going for yourself is - you can basically skip high
school math and jump right in for the good stuff.

I’d recommend getting great and cheap russian recap of mathematics up to 60s
[1] and a modern coverage of the field in relatively light essay form [2].

Just skimming these will broaden your mathematical horizons to the point where
you’re going to start recognizing more and more real-life math problems in
your daily life which will, in return, incite you to dig further into aspects
and resources of what is absolutely huge and beautiful landscape of
mathematics.

[1] [https://www.amazon.com/Mathematics-Content-Methods-
Meaning-V...](https://www.amazon.com/Mathematics-Content-Methods-Meaning-
Volumes/dp/0486409163)

[2] [https://www.amazon.com/Princeton-Companion-Mathematics-
Timot...](https://www.amazon.com/Princeton-Companion-Mathematics-Timothy-
Gowers/dp/0691118809)

~~~
gradschool
The Princeton Companion to Mathematics is a good resource consisting of a huge
collection of detailed articles on many mathematical subjects by knowledgeable
contributors. It requires no specialized background and is curated by Fields
Medalist Tim Gowers. Whoever reads it from cover to cover is my hero, but
failing that there's always an interesting article to jump to.

Don't just be a consumer but write something as soon as you're inspired. I
wish there were more emphasis on writing mathematics in school prior to the
graduate level. Leslie Lamport says if you're thinking but not writing you're
not really thinking; you only think you're thinking. For Feynman the act of
discovery wasn't complete until he had explained it to someone. There's also
the rule of thumb that if you can't explain a mathematical concept to a ten
year old, you don't understand it yourself.

Edit: typo

~~~
synthmeat
> The Princeton Companion to Mathematics is a good resource...

I think Princeton Companion to Physics curated by Frank Wilczek, a Nobelist,
is due to be published this year.

> Whoever reads it from cover to cover is my hero...

Yeah, I'd die an accomplished man if I would grok just a few books I treasure,
amongst which are TPCTM and MICMAM.

> Don't just be a consumer but write something as soon as you're inspired.

Absolutely. That's why I recommend just a small amount of comprehensive
resources. It's hard to get motivated by a pile of books complemented with
synthetic problems related to a particular chapter. The idea is to just go
about your daily life and start to slowly see more and more math problems
everywhere around you; it does wonders to motivation.

~~~
mtreis86
What is MICMAM?

Whitepapers, lectures, and speech transcriptions are also good motivation, and
useful resources. Sometimes overwhelming, especially if reading mathematical
text is as a foreign language. And sometimes it takes you down a rabbit hole.

My biggest block for learning math has really been all the unlearning. After a
while ideas like negative numbers and zeros and processes like addition and
subtraction stop making as much sense as I thought.

Here is my favorite rabbit hole:

[http://www.turingarchive.org/viewer/?id=465&title=01](http://www.turingarchive.org/viewer/?id=465&title=01)

leads to:

[http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002278499...](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002278499&physid=phys735#navi)

leads to:

[http://www.jamesrmeyer.com/pdfs/godel-original-
english.pdf](http://www.jamesrmeyer.com/pdfs/godel-original-english.pdf)

Where to go from there - philosophy or computation? Lambda calculus is only a
couple clicks away. Lisp papers, perhaps?

~~~
synthmeat
> What is MICMAM?

Check my [1] at root.

> Where to go from there - philosophy or computation?

For me, there's plenty of fun in mathematics without venturing even near the
edges of it. Maybe one day I'll grow bored of it, who knows - it's a lifelong
process.

~~~
mtreis86
I didn't mean to insinuate that the process I describe is one to be taken out
of boredom. Let me try to explain what I am thinking:

I have been studying lisp and wanted to understand more about the origins. So
I went back to the beginning of the language and read the various McCarthy
papers. But what he was thinking is not entirely clear to me. So I wonder,
what papers was he studying himself when he wrote this? That is easy to answer
as he put the references right there in the back of the paper for me to track
down. So I start reading papers written by Church and Godel. I repeat this
process recursively while looking for shared references. That network of
interconnected papers is a treasure trove of useful information. Reading the
same papers an author was reading during their writing process is a valuable
way to expand your understanding of their work.

------
KodiakLabs
The simplest approach I think would be to start with Khan Academy. Well spoken
clear and concise. You can go from a Highschool level towards subjects from
first year university. Once there, it should be easier to self teach from
books.

~~~
hultner
I can recommend this path, studied almost all my pre-engineering math this way
2011 and were better equipped for initial engineering courses then most of my
peers. However it kinda capped out at high-school level (or atleast Swedish
equivalents).

~~~
KodiakLabs
Interesting. Truth be told, I remember when it was simply him on YouTube. I
guess to carry on, one would have to go for 3Blue1Brown to get the
fundamentals of university math.

------
laichzeit0
Book of Proof is hands down the best book to start with.
[https://www.people.vcu.edu/~rhammack/BookOfProof/](https://www.people.vcu.edu/~rhammack/BookOfProof/)

I’ve worked through the whole book twice because I loved it so much.

------
rizn
I was in your position as well and my recommendation is to buy a good book.

I personally chose Precalculus by James Stewart and it works for me. It's a
thick 1000 pages book with excercises and tests.

It quite well explains all topics, which you would have in high school (from
basic arithmetics to everything you need to start calculus).

I do maths in my spare time (a few hours a week) and I completed 700 pages
over past 3 years.

This year I should complete the book and be ready to do more advanced
mathematics.

95% is self explanatory (if you focus and re-read) and explains well proofs.
When I didn't understand something I found answer on google or asked a few
questions on math stack exchange.

My point. You can absolutely do maths on your own. You don't need classes with
a teacher, but it only depends what kind person you are and what works for
you.

EDIT: Do all exercices and never skip to the next bit if you don't understand
something from the previous part.

~~~
budadre75
You are very persistent! But the efficiency is too low, not to put you down
but precalculus is not something you should spend so much time on, there are
so much more to learn and so much more fun after precalc. At this rate, it's
probably going to take you 10 years to complete calculus and differential
equations, which typically college students take a year to finish and start
applying. Plus there are abstract algebra, discrete mathematics, etc. I know
you are probably having fun with the precalc topics, but trust me it's more
fun afterwards. Imagine what you are learning now is from times before 17th
Century, and you wanna catch up to all the fun today at that rate, it doesn't
sound fun.

~~~
rizn
Thanks for your input :)

Firstly, maths is just my hobby. I didn't pay much attention in my school
years, but later in life I wanted to know more about maths.

I'm a typical code monkey, which does business software and hardly needs any
maths, so agian maths is just my interests and I don't foresee using it in my
career. I might do more maths when I retire as maths is such a broad subject,
it will keep me as a hobby to the rest of my life :).

I'm taking it slow as a) I'm not in hurry, b) I want to have solid foundations
in maths. I want to ensure I understand well basics and proofs and where they
come from. I also do lots of exercices based on acquired knowledge.
"Productivity" is not my main concern :)

Interesting thing I found maths helps me with my work indirectly. In maths we
encounter problems such as "a plane flew north at 300km/h and side wind east
was 30km/h. After 2h, the plane changed direction 30deg...". You need to find
all data and best formulas for the problem. It's like translating
customer/project owner issues to technical ones. It's fun!

I wish to progress my maths faster, but I can do only x hours a week and want
to do it properly. I'm looking forward to more challenging 21st century maths!

------
RossBencina
I agree with other comments that "learn maths" is too broad. You can take a
university degree in maths and still be just at the beginning of "learning
maths." I recommend refining your goal somehow: perhaps to learn math related
to certain applications that you're interested in, or learn math in a certain
area (e.g. high-school algebra, geometry, probability, discrete math, graph
theory, calculus, pure math, abstract algebra, topology, etc).

If you have not mastered high-school algebra and other pre-calculus subjects,
you should start there; most other maths subjects will assume that you know
these things. Calculus takes up a lot of space in upper high-school and early
university courses -- but if you're a developer there may be other subjects
that are more immediately useful to you (e.g. discrete math, linear algebra).

I set out to "learn maths" (that's verbatim what it says on my personal Kanban
board). In the end I took some university classes. For me they provided the
structure and teachers to help me learn. Also, there is a difference between
having an idea about what some math-thing is, and being able to pass an 3 hour
closed-book exam in that topic.

I agree that Khan Academy is a good learning resource that will provide
structure to your learning:

[https://www.khanacademy.org/](https://www.khanacademy.org/)

Purplemath is another good resource:

[http://www.purplemath.com/](http://www.purplemath.com/)

YouTube is full of videos of people running through problems on any
conceivable topic. Definitely search there for help.

Once you've worked your way through the high school prerequisites, I'd
recommend Linear Algebra as a good next course. It has many practical
applications, and is also an entry point towards pure math subjects like
Abstract Algebra. Also, you don't need to know any calculus to study linear
algebra. I like Gilbert Strang's OCW course:

[https://ocw.mit.edu/courses/mathematics/18-06-linear-
algebra...](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-
spring-2010/video-lectures/)

Finally, mathematics is HUGE. The following will give you a bit of an idea:

The Map of Mathematics
[https://www.youtube.com/watch?v=OmJ-4B-mS-Y](https://www.youtube.com/watch?v=OmJ-4B-mS-Y)

------
nhaehnle
I'll make an unconventional suggestion that, in addition to just brushing up
on high-school mathematics, you should read Gödel, Escher, Bach by Hofstadter.
It's a very meandering book, but it contains a lot of interesting ideas
related to math and probably one of the best ways of teaching you about formal
systems, which is really crucial to the axiomatic approach of "real" math.

------
chx
A lot of books listed here can be used to scare anyone away from maths. Too
dry for starters.

I would say you must start with Rozsa Peter's Playing with Infinity
[http://a.co/6MMCE5g](http://a.co/6MMCE5g) to quote an Amazon review

> This book is a gem. I read it as a highschool student, and it played an
> important role in enticing me to become a mathematician. Its emphasis is not
> on practical applications or on solving funny problems: instead, it is an
> inspiring introduction to some of the great intellectual challenges in the
> history of mathematics.

Another in a similar vein
[https://en.wikipedia.org/wiki/One_Two_Three..._Infinity](https://en.wikipedia.org/wiki/One_Two_Three..._Infinity)

You can go and study the textbooks after.

------
ska
The simplest thing I can offer is that you cannot learn mathematics by
reading, watching, or hearing about it - you have to do it yourself.

The way most people run into trouble is by skipping over new concepts quickly
thinking “I get that”, and then ending up in a real muddle with a later
concept that builds on it.

There are better books and lectures and weaker ones, but none are a
replacement for working problems.

------
kraitis
You want to acquire and shoot for the so-called mathematical maturity. More
precisely: to become an autonomous problem-solver and have the know-how to
solve (non-)trivial proofs. Typically this means bridging the gap between
computationally based maths which one is exposed to in pre-school to high-
school years and sometimes in the first year of college/uni, and proof-based
maths which involves and demands a good command of sets and operations on
sets, quantifiers (universal, existential), logical operators (not, and, or,
material conditional, biconditional), and proof methods (direct, indirect
a.k.a reductio ad absurdum, induction, pigeonhole principle, etc.)

A good series of books aimed for pre-school and high-school students to
accomplish just that is The Art of Problem Solving. Google it.

------
elcapitan
Method-wise it could be helpful to get a (lightweight) computer algebra
software and learn how to use it and how to explore knowledge using it. One
thing you won't have when you're out on your own is a method to just try out
stuff and verify that it is correct, or to get better visualizations quickly.
Often you will get stuck with something and need a different angle (which
teachers or other students could normally provide). Then you can just open the
software and play with it.

One place to do that for free on a basic level would be Wolfram Alpha:
[https://www.wolframalpha.com/examples/math/](https://www.wolframalpha.com/examples/math/)

Edit: (I mean this in addition to the learning resources like books and
videos)

~~~
mkl
Personally, I wouldn't recommend Wolfram Alpha, or anything proprietary. If
you have some programming knowledge (as OP is likely to), Sympy and Matplotlib
are much more capable and controllable (and free!).

I think if you're building up skills and knowledge, it should be towards
something you have control over and can use in any situation without worrying
about licences, cost, etc. At my university, we teach undergrad engineers
Matlab, and it just seems like an expensive clunky dead end to me (though
their numerical methods knowledge should be transferable).

~~~
elcapitan
I think for starting from scratch these software library interfaces (that is
what they are) create way more problems than they solve, if OP is not
experienced enough to use them. That doesn't mean they can't get around to use
them finally, but for exploring a space a tool like Alpha that is both
evaluating as well as explaining seems much more useful. Interacting with
Sympy and Matplotlib seems like something you would do when you have already a
solid understanding of what you want to achieve.

~~~
mkl
I'm not so sure. I use Sympy, Matplotlib, Numpy, etc. even when I am just
exploring, and have little idea what I want to achieve.

I do use and recommend [http://desmos.com](http://desmos.com), which is
proprietary (but pretty limited and easily translated to Python). It doesn't
lock features behind a paywall like Wolfram Alpha.

~~~
elcapitan
I was just guessing from OPs post saying that they don't even have basic
highschool math knowledge, from which I would assume that translating
mathematical concepts into concepts in Python would probably create more
difficulties than necessary, whereas in Alpha you can just dump an equation
and then get an interpretation. Matplotlib in particular is kind of an expert
interface for a library, at least when I last checked it out.

Desmos looks interesting.

~~~
mkl
I use Desmos all the time when teaching. It's accessible to students with no
programming knowledge, and the killer feature is that I can just give students
a link so they can interact with a demo I've made.

For example:

Quadratic through 3 points (drag the points around):
[https://www.desmos.com/calculator/tf1f80zgug](https://www.desmos.com/calculator/tf1f80zgug)

Pythagoras's Theorem (drag the sliders):
[https://www.desmos.com/calculator/dbjxbeuzk7](https://www.desmos.com/calculator/dbjxbeuzk7)

Why shearing preserves area (drag the slider):
[https://www.desmos.com/calculator/1vykaqf7je](https://www.desmos.com/calculator/1vykaqf7je)

------
lalala1995
I am not a maths major, however as I currently self-studying Mathematics, so I
hope this would come as a good reference points for you.

I think we both should prepare for a long journey, cause it is the nature of
maths.

I prefer formal and classic textbooks/notes as I think they are the best
resources. Mathematics has been around for a long long time, keeping things up
to date isn't really what should most concerns you.

[0] : [https://www.quantstart.com/articles/How-to-Learn-Advanced-
Ma...](https://www.quantstart.com/articles/How-to-Learn-Advanced-Mathematics-
Without-Heading-to-University-Part-1) This article aims at kick-starting a
career in quant, but the bullet points are really similar to any undergraduate
program.

[1] : Schaum's series Really good textbooks on basic maths, helped me a lot on
those maths modules during my study.

[2] : Any Massive Open Online Course of your choices. I am currently using MIT
OCW. They are basically an Undergrad Course minus interaction with lecturer.
You should ask some of your maths friends to help you out. Good, intuitively
explanation in person helps a lot.

[3] : And last but not least, have fun while doing it. You can participate in
maths competition, watch Youtube videos( 3Blue1brown / Numberphile) Read
Magazines and Journals too, admires the Apollonian aesthetic of Mathematics.

Maths is one of the few subjects where nature > nurture, I think ( and
observed). But take heart.

------
mkl
What do you know already? What kind of stuff do you want to learn about? What
do you want to do with it? Maths is _big_ , and cumulative.

Edit: Re your experience edit, I second the recommendation of Khan Academy.
I'd also recommend the book _Measurement_ by Paul Lockhart.

~~~
RossBencina
> Maths is big, and cumulative.

The cumulative part needs to be emphasised. Almost every topic in math, from
grade-school on up, has pre-requisite knowledge. If you miss key knowledge you
will easily get lost, so it's important to take things step by step.

------
richardjdare
I guess I'm similar in that I left high school with very little mathematical
knowledge. I've struggled with _many_ _many_ maths books over the years which
usually assume you have a certain background, or don't explain things very
well, or only explain things using notation which they don't explain.

The books that helped me the most are "Mastering Mathematics", and "Mastering
Advanced Pure Mathematics" both by Geoff Buckwell. They will take you through
UK GCSE and A-Level maths, from nothing to calculus. They have plenty of
examples and exercises to work through. Just start at the beginning and work
through them.

They are based on a UK curriculum though, so that may or may not be what you
want.

------
castle-bravo
I suggest you take a look at Project Euler [0]. It's a bunch of math puzzles
that usually require programming to solve. In order to solve most problems in
a reasonable amount of time, you'll need to use results from number theory and
other areas. Once you've used a result, you can try to prove it or to
understand the proof.

Project Euler emphasizes number theory, which is the most approachable field
of mathematics for total beginners because the background you need is just
addition and multiplication. You should be able to make progress in number
theory much faster than by taking the traditional route through calculus.

Another advantage of the Project Euler approach is that you'll learn how to
put math into code, which is fun and tremendously valuable.

Another thing I recommend is learning geometry [1]. The way to do this is to
use a ruler and compass to draw various shapes and then prove that those
shapes have certain properties (e.g. prove that an angle really is a right
angle). I think this approach also has more merit than the traditional
approach, because you learn how to write proofs without driving yourself to
exhaustion and frustration with calculus exercises. Geometry is really fun if
you have a visual bent.

I also suggest learning linear algebra before calculus, because it's more
useful to programmers and more accessible. The way to learn linear algebra is
to study OpenGL and OpenCV with an emphasis on graphics and machine vision
theory. Making things work in OpenGL is more rewarding than just doing
exercises out of a textbook.

At a certain point, you'll find that you can't progress any further in number
theory or geometry without calculus and complex analysis, at which point
calculus should be a fun challenge for you instead of a tough slog. You'll
need multi-variable differential calculus and linear algebra to understand
neural networks.

In summary: Have fun! Math is fun! Learn to write proofs early on! Watch
Numberphile [2]!

[0]: [https://projecteuler.net/](https://projecteuler.net/)

[1]:
[https://en.m.wikipedia.org/wiki/Euclidean_geometry](https://en.m.wikipedia.org/wiki/Euclidean_geometry)

[2]: [http://www.numberphile.com/](http://www.numberphile.com/)

------
IvanK_net
Whenever I tried to self-learn anything, it was a very bad idea. Some parts of
the subject I enjoyed and other parts I hated. I tend to make my own
conclusions, which parts are useful and wich are a waste of time (so I tend to
skip them). I tend to filter out the material this way, in order to make
learning less painful and more fun.

My conclusions (what is useful and what isn't) were always wrong and I ended
up not learning anything properly, not getting a proper understanding of
anything.

Please, if there is still any such option for you in your country, always
choose a proper school education instead of self-learning. It is really great,
when there is some leader with a proper understanding of the subject (a
teacher) and others, who are having "the pain" with you (classmates), so you
can see you are not "suffering" alone, and you don't start making your own
conclusions (since you would see, that others are taking seriously what you
wanted to call a waste of time). Classmates also help each other during the
learning process.

So personally, I think a person gives up self-learning as soon as it becomes
too painful / boring. The best way to overcome it is to see other people
around you going through the same process, or to see somebody who you admire,
who has already gone through the same process (it could be your teacher, your
parent, your role model etc.). You could call that "the motivation".

~~~
Zeule
This answer is absurd, and only true for people with little motivation. Self-
learning does require more motivation and "grit" to keep going, and some
planning early on, and there is a higher dropout rate. But in many cases, it
is the best option. I've found that this is especially true for many parts of
STEM fields. It is a far more productive and effective use of time to work
through problems than listen to lectures. Of course, it is also helpful to
have people you can ask questions when you don't understand as comes with a
class, but that can also be found online. I also find that I often understand
things I learn on my own more deeply because I can go at my own pace and have
more time (time not spent listening to lectures or preparing for exams on a
fixed deadline) to draw from a variety of resources.

One of the most important skills someone can learn is how to learn, and
especially how to solve problems and keep going when it is difficult.

For math through high school level/early university, I'd suggest Art of
Problem Solving (if you can handle it). It teaches by having people solve
problems rather than presenting mathematical techniques to memorize. Some of
them are straightforward, but many are tricky problems and fun puzzles with
elegant solutions. It helps you gain a good sense for numbers and problem
solving, and an appreciation for the beauty of math. The teaching method helps
you intuitively understand rules rather than memorize them. They also have a
nice gamified online practice system (Alcumus) to go along with the first half
of their books.

For some higher level, more applied areas like linear algebra there are some
good coding-based courses like codingthematrix.com. Project Euler is also
another good option for practicing math with programming.

~~~
pacnw
That being said, there is something for the pressure cooker environment that
forces you to consider and learn the hard topics. I have gone back to school
twice as an adult, first for a pure math degree, and now for comp sci -
machine learning. Both times, the amount of pain it takes is just not
something a person with average or above average motivation would go through
on their own.

~~~
Zeule
True, though on the flip side sometimes it is easier to take the time needed
for deeper understanding outside the pressure cooker environment of classes.
Pressure cooker environments can be motivating to keep moving forward, but
sometimes at cost of depth and intuitive understanding.

------
candu
There are several amazing suggestions in this thread already.

I'll toss in vote N+1 for "How to Solve It" by Pólya: once you get past the
hurdle of just understanding notation / language and some of the basic
concepts, mathematics becomes much more about problem-solving.

Aside from that...

Oliver Byrne's Euclid:
[https://www.math.ubc.ca/~cass/euclid/byrne.html](https://www.math.ubc.ca/~cass/euclid/byrne.html)
\- a graphical treatment of Euclid's Elements. Much, much more accessible than
earlier renditions, and a great introduction to methods of proof.

Vi Hart's videos:
[https://www.youtube.com/user/Vihart](https://www.youtube.com/user/Vihart) \-
she does a wonderful job of conveying the wonder of mathematics in a clear,
informative manner.

A Mathematical Mosaic: [https://www.amazon.ca/Mathematical-Mosaic-Patterns-
Problem-S...](https://www.amazon.ca/Mathematical-Mosaic-Patterns-Problem-
Solving/dp/1895997283) \- this was one of the books that got me excited about
mathematics as a kid. The material is advanced by high school standards, but
presented in a way that invites you to think / learn / generalize.

------
ll350
I almost lol'ed when I read this, but I think I have good suggestion for you.
"Maths - A Student Survival Guide" by Jenny Olive:
[https://www.amazon.com/Maths-Students-Survival-Self-Help-
Eng...](https://www.amazon.com/Maths-Students-Survival-Self-Help-
Engineering/dp/0521017076) She basically starts out with the simplest algebra
(fractions) and gradually works up to topics in 1st semester Calculus. And she
starts each chapter with a short quiz to test yourself and skip ahead if
already know the material. This book is great for what you are describing, if
I'm understanding you. I picked it up when I was preparing to return to
college after being away for many years. I supplemented it with another book I
highly recommend: "Mastering Technical Mathematics" by Stan Gibilisco and
Norman Crowhurst: [https://www.amazon.com/Mastering-Technical-Mathematics-
Third...](https://www.amazon.com/Mastering-Technical-Mathematics-Third-
Gibilisco/dp/0071494480)

I found that Jenny Olive's book was well designed and preferred it's style to
any math textbook I have ever used. Even so occasionally I would get bored
while working thru it. That is when I flip thru the Stan Gibilisco's book,
which was full of interesting looking problems and examples. When I would try
to solve one of them, it would become apparent that I still need to work on
the fundamental concepts that were prerequisites for solving the problem. Thus
I would return to Jenny Olive's book right where I left off, re-energized by
the desire to master those fundamentals that she covers so well.

------
jimnotgym
I studied my 'college level' maths with the Open University. The materials
were excellent, being designed specifically for self learning and they offer
access to a tutor too. OU courses also give you proper qualifications. I am
not sure how it works for non UK students. open.ac.uk

However I did mine before the cuts to higher education in the UK and the
courses are much more expensive now. This is very sad as it enabled me to
change careers.

~~~
IndrekR
Second that! OU math courses were really good. This involves the materials and
tutor support. Took them in 2003..2004, so things may have changed.

------
james_niro
Sign up for pre-calc class on edx.org it is free and offered through ASU.
First you take the test to see where your knowledge stands then you learn
based on that test.

They use ALEKS learning which is a great tool to learn online. Make sure to
take notes and do the problems.

My advise don’t pay for math at community colleges because they use the same
tools but you have to pay somewhere between 300 to 500 for courses which is
waste money.

------
trengrj
I would select books based on your interest. I find the Dover publications
good because they are both cheap and slightly older, this means they are less
focused on undergraduate monetisation (version hopping, not supply answers to
problems, glossy print), and more focused on proofs and algorithms. You can
see them here [http://store.doverpublications.com/by-subject-
mathematics.ht...](http://store.doverpublications.com/by-subject-
mathematics.html). I particularly liked Probability: A concise course, and
Number Theory by George Andrews.

Amazon used to have a great number of graduate preparation book lists which
always included books such as Rudin's Principles of Mathematics Analysis, and
Halmos' Finite Dimensional Vector Spaces. These classic maths books are
brilliant but usually easier to understand if you already have some experience
with the material.

Final advice is to find a study partner as it can be hard to track how you are
going and keep motivated, especially without the instant feedback loop you get
with programming.

------
cybernoodles
If you love math, read the books that interest you most, and read about math
in the context of your interests. You will get much more out of that than
reading books that others told you about. And you will also stick to it and
turn it into something you enjoy rather than feeling guilty for not reading
enough of a book you have less interest in.

------
watwut
I would suggest to browse through coursera and pick something free and easy
(since you labeled yourself pre-highschool). If it turns out too difficult,
dont worry, unsign and pick something else. I never tried Khan Academy, but
people seem to praise that too. Moreover, maybe just taking high school math
book and exercises book would be fine.

Most importantly, at stage you are at, learning math should consist of doing a
lot of exercises - with increasing difficulty. Just like with sports, you cant
learning it by reading theory only. Pick up book with a lot of exercises and
do homework if you sign to some course.

The rule of thumb is, that if you can solve all exercises without having to
think or being occasionally frustrated, then the exercises book is too easy
for you. If you have difficulties, then it means that you are learning. (If
you end up completely stuck then you need something easier.)

Videos and such are fine, but really really focus on exercises.

~~~
alexdowad
I found Khan Academy really helped with high-school algebra, single-variable
calculus, and especially multi-variable calculus (the visualizations were
great). Khan Academy's linear algebra course was awful and I dropped out
there.

------
abcve
If you want learn Mathematics from bottoms ups I'm think this book[1] is for
you. This list of Mathematics books[2] too is awesome.

[1] Mathematics: From the Birth of Numbers by Jan Gullberg [2]
[https://mathblog.com/mathematics-books/](https://mathblog.com/mathematics-
books/)

------
kuroshit
Firstly, don't use those books that suggest to teach you more than one subject
at a time, in a traditional way. In mathematics those kind of books do not
work, at least in my honest opinion that is. What I mean by this is that, you
will have some knowledge over the subjects taught by the book but you will
have no _understanding_ what so ever, or it won't be good enough in the long
run.

As you said, the problem was that you didn't pay any attention while in high
school. It was because you had no interest, and if you try to self-learn
mathematics the same way you tried learn basic maths in school, then you will
also fail.

The answer to all this is better books or a different kind of approach. I
should mention that this is all from my own ongoing experience. There are
traditional books that cover high school math[0]. And there are bad ones and
good ones. The good ones still throw definitions and theorems at you, but it's
more clear and concise, and most importantly understandable.

Now comes the new kind of books which try a different approach to the subject
at hand. They try to give more understanding that anything else. I only read
"Burn Math Class" by Jason Wilkes and "Math, Better Explained" by Kalid Azad.
These books lack exercises, which are, in my opinion, as important as
understanding. But one doesn't work without the other.

As for the future just follow this[1] so that you won't get information
paralysis or other difficulties that come when there are a lot of choices.

[0] [https://www.physicsforums.com/insights/self-study-basic-
high...](https://www.physicsforums.com/insights/self-study-basic-high-school-
mathematics/) [1] [https://www.physicsforums.com/threads/micromass-insights-
on-...](https://www.physicsforums.com/threads/micromass-insights-on-how-to-
self-study-mathematics.868968/)

------
cyphar
Khan Academy is pretty good, though in my experience most of the videos focus
on the way high-school courses are structured rather than teaching in a method
that is the most intuitive (if you're planning to dedicate more time than an
average high-school class timetable then it would make sense to learn in a
more methodical way than the "scatter shot" that most high school curricula
use).

Unfortunately a lot of the good "starting out" maths textbooks I know of are
basically university level (though it should be noted that first-year of
university mathematics is basically re-learning all of your previous
mathematics knowledge but with new insights). While I wouldn't stop you from
trying to read a university-level textbook, most of them are structured in a
way that requires some familiarity with the topic before reading.

------
ZenoArrow
I'd like to learn vector calculus (to better understand Maxwell's Equations).
Does anyone have any learning resources they could recommend? I've got this
video in my Watch Later queue on YouTube, but if there are any other resources
that give a clear introduction to the material I'd be interested in them. To
give a bit of background, I learnt the basics of calculus back in school but I
haven't used it in over 15 years, so I'm likely to need to get the basics down
first.

[https://www.youtube.com/watch?v=wsOoClvZmic](https://www.youtube.com/watch?v=wsOoClvZmic)

~~~
RossBencina
Edward Frenkel's UC Berkeley course:

[https://www.youtube.com/playlist?list=PLaLOVNqqD-2GcoO8CLvCb...](https://www.youtube.com/playlist?list=PLaLOVNqqD-2GcoO8CLvCbprz2J0_1uaoZ)

------
urmish
A good way to start is learn set theory (start of with naive set theory Dover
books are cheap:
[http://store.doverpublications.com/0486814874.html](http://store.doverpublications.com/0486814874.html))

And some introductory mathematical logic.

From this you can immediately move to Analysis on the real line (up to Reimann
integrals)

Linear Algebra is also something you can start (Kenneth Hoffman and Ray Kunze
[https://www.amazon.com/Linear-Algebra-Kunze-
Hoffman/dp/93325...](https://www.amazon.com/Linear-Algebra-Kunze-
Hoffman/dp/9332550077))

Once you are comfortable with Riemann Integrals, you can start attacking
Complex Variables (John Conway has excellent springer texts: Functions of One
Complex Variable Vol 1 and 2)

Some texts you should look at after you understand basic set theory:

Analysis:

1\. [Michael_Spivak]_Calculus - good book for introduction to real analysis

2\. [johnsonbaugh,pfaffenberger]_Foundations_of_Mathematical_Analysis - Dover
publications

3\. [Vladimir_A._Zorich]_Mathematical_Analysis_I - well written but less
known. Recommend checking it out.

4\. [Gerald_B._Folland]_Real_Analysis_Modern_techniques_and_their_applications
- My top pick, but a tough read.

5\. [Rudin_Walter]_Principles_of_Mathematical_Analysis - classic book

Linear Algebra:

1\. [David_Lay]_Linear_Algebra_and_Its_Applications -

2\. [Friedberg,Insel,Spence]_Linear_algebra - Undergraduate level text

3\. [Hoffman,Kunze]_Linear_Algebra_2nd_edition - Graduate level text. My top
pick.

4\. [Gilbert_Strang]_Introduction_to_Linear_Algebra - Undergraduate level
linear algebra. Same guy has MIT OCW lectures.

5\. [Peter_D._Lax]_Linear_Algebra_and_Its_Applications

Complex Variables/Complex Analysis:

1\. [John_Conway]_Functions_of_One_Complex_Variable_I - My top pick

2\. [Lars_Ahlfors]_Complex_Analysis_(Third_Edition) - Classic. Not a big fan
though.

------
bryanph_
There is no such thing as "bottoms up" in mathematics, I think a relatively
broad examination of mathematics might be useful so that you can discover what
is relevant to the subjects you are actually interested in. I learned this the
hard way. Here you have Feynman talking on this:
[https://www.youtube.com/watch?v=YaUlqXRPMmY](https://www.youtube.com/watch?v=YaUlqXRPMmY)
Finding a real world use case for the math you learn will be crucial if you
self-study to keep yourself motivated to keep going.

------
NiklasMort
I'd start with this video to get an overview of all the topics and areas that
mathematics entails (some might be unknown to you)
[https://www.youtube.com/watch?v=OmJ-4B-mS-Y](https://www.youtube.com/watch?v=OmJ-4B-mS-Y)
. Then you go ahead and research a bit what sounds interesting to you and then
you might google that topic and add "foundations" to that google search. It's
just that most school/university math is heavily focused on analysis and
algebra but there is so much more!

------
dmurthy
I was in a similar situation a few years back but my weakness was calculus. I
really had no grasp of what it was used for until I had to learn as part of my
work. I was on the brink of losing my job if I hadn't. I had a very patient
boss as mentor who spent hours teaching me the practical applications of it -
remotely over phone. Now I love calculus.

So I believe the best way to learn math is by finding areas where you are
forced to apply it. And it is never too late. I learnt most of it after
turning 30.

------
tel
Probably not a perfect book for someone at the HS level, but just a bit past
that I would HIGHLY recommend MacLane’s Mathematics Form and Function.

It’s a revisionist history of mathematics aimed to demonstrate how ideas flow
from one to another. He eliminates a lot of dead-ends and takes a perspective
as to one “ought” to move from one subject or discovery to another. OTOH, the
historical perspective is both readable and often a missing piece which makes
other math tougher.

------
asdf1234tx
By your early 20s you should know if you need a teacher, or just a guide, or
no one but yourself(plus textbooks&internet), for medium level difficulty and
down. If you have to ask, I'm pretty sure you need a teacher.

For example, by the time I was in my last year of college, I regularly skipped
90% of class time, and showed up to take tests. I handed in assignments at
instructor's office.

I don't process auditory information all that great, but I'm extremely
efficient and robust in my processing of visual information including the
written word, graphics, pictures, and video.

On some topics I have sought a mentor, with whom can provide a little
guidance, and bounce ideas off of. But his is extremely far from the
traditional instructor role.

So in conclusion, most people do far better in class. But for me it's slow,
tedious, distracting, and out of sync with my normal learning processes. The
few other people that I know who are like me, also skip the classes and have
done well. You'll know because from birth, you'll be an excellent test taker,
get bored silly in classroom environments, and have likely read 1000 books or
more in just two or three years. We never asked "do I need a class", we just
stopped going.

------
schlinb
I've been using YouTube videos. One recommendation -- watch the videos at 2x
speed. The information is actually easier to absorb when it's presented
quickly.

~~~
ivan_ah
+1 for this. The chrome extension [1] is even better since it allows for 0.1
playback speed increments, e.g. 1.8x speed, or 2.3x speed.

I've watched some lectures with really slow speakers at 3x and was still able
to understand, and really wondered what the people watching the lecture live
at 1x must have felt...

[1] [https://chrome.google.com/webstore/detail/video-speed-
contro...](https://chrome.google.com/webstore/detail/video-speed-
controller/nffaoalbilbmmfgbnbgppjihopabppdk?hl=en)

~~~
rdc12
In person and by video of the same material experience can be quite different.
One of my current lecturers, is very engaging, with great natural flow in the
lecture room, but watching via video somehow fails to capture this.

That is for lectures that I miss due to a clash, not re-watching.

------
dprophecyguy
I know people are recommending a lot of books here but I want to say this, I
know a lot of you guys might going to shit on me but telling someone about 5-6
book in order to self-learn maths is never going to help. I see that when
someone people ask for help instead of relating what he is really asking for
that he wants to understand and learn math people start telling the names of
these books that they knew about not thinking about the effect that straight
up throwing 6 book titles will do no good to the person. So now I have defined
the problem I will tell you only one resource I know its bit of understatement
but I think Learning math from Khan Academy would be sufficient for you. And
once you find something on Khan Academy and you are done with it. I will
recommend you this site www.brilliant.org And if you still want to practice
just search for test question papers and cheat sheet on the topic you want to
practice. Print the cheat sheet beside you, and do as many questions you can
with the help of cheat sheet.

To the person who is asking the question and people who are writing the
answers I just want to say that knowing a lot of good resources to go through,
is not the learning. Now I see this thing happening everywhere, people want to
know about the process so much that instead of doing what needs to be done
they kind of start storing this metadata of the process and this thing is
happening a lot on the internet. People know a lot of resources, a lot of
tutorials and video and a shit load of things. But when it comes to execution
and practice I can hardly say only a very few might have gone to complete what
they have started. I am saying all this because I have gone through this cycle
myself I have wasted 2 years of my life. Collecting resources related to ML,
web development, Math, Psychology, Philosophy you say whatever you find
interesting I will tell you some famous book or MOOC course on that. So I will
ask everyone this question take a look on 1 back of your life, if you guys
were trying to learn anything do a retrospective whether you really have
learned anything, write things that are going right, right things going wrong
and start doing things, making project, solving problems really doing the
things not just trying to perfect the process. I can go on but I think I have
made the point if I keep writing more I think I will contradict myself that
it's not about what and how info you get it's about you get something
actionable out of something. If some 2-minute video gives you something
actionable to do rather than going through a 2-hour chapter in textbook there
is no point of going through 2-hour chapter. Knowledge is all about applying
not learning the facts and saying it around to your friends I know it feels
good but nothing comes out of it in real life.

~~~
dsacco
I agree with the basic thrust of your comment, which as I interpret it boils
down to saying it's unhelpful to throw book titles at a person without any
context or guidance. However, I disagree with this point:

 _> If some 2-minute video gives you something actionable to do rather than
going through a 2-hour chapter in textbook there is no point of going through
2-hour chapter. Knowledge is all about applying not learning the facts and
saying it around to your friends I know it feels good but nothing comes out of
it in real life._

There are optimal and suboptimal ways to learn things, sure. But some things
legitimately cannot be reduced from a complicated textbook chapter to a
straightforward YouTube video. You can improve exposition, but that comes with
its own time efficiency trade-offs, and you won't meaningfully simplify the
material without compromising significantly in depth of coverage. In
particular, even extremely good video series like 3Blue1Brown's Linear Algebra
have neither the breadth nor the depth to replace the material in any given
chapter of e.g. Axler's _Linear Algebra Done Right_. The exposition is
certainly clear, and you can get a nice overview, but you're actually _not_
learning enough to apply the mathematics if you watch a video on it.
Ironically, such a video is much more likely to leave someone able to talk
about the subject but woefully incapable of actually _doing_ it. The videos
that _can_ be used for learning are mostly lectures.

More importantly I think your conception of working through a chapter is
incorrect. If you're spending a significant amount of time working through a
textbook chapter, you're either 1) actively learning the material and _doing_
the mathematics, not just passively reading it, or 2) you're not prepared for
the material, and you're not learning efficiently. Finding a video to simplify
the material doesn't resolve #2, it just disguises it. When you learn
mathematics from a textbook, you _should_ be applying the material by working
through the exercises. You can't immediately jump into e.g. data analysis
after learning linear algebra, and you certainly can't do so by watching a
video on the subject.

I'm fully with you that replying to these questions with textbook
recommendations is unhelpful, but not because they're textbooks. Textbooks are
great! The hard reality is that you can't simplify most mathematics into an
easily, quickly digested format, _especially_ if you want to apply it. There
are simply too many prerequisites for most material and too many unknown
unknowns that can leave glaring blind spots. "There is no royal road to
geometry" is a saying for precisely this reason.

~~~
Chris2048
> it's unhelpful to throw book titles at a person without any context or
> guidance

Here's one reason it's unhelpful; After 4 or 5 people list a bunch of books,
you now have a long list of books, and no way to know which to start with,
unless there is a lot of overlap - you could just as well google/search amazon
and look at ratings, which would give you far better results.

Furthermore, many people tend not to read multiple math books on the same
subject, so they just recommend what they know w/o having any knowledge of how
that book fares against other suggestions.

If they expand on either their own credentials, and reading on the matter, so
you have the context of their knowledge; or give reasons _why_ that book is
good, this gives a better basis for comparison.

As an aside; I like the coursera structure of following video lectures with
content summaries of what was covered in the video. The advantage is 1) quick
reference of material without having to search through a video; 2) if you
already understand the topic, you can often just read the summary, and skip
the video if you think you already know the material.

Another aside; one problem with books versus videos is often books are long
compendia of a field, where as videos are shorter. if you already know topic A
and want to learn B, then there may already be a video on topic B, where as a
book might cover B in some chapter. For these cases, it might be good to
discuss individual chapters/sections of book, but that assumes you can ready
them somewhat independently given prior knowledge.

------
forapurpose
Jeremy Kun (who is worth reading beyond this article) has a post with very
useful insights:

[https://j2kun.svbtle.com/mathematicians-are-chronically-
lost...](https://j2kun.svbtle.com/mathematicians-are-chronically-lost-and-
confused)

It begins (and there is far more of value in the post):

 _Many people who are in this position, trying to learn mathematics on their
own, have roughly two approaches. The first is to learn only the things that
you need for the applications you’re interested in. There’s nothing wrong with
it, but it’s akin to learning just enough vocabulary to fill out your tax
forms. It’s often too specialized to give you a good understanding of how to
apply the key ideas elsewhere. A common example is learning very specific
linear algebra facts in order to understand a particular machine learning
algorithm. It’s a commendable effort and undoubtedly useful, but in my
experience this route makes you start from scratch in every new application.

The second approach is to try to understand everything so thoroughly as to
become a part of it. In technical terms, they try to grok mathematics. For
example, I often hear of people going through some foundational (and truly
good) mathematics textbook forcing themselves to solve every exercise and
prove every claim “left for the reader” before moving on.

This is again commendable, but it often results in insurmountable frustrations
and quitting before the best part of the subject. And for all one’s desire to
grok mathematics,_ mathematicians don’t work like this! _The truth is that
mathematicians are chronically lost and confused. It’s our natural state of
being, and I mean that in a good way. ..._

------
cybernoodles
If you love math, read the books that interest you most, and read about math
used within the context of your interests. You will get much more out of that
than reading books that others told you about. And you will also stick to it
and turn it into something you enjoy rather than feeling guilty for not
reading enough about math in a context you cant strongly relate to or be
attracted to. Take advantage of your interests.

------
hugocbp
So, I decided to go back to study Math from the very basics to try some more
advanced Data Science stuff later and I'm really enjoying Krista King's
courses on Udemy.

They are very clear and straight to the point. She currently has courses on
Fundamentals of Math, Algebra (I and II on the same course), Calculus (I, II,
III, in separate courses) and Geometry and Trigonometry. She is also preparing
some Probability & Statistics course and later a Linear Algebra one.

That is the only way I could self-learn Math without quitting after more than
12 years without doing math at all in real life. I was originally a Lawyer so
I didn't care much for Math in school and didn't have anything relate to Math
for all those years until now.

I tried a bunch of other paths, like challenges, For Dummies books, more
advanced University open courses... Nothing stuck until her courses.

They are not free, but you can find basically permanent Udemy coupons to buy
each course for USD 10.

Here is her profile:
[https://www.udemy.com/user/kristaking/](https://www.udemy.com/user/kristaking/)

------
baby
Go in a store, look at math books and find out one that looks like it:

* teaches you things you don't know

* relies on things you know for the most part

* do it well

You're in luck, a few years ago pedagogy in math was not really a thing. Now
we have khanacademy and plenty of other courses (Coursera, Udemy, ...) and
youtube videos to help you.

There are different goals when you're learning math. Do you just want to get
more exposure? Then focus on youtube videos (subscribe to some math channels)
and novel-type of books.

You want to get a deep understanding of some area in math like algebra or
number theory? Then you're going to have to buy these books I told you about
and do the exercises in them. If you've chosen your book well, they will have
good explanations on how to solve these if you get stuck. In any case,
exercising is the way to go.

For those who recommend a good teacher, I think this is only a good advice if
you're just starting and too many concepts are scary to you, but if you're
already have a good basis then good videos, online courses and text books are
enough and even better than a teacher.

------
nickpsecurity
The last time people asked, I collected the responses so I could do the same
thing as you. Note that I'm wanting to learn it in a way where I can do
proofs. So, I have general-purpose books and stuff for that. I just ordered
the three books I've seen pop up the most. Although 2 are in the mail,
Concepts of Modern Mathematics by Stewart just got here yesterday. It had an
_awesome_ opening that made me wish the math I was taught in school was done
like this back when I went. Makes newer stuff make a lot more sense, too. I
included a link to Dover that has a Google Preview button on it where you can
read full, first chapter for free to see if it's what you like. Other two are
more about exploring and proving things which may or may not interest you. I
added them in case anyone is reading your question to learn that stuff.

Concepts of Modern Mathematics by Stewart

[https://www.amazon.com/Concepts-Modern-Mathematics-Dover-
Boo...](https://www.amazon.com/Concepts-Modern-Mathematics-Dover-
Books/dp/0486284247)

Dover Version with Google Preview Button

[http://store.doverpublications.com/0486284247.html](http://store.doverpublications.com/0486284247.html)

Introduction to Mathematical Reasoning: Numbers, Sets, and Functions

[https://www.amazon.com/Introduction-Mathematical-
Reasoning-N...](https://www.amazon.com/Introduction-Mathematical-Reasoning-
Numbers-Functions/dp/0521597188)

How to Prove It by Velleman

[https://www.amazon.com/How-Prove-Structured-
Approach-2nd/dp/...](https://www.amazon.com/How-Prove-Structured-
Approach-2nd/dp/0521675995)

------
ecesena
Where do you live? In most Europe public university is free and you can attend
courses. I don’t know about US or other parts of the world.

If you just “show up one day”, you’ll be seen as pretty weird, but hey, it’s
math, there will certainly be people more weird than you. My recommendation
would be to introduce yourself to the teachers and be very
careful/polite/diligent in attending the course. Some teachers can be very
opinionated, just go with the flow.

Given your initial level, I’d recommend to focus on breadth instead of depth.
It’s likely the case that you don’t know enough yet to make an informed
decision.

For example, in my case I love algebraic geometry, number theory, and
cryptography. And I hate calculus with a passion, including analytical number
theory. None of this was clear to me until the 2nd-3rd year of university,
foundamentally because what you learn in high school (liceo, in Italy) is kind
ok fixed and biased towards calculus.

------
123212321
Work extremely slowly, possibly with a tutor through Principles of
Mathematical Analysis by Rudin. The book lays down the real foundations of
calculus and higher math. Learn the math as you come to it. Dont waste time
going extensively through lower mathematics. You will get bored and life is
too short for that.

------
dsacco
Rather than simply give you a list of resources or textbooks, I’d like to give
you a broad “map” of the various domains of mathematics, this way you
understand what you’re working towards. I’d also like to recommend how you can
maximally optimize your self-study, as someone who mostly self-learned enough
mathematics to be active in research. I think this meta-direction is just as
important as the resources you choose to learn from.

Mathematics, in my opinion, can be divided in two very major ways ways as
concerns pedagogy. First, most mathematics computation-based or proof-based.
Math research in general is about proving things, and most “serious” math
books after a certain level are almost exclusively about proving properties
instead of calculating results. On the other hand, most applied mathematics is
computationally inclined, and uses methods derived from research. Here is a
simple example: I can ask you to calculate the square root of 2 or I can ask
you to prove that it’s irrational.

It’s important for you to know what you want. Do you, for example, want to
achieve theoretical mastery of linear algebra that subsumes e.g. solving
linear equations, or do you just want to be able to execute the computational
methods proficiently? As you get into higher mathematics the line here blurs,
but different resources may still emphasize one approach or the other.

Now let’s talk about the domains of mathematics. Broadly speaking, we can
divide them into algebra and analysis. More accurately, we can divide their
methods into algebraic or analytic. Algebra is concerned with mathematical
structures and their properties, like fields, groups, rings, vector spaces,
etc. Analysis is concerned with functions, surfaces and continuity. I like to
say that in algebra, it’s difficult to identify what you’re studying and
whether it’s worth studying it, but once you do there is a lot of machinery
that’s relatively straightforward to prove. On the other hand, in analysis
it’s easy to find things worth studying, but difficult to prove interesting
things about them. For example, if you can prove that what you’re studying
satisfies all the conditions of a _field_ , you immediately can prove many
other things about it. On the other hand, the toolbox of analysis is widely
applicable to many things, but it often seems like you’re trying a hodge podge
of techniques, and the proofs can look kind of magical at first. For a
concrete example, try to prove that 1 + 2 + 3 + ... + _n_ = _n_ ( _n_ \+ 1)/2.

Now let’s take a tour of mathematics at the undergraduate level. In
theoretical (but not necessarily pedagogical) order we have: set theory,
calculus, analysis, topology and probability theory on the analytic side; and
set theory, linear algebra and abstract algebra on the algebraic side.
Analysis can be further subdivided into real analysis, complex analysis,
functional analysis, harmonic analysis, Fourier analysis, as you move from
foundational material to specialized material. Similarly abstract algebra
divides into group theory, ring theory, finite fields, Galois theory, etc.
Probability breaks down into discrete versus continuous random variables,
measure theory, statistics (on the applied side), etc.

Here is my concrete advice regarding learning the material. First, internalize
the idea that mathematics is “not a spectator sport.” You learn it by doing
it, not just by reading it. Every time you’re sitting down with a textbook,
attempt every exercise in good faith, and take an author’s lack of a proof as
an invitation to prove it yourself. The first time you read a chapter, read it
briskly, skipping over what you don’t know to get to the end of the chapter.
Let that material percolate in your mind a bit, even though you won’t
understand much of it. Then read the chapter again, but slowly. Write down
every definition, theorem and proof. Try to prove each theorem yourself before
reading the author’s proof. For anything unclear, search for different
examples of that concept or for different proofs of that theorem. Then attempt
at least half of the exercises at the end of the chapter. You will struggle _a
lot_ , and you will be demotivated _a lot._ It will feel frustrating and you
will be humbled continually. But I can promise you that if you keep
challenging yourself this way you will continue to improve. It’s not enough to
find the right textbooks or the right resources, you need to study them the
right way - in an active, focused way.

That brings me to my second piece of advice. There are many good books and
resources for any given topic. Different people respond more favorably to
different types of exposition. Sometimes you’ll receive a book suggestion and
realize it’s not for you - that’s fine! It might still be a good book. For
example, I rather like Rudin’s _Principles of Mathematical Analysis_ , but
please don’t try to learn from it without a teacher! For any given topic, find
four or five strong suggestions, preferably all at your level of capability at
the time. Then read the preface and the first 10 pages of the first chapter in
each book. Look at the table of contents to understand not only the coverage
of topics, but the pedagogical _arrangement_ of topics. Proceed with the book
you have the strongest affinity for, and use other books when the author is
unclear.

Finally, _now_ I will give you textbook suggestions:

1\. Set Theory: _Naive Set Theory_ , Halmos.

2\. Calculus: _Single Variable Calculus_ , Stewart; _Multivariable Calculus_ ,
Stewart; _Calculus_ , Spivak.

3\. Linear Algebra: _Linear Algebra and Its Applications_ , Strang; _Linear
Algebra Done Right_ , Axler; _Linear Algebra_ , Hoffman & Kunz; _Finite
Dimensional Vector Spaces_ , Halmos.

4\. Analysis: _Analysis I_ , Tao; _Analysis II_ , Tao; _Understanding
Analysis_ , Abbott; _Principles of Mathematical Analysis_ , Rudin.

5\. Abstract Algebra: _A Book of Abstract Algebra_ , Pinter; _Abstract
Algebra_ , Dummit & Foote; _Algebra_ , Artin; _Algebra_ , Hungerford;
_Algebra_ , MacLane & Birkhoff; _Algebra: Chapter 0_ , Aluffi.

Start with that, and once you've gained sufficient mathematical maturity look
for more targeted and specialized resources. I also recommend that you read
_Concrete Mathematics_ by Graham, Knuth, Patashnik; and _Mathematics: Its
Content, Methods and Meaning_ by Kolmogorov, Aleksandrov and Lavrent'ev. These
two, especially the latter, are good for covering a variety of mathematics at
once. They are good for both learning and mathematical "culture."

I can't stress this enough: it's important that you really optimize the way
you're studying and what your goals are, instead of trying to collect as many
book recommendations as possible.

------
BeetleB
One thing I did not do while learning mathematics that I wish I had done:

When you learn something new, actively look for interesting problems that can
be solved with it.

I mostly relied on the exercises (which are good to do), but since they were
not problems I invented, I've forgotten most of the material. However, if you
use mathematics to solve fun problems that _you_ invent, you are more likely
to remember it. And, to be honest, I suspect you also learn it better.

Most people take a problem and try to learn the math to solve it. I suggest
inverting the process. Learn the math, and seek out problems solved by that
math. Don't be afraid to make it an "overkill" solution. Even if the problem
has a simple solution, use your fancy new tools to solve it.

------
jbduler
In the videos series, Khan is excellent:
[https://www.khanacademy.org/math/multivariable-
calculus/mult...](https://www.khanacademy.org/math/multivariable-
calculus/multivariable-derivatives)

------
dahart
A lot of great resources and suggestions here, and I see some discussion of
books vs teachers.

Let me suggest a strategic approach: decide on a goal that is not math, but is
something you're interested in and requires math to achieve. Learning math for
math's sake is a bit abstract, and can go many different directions, so it
might help to pick a specific application of math as your vehicle, to be both
motivated and focused.

For example, I love computer graphics, and making pictures via programming
forces me to learn programming and math, and sometimes a bit of art too. It's
hard to play with ShaderToy, for example, without doing some math, and how
much math depends on what kind of picture you want to make.

------
undecidabot
I really enjoyed reading "Discrete Mathematics with Applications" by Susanna
S. Epp. It's very accessible. Having a good understanding of algebra should be
enough.

For programmers, Discrete Mathematics is arguably the most relevant discipline
of Math. Yegge wrote a long but interesting post on this subject [1].

[1] [http://steve-yegge.blogspot.com/2006/03/math-for-
programmers...](http://steve-yegge.blogspot.com/2006/03/math-for-
programmers.html)

------
no_wizard
I found [https://artofproblemsolving.com/](https://artofproblemsolving.com/)
to be invaluable and I was at a state you were a few years ago when I found
it. The online problems and the workbooks with the PDFs where awesome. I
really like it tho it can be cheesy sometimes it really covers all the way
through to calc very well there’s online quizzes/work problems as well as ones
in the books/PDFs and it comes with a video series

------
pdm55
I've just been watching Michael van Biezen,
[https://www.youtube.com/channel/UCiGxYawhEp4QyFcX0R60YdQ](https://www.youtube.com/channel/UCiGxYawhEp4QyFcX0R60YdQ),
and Braden Mann,
[https://www.youtube.com/channel/UCFS_EnwXR7m_c1IyePCPhYA](https://www.youtube.com/channel/UCFS_EnwXR7m_c1IyePCPhYA),
explain how to do Maths problems. These guys are good!

------
UncleEntity
I'm horrible at math, really, really bad.

When I need to do something math like I usually find some code that does
something similar, figure out what it's up to and adapt it to my needs. Pretty
pictures help a lot too.

Learned enough linear algebra this way to be productive, made a pie menu addon
for blender that got adopted by other people and eventually became builtin --
mostly to bughunt and find missing chunks of the python API though.

Actually, now that I think about it, I pretty much learn everything this way.

------
fao_
The OpenStax textbooks are exemplary:
[https://openstax.org/subjects/math](https://openstax.org/subjects/math)

------
justifier
find a problem without an answer(o) and try to answer it

research as much as you can about the problem until you understand it well
enough to explain it to someone else

when you come across new concepts, research them.. recurse ;p

if you like to program then try to write your own implementations of concepts
as you learn them

if you are unfamiliar with programming i would recommend trying it, you get to
build this math you are learning and when complete you can develop intuitions
by manipulating: constants, methods, models,..; as well as being a great
learning aide.. debuggers as standins for instructors

read original works: books,papers,..; where ideas are branched from

the engineered complexity of the state of the art can obfuscate underlying
concepts, and seeing how they were first formed can illuminate their present
state

these initial works can be very approachable because they often contain new
concepts that need to be defined within them, whereas a paper referencing that
initial work will often consider the concepts well known to anyone intending
to read them and omit comprehensive description

i think what is most important is that you research whatever is interesting to
you

(o)
[https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_m...](https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics)

------
jbduler
I recommend the excellent (and free) book from Jean Gallier @ the Department
of Computer and Information Science, University of Pennsylvania: Fundamentals
of Linear Algebra and Optimization. Everything you need to know, all the way
up to machine learning algs. Download here:
[http://www.seas.upenn.edu/~cis515/linalg.pdf](http://www.seas.upenn.edu/~cis515/linalg.pdf)
Enjoy!

------
goldenkey
Bill Shillito's lecture series for Project Polymath is by a mile, the best
introduction to higher level mathematics. It requires absolutely no
prerequisite knowledge.

Introduction to Higher Mathematics:
[https://www.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6Kz...](https://www.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6Kz..).

If you find he talks a bit slow, change your playback speed to 1.5x. Enjoy!
:-)

------
francasso
My own take on it:

 _Lang 's books about the basics I find lovely_

\- Basic stuff: Basic Mathematics - Lang

\- One variable Calculus: A First Course in Calculus - Lang

\- Multi variable Calculus: Calculus of Several Variables - Lang

\- Linear Algebra: Linear Algebra - Lang

\- Number theory: An Introduction to the Theory of Numbers - Niven, Zuckerman,
Montgomery

 _Some more advanced stuff_

\- Algebra: Algebra - Artin or Algebra, Chapter 0 - Aluffi

\- Complex Analysis: Functions of One Complex Variable I - Conway

\- Probability Basics: An Introduction to Probability Theory and its
Applications I - Feller

\- Real Analysis and functional analysis basics: Real Analysis - Folland

\- Basic Differential Geometry: Elementary Differential Geometry - O'Neill

\- Riemann Surfaces (algebraic take): Algebraic Curves - Fulton

\- Differential Topology: Differential Topology - Guillemin, Pollack

\- Riemann Surfaces (analytic take): Compact Riemann Surfaces, an Introduction
to Contemporary Mathematics - Jost

\- Modern Differential Geometry: Lectures on The Geometry of Manifolds -
Nicolaescu

\- Functional analysis: Fundamentals of the Theory of Operator Algebras I -
Kadison, Ringrose

\- Introduction to the Index Theory of which you actually have already seen
some in the Riemann Surfaces books with the Riemann-Roch theorem: Index Theory
with Applications to Mathematics and Physics - Bleecker, Doob Bavnbek

\- Homological Algebra: An Introduction to Homological Algebra - Weibel

\- Algebra for algebraic geometry: Commutative Ring Theory - Matsumura

\- Soft introduction to schemes: The Geometry of Schemes - Eisenbud, Harris

\- Algebraic geometry: Algebraic Geometry - Robin Hartshorne

This should get you up more or less to what was current in the '60s :)

 _Additional methodological note:_ I'm not suggesting going linearly through
all these books. Well, perhaps going linearly thorough the basics would be a
good idea, but after that I would follow my own interests.

The important thing is really to have pen and paper and work things out by
yourself, not just reading the book. I'm not saying you should try to prove
all the theorems yourself or do all the exercises, that would take an
unrealistic amount of time, but you can try to think about a theorem before
reading its proof to see if you have a sense of which road is more likely to
lead to a proof, and then try to reproduce the proof with pen and paper after
having read it to check if you actually understood it.

------
soniman
I've done this as an adult and what worked was Schaum's plus online videos.
The idea that you will work through long textbooks is not realistic. Also an
algebra cheat sheet esp the common algebra errors. Quick Calculus is also a
good book.

[http://tutorial.math.lamar.edu/cheat_table.aspx](http://tutorial.math.lamar.edu/cheat_table.aspx)

------
mathing
Khan academy. I also blundered my way through high school math, getting near-
failing grades and quitting as soon as possible. I got super interested in it
and while I was regularly studying khan academy was the best way of actually
learning the material. The exercises are a bit slow but ideally you'd learn
from the lectures and then supplement them with a book of just exercises.

------
buildfailure
May I recommend the NCERT math textbooks used in India. They are simple to
follow and nicely organized as different levels. They are available for free
on the NCERT website (website maybe a bit tough to navigate). The books cover
stuff from basic algebra, geometry and calculus. After this you should be good
to go for collegiate mathematics like analysis, multivariable calculus etc.

------
lordnacho
I'm looking for recommendations for the same question, but for someone who did
graduate Engineering and Econ, and does a lot of coding.

There's a lot of mathematical fundamentals they don't teach you while you're
learning a whole variety of applied math.

Off the top the pieces we did:

\- Linear Algebra

\- Computation (big-O etc)

\- Vector calculus

\- Finite Element methods

\- SDEs, PDEs

\- Discrete math (graphs, cryptography, number theory, etc)

\- Complex analysis (used a lot in EE)

------
bochengkor
3Blue1Brown

~~~
hiaux0
+1

And definitely follow through his "pause and ponder" sections. If you want to
build up your maths skills, it is crucial to learn how to think in the maths
way. Like becoming a good programmer involves writing lots of code yourself,
or to become a good dancer you need to practice your steps. For maths it's
abstract thinking. Appreciation of maths is one thing, having the discipline
to self-study a whole other.

Edit. Regarding your 2nd Edit: His videos are made for the broadest audience
possible. I'd recommend picking any video whose topic interests you the most
at the moment. You will see what knowledge you lack (take notes of these!) and
can expand from there. Be it to watch his maths fundamental ((1)) series
[0],[1] or just rewatch.

((1)): As in any other things, knowing your fundamentals is significant to the
understanding of a topic. It won't help you at all if you can apply (copy
paste) some machine learning techniques if you don't know about linear algebra
at all.

[0]:
[https://www.youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQ...](https://www.youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab)

[1]:
[https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQ...](https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr)

------
ssvss
[https://www.reddit.com/r/math/comments/652p2a/what_are_your_...](https://www.reddit.com/r/math/comments/652p2a/what_are_your_favourite_maths_lectures_on_youtube/)

Related thread in reddit, which I found useful.

------
oh-kumudo
What math are you interested in learning? Math is just too broad to be a
single subject if you want to dig.

------
sassy_samurai
This might be what you're looking for:
[https://www.quantstart.com/articles/How-to-Learn-Advanced-
Ma...](https://www.quantstart.com/articles/How-to-Learn-Advanced-Mathematics-
Without-Heading-to-University-Part-1)

------
iamwil
Do the homework if you want to really understand it. Reading math isn't like
reading fiction. You need to go slow, and you need to do the problems and the
proofs to get it. Yes, ask for help when you get stuck. But there's really no
way around grinding through the problems.

------
desio
Check out my study plan here:

[https://github.com/desicochrane/data-science-
masters](https://github.com/desicochrane/data-science-masters)

It's still evolving especially towards the latter parts - but the earlier math
progression is pretty solid.

------
your-nanny
A general comment: a teacher or tutor who will answer your questions is best.
You can learn math on your own, but someone to keep you on the right path will
save you so much time. Also, for textbooks, work through every example, step
by step, and then do all exercises. Essential.

------
8bitsrule
Depends on what kind of math you enjoy doing. If you're interested in
practical applications, start by becoming an algebra ninja ... so good that
just looking at a problem makes it fall apart.

If you find something that's 'beautiful' (I never did :-( ), then scratch that
itch.

------
softwarefounder
\- Hook yourself into online communities \- Take a college course, PRIMARILY
if you need to be pressured to finish your work (i.e. pass/fail). Otherwise,
it might feel like a waste if you can understand the concepts through your own
learning + communities.

------
agumonkey
I posted this thread on
[https://www.reddit.com/r/learnmath/](https://www.reddit.com/r/learnmath/) ..
and now I realize that there are probably useful answers for sidyapa and
others.

------
HaoZeke
I think you need to first decide your application before starting such a
thing.

There's no way to remember the entirety of mathematics, even if you do rote
memorize and practice your way through it.

You need to figure out which sub-discipline you can apply and follow through
with it,

------
smnplk
This one is great for start. It has many exercises too.

[https://www.amazon.com/Maths-Students-Survival-Self-Help-
Eng...](https://www.amazon.com/Maths-Students-Survival-Self-Help-
Engineering/dp/0521017076)

------
ymgch
I am studying from this MOOC, really good!

[https://courses.edx.org/courses/course-v1:ASUx+MAT170x+2T201...](https://courses.edx.org/courses/course-v1:ASUx+MAT170x+2T2017/course/)

Next month will go for Calculus.

------
sonabinu
I started by working through some elementary math books meant for middle
schoolers and high schoolers. Then took classes at community college and now
learn on my own. At the initial stages, I needed a teacher and structure,
especially with Calculus

------
swframe2
For me "learning math" is a language translation process. The math text uses
terms and symbols that are hard to remember.

I just rewrite the math text to get it into a form I can read without stopping
to figure out an unfamiliar term or symbol.

------
naveen99
Herb Gross and Pavel Grinfeld are my favorite:

[http://lem.ma](http://lem.ma)

[https://m.youtube.com/watch?v=rXOGLlKuvzU](https://m.youtube.com/watch?v=rXOGLlKuvzU)

------
whitepoplar
This book: [https://www.amazon.com/Mathematics-Content-Methods-
Meaning-D...](https://www.amazon.com/Mathematics-Content-Methods-Meaning-
Dover-ebook/dp/B00GUP46MC)

------
sidcool
I asked a related question in another thread:
[https://news.ycombinator.com/item?id=16582762](https://news.ycombinator.com/item?id=16582762)

------
topologie
All you need is here:
[http://math.ucr.edu/home/baez/books.html](http://math.ucr.edu/home/baez/books.html)

------
tensor_rank_0
take pre-calculus and do every assignment immediately as it is assigned. then
review before the next class. pre-calculus is designed to give people like you
the tools to move onto the maths you want to learn.

after that, echoing another top-level comment: take calculus and physics. take
2 semesters of each.

after those 3 classes, you'll know more math than 90% of the world's
population. more importantly you'll be prepared to move on to differential
equations, linear algebra, and vector calculus.

------
duiker101
A lot of good resources here but does anyone have a place where I can find
exercises for algebra and calculus? Even better if it were an android
application.

~~~
laughingman2
[https://brilliant.org/](https://brilliant.org/) has practise modules. Other
than that I think [https://www.khanacademy.org/](https://www.khanacademy.org/)
has practise sessions too.

------
scranglis
To cover pre-high school through undergrad:
[https://brilliant.org](https://brilliant.org)

------
tzs
> My current proficiency level is pre-high school mathematics as I didn't pay
> much attention in high school, learning effectively nothing.

I took a look at your other comments and submissions, to try to get an idea of
whether or not you are a programmer. The fact that you are on HN makes the
odds pretty good that you are or were, but there are enough people here who
are not that I didn't want to assume. It's relevant because programming
requires thinking in ways that overlap a fair bit with the kind of thinking
required in mathematics, so whether or not you are/were a programmer would
affect my recommendations.

What I found puzzles me, particularly concerning your current "pre-high
school" proficiency level. In a post a few days ago, you said that you are
"graduating a university with a civil engineering degree in a few months" [1].

Most sources say that civil engineering requires calculus, differential
equations, linear algebra, complex analysis, and probability and statistics.

I'm having trouble understanding how you could have gotten into a university
civil engineering program in the first place if you math proficiency level is
really just pre-high school, let alone got to within a few months of
graduation.

Also in that post, you are asking a quite similar question to this one, except
it is about CS fundamentals instead of about math from the ground up:

> Hi, I am a front-end developer since 2 years freelancing for local clients.
> A week back I fell in love with computer internals and now want to learn CS
> fundamentals and become a full time software engineer.

...

> Could you give me a roadmap for how I should go about learning CS
> fundamentals? What books and papers should I read? How did you learn, what
> step and approaches did you take?

I'm not normally suspicious of these types of questions...but something seems
off here.

If the question is on the up and up, and you really do only have pre-high
school math, take a look at ck12.org [2]. Go to one of their subjects on that
page, such as algebra, and click it. On that page, click where it says
"FlexBook Textbooks". That will take you to some freely available textbooks
meant for K-12. Click where appropriate to get the high school ones. These
should give you the high school math you are missing. I think you can also
find on that site material for teachers, which could help for self-study.

Get this material down cold...it's not all that college level stuff is going
to assume you know it well, so push through even if it sometimes is not a lot
of fun. If you need something fun while going through the high school stuff,
take a look at the books that collect Martin Gardner's old "Mathematical
Games" columns from Scientific American.

[1]
[https://news.ycombinator.com/item?id=16506771](https://news.ycombinator.com/item?id=16506771)

[2] [https://www.ck12.org/student/](https://www.ck12.org/student/)

~~~
synthmeat
> I'm not normally suspicious of these types of questions...but something
> seems off here.

Really doesn't need to be off. Some shitty decisions on where to put attention
in past and now probably in position where one can play catch-up with
ambitions or just doesn't play well with authority and wants to conquer things
with a more individualistic touch...

..or really any number of scenarios that can explain this not being "off" at
all.

------
splitknot
You can self learn math to a certain point but eventually you will reach a
subject where its introductory text requires a context not written plainly.
This is easily observable in research. Researchers do not write papers with
the goal that everyone understand their work but rather that those in their
field understand so the work can be verified. You are then stuck either having
to derive the mathematical principles you need or you will have to ask others
for help.

------
timkofu
Khan academy. I decided to relearn all math from scratch and now I'm almost
done with Algebra 1. It's pretty good.

------
halayli
most comments here recommend books, but what's probably more important is how
you learn math from these books.

------
michjedi
what is your current level?

------
mamazaco
We've got just the right course for you at passyourmath.com

------
sidcool
What is people's view on Mathematics for Computer science?

------
rorykoehler
MIT Open Courseware is great

------
razib
depends on the level or the depth

------
Dowwie
Khan Academy

------
hamad
haskell(common knowledge) -> category theory(compare it with set theory here
you will read about math foundation crisis) -> type theory(curry howard
correspondence or logic == program)

personal perspective

~~~
hamad
I highly recommend Bartosz Milewski series in category theory
[https://www.youtube.com/watch?v=I8LbkfSSR58](https://www.youtube.com/watch?v=I8LbkfSSR58)

~~~
elcapitan
Independently of whether this is a helpful answer for the original question,
thanks for that link, was looking for some good lecture introduction on
category theory and this looks really nice.

~~~
hamad
it's base on a fantastic book
[https://bartoszmilewski.com/2014/10/28/category-theory-
for-p...](https://bartoszmilewski.com/2014/10/28/category-theory-for-
programmers-the-preface/)

~~~
elcapitan
That looks great, thanks!

