How to Study Mathematics (2017)
math.uh.edu
math.uh.edu
Find out where the mathematics you're learning comes from : who first developed it? what problems where they trying to solve? why were they trying to solve those problems? what problems does it solve for us today?
In my mathematical education I noticed that I had a more pleasant time and felt more motivated to learn the material when the teacher gave us this kind of background story. Since most teachers don't do so today, the student typically needs to get on the Internet to do their own research.
For some of us, this is incredibly useful advice re: math study habits. I also figured out that once I had the context and history, it was much easier for me to learn and apply.
At times he went a bit overboard, but he makes some valid points in this lecture: https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
so it's always been irritating to me how much introductory or "applied" texts bias themselves towards physics and engineering. like how a lot of multivariate calculus/introductory real analysis texts pretty much limit themselves exclusively to three dimensions and teach notation to match.
it even plagues more advanced books; i'm struggling through Villani's text on optimal transport and nearly every example is from physics even though the theory was developed for economics and has extremely broad applications across many disciplines. half of the "motivating examples" i can't even make sense of what they're talking about.
as much as I struggle with the hyper-abstract bourbaki-style stuff, at least I have a hope of understanding it without needing to dip into an irrelevant discipline where i have no background or historical context.
And for example, a real revelation with regard to infinity came, when I read (or heard somewhere, it was long time ago) that infinity can sometimes be few millimeters or even micrometers. Up until then I always imagined some really large number, but at that point I realized, you need to put problem before you need to put problem before you into perspective.
For me, the definition "A group is a set of transformations on an object such that..." is so much more enlightening than "A group is a set G together with a binary operation * such that..."
Only yesterday I was trying to learn about differential forms. Most of the notes I found online introduce the wedge product in a deeply unhelpful way, by listing some axioms that it satisfies and deducing results from the axioms. It takes hours of work to understand why those specific axioms were chosen. For me - and maybe I'm wrong here - that's Bourbaki's formalist approach in a nutshell.
It was only when I found Terry Tao's notes [0] and Dan Piponi's notes [1] that I could actually see the use of differential forms. It's an unfortunate state of affairs for the discipline that in order to learn about X, you have to google "X intuition", since it's not given to you as a matter of course.
[0] https://terrytao.wordpress.com/2007/12/25/pcm-article-differ... [1] https://github.com/dpiponi/forms/raw/master/forms.pdf
30+ years later that remark is still having an effect in that when I took my kids to the top of the tour montparnasse I made them look down at the cemetery to point out that Galois is down there (somewhere).
I think that's actually an example of the problem. Galois's politics really had nothing to do with the problem he was trying to solve. Historical features are thrown in as random info with no point. The point of making mathematicians real people is so that students can say "Oh! That's how they came up with that problem! That's why they solved it this way! I could have done that!"
Rather instead, we further mythologize the individuals. You don't need to join radical political organizations and get yourself killed in your 20s to be a great mathematician.
Now every evening I find myself saying out loud, "why the hell didn't the math teachers explain it like this." For example, we all learn about the Pythagorean Theorem, but why didn't we learn about the Pythagorean Brotherhood[1] a little bit? And feel just a little bit of the excitement that Pythaogoras did when he found an exposed wire of the universe and figured out how to work it.
Fun fact: Pythagoras had to pay his first pupil to listen to him. No one gave a shit, probably not too different than how kids don't give a shit today. But the kids we're teaching don't even get paid, they're told their true reward is something intangible or some vague math-fu. All this while they have a black hole for their mind in their pocket where at least the fall is comfy.
He does mention in the book that the only math classes he had any success with were AP classes. I think it's dedicated to his teacher. But really it makes no attempt to disguise that it's mostly the math angle of his uneasiness with the world and yep I agree with you on the 20 pages a day.
Thankfully(?) I was taking physics at the same time, so I did get to see how double and triple integrals and differential equations could be applied. But it frustrated me to no end that the person who was driving my math education cared so little about helping us understand it.
I generally buck the standard advice and memorize first, before trying to understand. Understanding is much easier for me if I can easily hold everything in my working memory.
That is why we can have databases that do not fit in 'memory'. Do you see the analogy here?
If anything, I saw those coming from less-advanced math founder on advanced math (arithmetic to algebra, algebra to calculus, calculus to advanced subjects) because they attempted to deal with the subject based on memorization rather than grasping the basic point.
Generally, I did wind-up committing a lot of content to memory but it was and is much easier, even "effortless", when I knew the reason for each part of the content.
Of course, language is slippery and we might be talking about exactly the same thing. So I wouldn't give blanket advise - each person has to find their own learning style (or styles, if it varies by discipline).
A basketball player has memorized how to shoot a free throw, not by cramming by rote, but by practicing.
The equivalent for mathematics is solving problems that you find personally relevant and interesting and which are at the limits of your current abilities. NOT just doing piles of rote calculation. Often solving a hard problem requires a big pile of intermediate calculations, but those a tool in service to some meaningful purpose, not an end in themselves.
Forcing students to work through thousands and thousands of trivial calculation exercises is horrible pedagogy (and a very inefficient use of time).
It’s like doing nothing but shooting drills and getting quizzed on the NBA rule book for years before ever trying to play a game.
No, they start by joining a club where they get professional training, from a very young age. Just playing the game all the time won't make you any better at it. There are millions of children that spent significant part of their childhood playing basketball for fun, and never went past amateur level.
This is just a conclusion from Ericsson famous paper[1]: you need DELIBERATE practice, not just recreational playing.
Playing for fun will just develop bad habits, which you then will have to unlearn, so recreational playing might actually be an obstacle to learning.
[1] http://www.nytimes.com/images/blogs/freakonomics/pdf/Deliber...
But there are many aspects to any complicated skill, including tiny subskills, combination of various subskills, and at a higher-level meta skills like knowing when to change strategies, etc.
The problem with only sport games as a mechanism for feedback is that (a) there’s not necessarily a good way to correct and try again right away if you want to practice one particular thing, (b) situations that arise are not homogeneous enough to compare feedback across trials, (c) there are periods of waiting in between direct practice, (etc.?).
As a result, people who are training to be basketball players spend only like half their time playing realistic games, and the rest of their time on a variety of drills and workouts including general aerobic and strength training, practice running sideways and jumping, shooting / passing / dribbling drills, practice setting screens / rebounding / driving / inbounding the ball, larger-scale coordination drills, and so on. These let the players focus their conscious attention at one level at a time, and repeat a situation with analytical feedback in between.
But those who improve fastest are the ones who figure out what they need to work on (or have a mentor/coach to figure out what they need to work on) and then get lots of feedback about that particular area, in an intentional way. This can even be done within the context of a realistic full game by choosing what to direct attention/focus to.
But you’re missing my basic point, which is that all the skills in basketball are situated within a context. It’s obvious how they fit into playing a game. Students start with the high-level context (by watching / playing full games), and then break it down into small pieces they can work on separately or jointly, and then frequently practice putting those pieces fully back together in the context of the game. I doubt you’ll find anyone seriously training to play basketball who doesn’t play in a semi-realistic game-like situation at least once per week.
In classroom education, often the high-level context is completely missing, and students end up memorizing / drilling on a pile of atomized facts and trivial very low level skills without ever practicing combining those in larger settings, without ever getting a very fast or effective feedback loop going, and without ever having any motivation for going through the practice.
In the case of mathematics education, someone being taught 1-on-1 by a skilled tutor can move I’d say about 4–10 times faster than a student only watching lectures, doing completely independent book reading / trivial homework exercises / quizzes, and then getting back graded homework/quizzes a week afterward. That student can be assigned dramatically harder problems, can be helped along by limited (but well placed) hints, and can be given direct feedback/corrections at every level (from basic arithmetic computations through higher-level strategy to meta skills like strategy selection/evaluation). This is something that can’t be done in basketball because basketball is a group activity; you can’t watch a single player go through the game, and pause/backtrack everyone to the beginning of a situation whenever that player makes a mistake or suboptimal play.
That was sometimes a problem for me, if I picked up something quickly I often found it harder to retain, whereas if I had to work hard to grasp a concept then I would remember it better.
These videos do an incredible job of illustrating how to intuitively arrive at an answer by composing many of the parts you need to build a proof for more complex topics.
To pick an example:
Intuition is great for understanding the intermediate value theorem. If the temperature was 40 degrees at 8 am, and is 60 degrees at noon, it must have been 50 degrees somewhere in between.
Why, though? How can you guarantee that there will always have to be a time where it was 50 degrees[1]?
[1] provided that temperature is real-valued and temperature change is continuous, the debate of which I'll leave to the science people
What I believe to be true is that a solid intuitive proof will be sound, but not necessarily transcribable until formalized. Formalization of the proof is essentially the matter of making a proof communicable.
Putting it one more way: purely-intuitive proofs would be just fine if one could share one's mental state with another.
What do you think?
> Am I correct in understanding that it's not that intuition is invalid, but that it's insufficient for communicating a proof to another mathematician.
is close to correct. It's a very useful tool, but it can sometimes lead you into incorrect assumptions.
However, the rest of it incorrect. Math, and in particular, probability and statistics, are full of things that seem intuitive and easy at first glance, but are actually a bit more nuanced.
Take for example, the Monty Hall Problem. It's probably familiar to most here, but essentially you have three doors. Behind one, is a car or some other desirable object, and behind the other two are goats or something undesirable. Select a door, and you get whatever is behind that door. What's the probability of getting the car in this scenario?
Easy enough right? It's just 1/3. However, if Monty Hall opens one of the doors after you select a door, he always reveals a goat, and he offers to let you switch your door now, should you switch or keep your door?
What's the probability that you get a car if you switch? Is it 1/2 or 1/3? What's the probability that you get a car if you stay? 1/3?
Well, the somewhat surprising answer is that you have a 2/3 probability of winning the car if you switch, and a 1/3 probability of winning the car if you stay.
Often things will break down in the limits as well, so if you try to apply your finite dimensional intuition to something that corresponds to an empty object or an infinite number of objects, you'll find that you're often wrong.
No, I have had purely intuitive proofs myself that turned out to be wrong.
However what you may have picked up on is we disstain people who insist they have a brilliant idea for which someone should prove they are right. That’s like saying a car engine consists of cyclinders and piston; someone else just needs to do the work to put the pieces together.
https://terrytao.wordpress.com/career-advice/theres-more-to-...
I recommend reading the entire post - it's not too long, and links to other great materials - but this is a good relevant quote:
> The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition.
They do this for good reason. I had to take, as part of my philosophy concentration, a bunch of mathematical logic classes. First Order and Second Order logic, as you might imagine, are pretty simple. The rules make sense in a very intuitive way. I'd get 100 on exams just because I'm good at programming. But when I had to study metalogic, model theory, Henkin Proofs, and Godel's Theorems, that intuition quite literally flies out the window. I guess my point is that most interesting stuff is rarely intuitive.
But respectfully, ZFC came about because of logical paradoxes that couldn't be accepted as consistent.
You can create a new number system and derive all kinds of consequences but the truth is, most mathematicians care more entirely about prime number theory on the naturals that are entirely based on counting.
Most of modern math is based on the natural numbers. You can't remove all intuition. The thread that holds our love for math is also the same one that tells us we are exploring consequences that tell us a dearth about our universe.
If you think that coming up with theorems about, e.g. infinitary algebra, is intuitive, then I guess you're just a lot smarter than I am. Infinities always throw a wrench into our intuitions. Something being a "logical consequence" of something else is an incredibly simplistic way of looking at things, not to mention that I'm not exactly sure what that has to do with intuition anyway.
That 'non-intuitive' aspect of high-dimensional spheres having very small volume relative to a hypercube becomes a point that builds intuition for how those spheres behave, once you're familiar with it. And then you can start throwing out things that seem intuitively wrong, and following intuition towards new ideas.
> the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference
There is such a thing as developing your intuition.
I think your meanings are not qualitatively different.
The trick is that you need good data to feed it that makes the patterns clear. You might have not gotten a lot of that on the topic you had no intuition of. On some topics, there hasn't been much of an attempt to do that since people expect it to be hard with everyone forced to do the methods that are non-intuitive. Lots of traditions are like that. Then, there's the possibility something can't be intuitively handled at all. I don't know if those exist but they might. That many activities of people doing math seem to get an intuitive boost during their day-to-day work makes me lean towards intuitive possibilities for about everything that can be approximated (esp with heuristics).
Let's claim that writing a proof is similar to writing a program. You start from a state, your premise, and have your assumptions, input, which you process through a series of steps, maybe taking cases in between (branching) and return an output (hopefully True).
Then considering this correspondence, it will be fine to say that you, as a programmer, can guess the output/behavior/semantics of any program without running it. Similarly, a mathematician would be able to reach the conclusion that the theorem holds, without going through all the details of the proof.
Sure, in most cases you can, and as you get more and more experience and knowledge as a programmer you can say you improve your guesses. But this ends up backfiring with a good probability, when our model skipped a step, e.g. the interaction of two concurrent data structures.
Thus I don't think it is right to say "as you are able to know all axioms and rules of inference, you should be able to achieve perfect intuition". You can know the fundamentals of each layer on your stack and you would find it insane if one asked you to guess the output of any non-trivial terminating program.
Note also that the way of inference (logical consequences) can and does change (e.g. ZF vs ZFC).
() I may be misusing the word model here -- it is usually defined (at least to things I work with) as the interpretation of a theory that satisfies all theorems (inferences). () constructive proofs yes. No flame wars please :)
citation?
3Blue1Brown is better than most but far from best.
I've never been able to find a good explanation online, only in some older textbooks. I've begun to fear that there are few people who are able to read equations while the others interpret the explanations of those people, where then those same old explanations gradually mutate as they propagate without reference to their source. This Stanford lecture is the best I could find: https://www.youtube.com/watch?v=1rqJl7Rs6ps
"The Analytic Theory of Heat" Joseph Fourier who explains the realization at length.
3b1b's videos are excellent, but if they were interactive, open-source and in a similar model to Wikipedia, people could contribute and improve upon content that is already great.
There are great examples of what is possible with explorables, see Bret Victor, D3.js, distill.pub, observablehq etc. Just no encyclopedic model yet.
write one and maybe include that fact as a resume / interview item :)
- Kevin Houston "How to Think Like a Mathematician"
- Keith Devlin "Intro Mathematical Thinking"
- "How to Study as a Mathematics Major" Lara Alcock (for some reason, she/Oxford Press has 2 books with seemingly identical content under Math Major and Math Degree titles)
Mason, Burton, & Stacey, Thinking Mathematically
Pólya, How to Solve It; Mathematics and Plausible Reasoning (2 vols); Mathematical Discovery (2 vols)
Schoenfeld, Mathematical Problem Solving
Larson, Problem-Solving Through Problems
In my late 20s I decided to try again, but jumped straight into calculus. And at first regretted that decision. However, I got lucky by stumbling upon this book:
https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/...
It "reads" like a book, with the ideas given context. I had an "ok" connection with Algebra, and the book explained the rest well enough for me.
In school, the textbooks were loaded with symbols, but not enough description -- I guess they relied on bored teachers making minimum wage to do that part. I went to a school with poor academic showings (but connections to state superintendent of ed got them a grant for football facilities).
Coincidentally, this book goes well with the technique described here:
http://www.pathsensitive.com/2018/01/the-benjamin-franklin-m...
If Susan Can Learn Physics, So Can You - https://www.susanjfowler.com/blog/2016/8/26/from-the-fledgli...
and
So You Want To Learn Physics... - https://www.susanjfowler.com/blog/2016/8/13/so-you-want-to-l...
According to Susan's account, it took her 1.5 years to go from no college-level math to a full college-level physics curriculum. At the school I studied at, the Calc series (I, II, and III) takes 1.5 years. The intro physics (I and II) takes a full year, and the upper physics take at least a year (Mechanics I & II, E&M I&II, Relativity I&II, Quantum I&II, Lab I&II). (And you had to take intro physics before higher-level physics, and most courses had dependencies.)
That's approximately 45 credit hours. If you took 18 credits a semester, that would take at least 2.5 years. And those are all tough courses -- no 3 credit gen ed "filler" courses. Even at 15 credits a semester, that's still 3 years.
So, to get through all of those courses in 1.5 years would be like learning about 30 credit hours / semester. And add in the physics electives, like thermo, waves, statistical mechanics, quantum field theory, additional math courses, etc... that's a lot on top.
I guess my point is that if Susan's account is discouraging, note that a lot of people take far more than 1.5 years to get through undergraduate physics.
Maybe not who your thinking of, but similar story.
See: https://barbaraoakley.com/books/a-mind-for-numbers/
She also has a great Coursera class called learning how to learn
It occurred to me that while I learned in college how to show, using Galois theory, that quintic equations are not generally solvable by radicals, I had no idea how Galois theory really relates to the process of finding roots. So I went back and derived the quadratic formula, the cubic formula, and sketched the quartic formula to see how the process used the ideas formalized by Galois theory and where it breaks down. I've tried to write the result up in a motivated and understandable way, instead of like a math textbook: https://alexcbecker.net/mathematics.html#the-quadratic-equat...
For
> To the mathematician this material, together with examples showing why the definitions chosen are the correct ones and how the theorems can be put to practical use, is the essence of mathematics.
Is good, but more is needed.
(1) Plan to go over the material more than once. The early passes are just to get a general idea what is going on.
In such passes, for the proofs, they are usually the near the end of what to study and not the first.
(2) When get to the proofs, for each proof and each of the hypotheses (givens, assumptions), try to see where the proof uses the hypothesis.
Next, try to see what are the more important earlier theorems used in the proof. So, sure, in this way might begin to see some of how one result leads to or depends on another and have something of a web, acyclic directed graph, of results.
And try to see what are the core, clever ideas used in the proof.
(3) For still more if you have time, and likely you will not, can use the P. Halmos advice, roughly,
"Consider changes in the hypotheses and conclusions that make the theorem false or still true."
(4) But, note that to solve exercises or apply or extend the theory, need some ideas. So, where do such ideas come from? In my experience, heavily the ideas come from intuitive views of the subject.
So, my best suggestion is to try to develop some intuitive ideas about the material. Definitely be willing to draw pictures, maybe on paper, maybe only in your head.
In the end, a solution or proof does not depend on intuitive ideas, but finding a solutions or proof can make use of a lot in intuitive ideas.
For research, most of the above applies, but IMHO there are more techniques needed.
https://news.ycombinator.com/item?id=16390046
Also a lot of books these days have wolfram mathematica code and it works surprisingly well even for some more abstract parts of math, To get a good intuition.
Put the book down when you get to the example or proof of a theorem, keep all the theorems and examples before it in your head, and see if you can reason it out yourself. Can't promise that your head won't hurt afterwards
Google the title + PDF
I don't think I've ever seen a "How to study applied mathematics". How do applied mathematicians, physicicst and engineers (who apply mathematics to real world problems) study mathematics, when they use it as a tool? How much or little emphasis do they give to proofs and theorems?
A dump example, consider the problem of detect if two date ranges overlap (for a calendar app perhaps). By recognizing a date range, DateRange(a, b) as an interval on the real line: [a, b] or a set {x | a <= x <= b}, the problem becomes find if two sets' intersection is empty. Set {x | a <= x <=b } and {x | c <= x <= d}'s intersection is {x | (x >= a and x >= c) and (x <= b and x<=d)} which simplifies to {x | max(a, c) <= x <= min(b, d) }. Therefore, the set is empty iff max(a, c) > min(b, d).
Translating the problem into mathematical structures here (intervals), and manipulate them with math tools (inequalities) would count as applied math I guess. By having a repository of math knowledge and an eager attitude to look for opportunities to apply them, one can find success in solving practical problems. In this context, rigor and formal proofs are less important and math intuition is paramount.
An real world example I like to motivate myself with is Richard Feynman and connection machine[1], a story about Feynman using partial differential equations to model on boolean circuits. I do believe even with a limited set of mathematical tools, by trying them on every problem you encounter, you can still get good results. Something like, if one squint hard enough, you can find a monad anywhere :)
[1] http://longnow.org/essays/richard-feynman-connection-machine...
A related thread from two days ago for those who missed it:
It's apparently just assumed that you're taking a course so the decision is made for you.
I did my PhD in math, in large part because I enjoyed his class on advanced linear algebra so much, and later on real and functional analysis.
1. Proper preparation. There are textbooks at even the graduate level which have no formal prerequisites and which are largely self-contained. Technically speaking, someone with no prior background but a strong mathematical maturity could tackle these, but it might take them an inordinate amount of time to really grasp the material. For example, if you understand things like mathematical induction and proof by contradiction, you can learn analysis before you've been exposed to calculus, or category theory without abstract algebra. But it's far from ideal because you'll probably need to go over the same material several times and struggle with it.
Furthermore, even the same subject within "advanced" mathematics can have wildly different depths of coverage depending on the author. Pinter's A Book of Abstract Algebra is probably approachable for anyone currently reading this comment, or even high school students. Dummit-Foote is a step beyond that, and appropriate for undergraduates who are already immersed in a math degree. But Lang or MacLane-Birkhoff would be significantly more challenging without first building up to them.
Sometimes this is not just a question of depth, but also of pedagogical style. You can get a lot of satisfaction by learning analysis from Rudin for the first time, but it's really a rough go of it if you're not prepared for the terse definition->theorem->proof->remark->definition->theorem->proof->remark style of writing. On the other hand, Tao's Analysis I and Analysis II are much more approachable (similarly, some writers, like Halmos or Munkres, are praised for their exposition in introducing otherwise complex material).
Ideally someone looking to study a subject should introspect about whether or not they are prepared for that subject overall. Once they've confirmed they are, they should read the first 10 pages of five or so well-recommended textbooks on the subject at their level, then choose to stick with the one that has the most approachable exposition style for them.
2. Proper study. When studying any given textbook (or videos, lectures, etc) it's really important to understand that mathematics is an active discipline. You cannot learn it by reading it. The process that has worked for me is the following: first, read through a chapter without taking any notes. Do so quickly, but not quite so quickly as skimming. When you come across things you don't know, compartmentalize them a bit and keep moving forward to the end of the chapter. The idea is to let the chapter's new material percolate a little before you begin actively tackling it.
Next, start over at the beginning of the chapter and write down every single definition and theorem as you read. Before reading the author's proof of any given theorem, try to prove it yourself for at least 10 minutes. Then compare your work to the author's, and copy their proof meticulously in order to learn the method. Continue on to the end of the chapter.
Finally, there will probably be anywhere between 5 - 20 exercises at the end of the chapter. Solve a meaningful fraction of these exercises, and don't look up the solutions to any of them until you've struggled with them for a good half hour or so (each). When you do look up the solutions, make sure you check multiple proofs for the same exercise so you can understand how the chapter's material can be applied in different, flexible ways.
Mathematics has always exemplified a central belief of mine, which is that humans learn under conditions of optimal struggle. Even though it feels like being mired in hopeless complexity while you're struggling to complete a particularly difficult problem, you're actively learning the subject by doing it. But it's a question of efficiency. You want to aim for a subject and a presentation of that subject which is difficult enough to be just out of your current capabilities, but not so difficult that you can't follow its exposition.
https://ocw.mit.edu/courses/mathematics/18-781-theory-of-num...
Introduction to Higher Mathematics: https://www.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6Kz...
If you find he talks a bit slow, change your playback speed to 1.5x. Enjoy! :-)
You can do the exact same thing you mentioned with math. Theorems aren't arbitrary; there's a way they're derived. If you learn how the theorem comes to be, even if you can't remember exactly, you can often remember enough to come up with the theorem again.
I think the way you learn is the ideal. Knowing the context behind the facts you learn gives them worth, and the redundancy of being able to derive the fact from the context also makes it better integrated with the rest of your knowledge and easier to remember in the long term.
Once you have the background knowledge, everything is a lot simpler. Learn whether each theorem has an easy proof or a hard proof. If it has an easy proof, you can forget it (deriving it when you need to; an example is the contraction mapping theorem [1]). If it has a hard proof, you can forget it and look it up when you need to.
[1]: https://www.patrickstevens.co.uk/how-to-discover-the-contrac...
That sounds odd for college-level math courses (note that I am just a first year engineering undergrad and therefore possibly know nothing). Do you have oral part in your exams?
[Edit: Oh, yeah. It's also against HN site guidelines. Saying such things repeatedly can get you banned.]
Or, look at it as a life-hack: Don't waste your time protecting the feelings of a non-existant group of people.
Source: have followed this technique indirectly for the past few months
window.document.body.style['max-width'] = '650px'This is a disgrace.
In any case. Definitions vary from book to book so mugging them up ad verbatim would hardly help.
I tried the Ivan Savov books and they seem great but I have no way to judge the quality of any resources.
Where do I start as a complete beginner? I want books not online resources like Khan etc.
I would eventually like to get into Geometry but I have no foundation whatsoever, just basic algebra.
On Udemy you can find nice courses as well for a good price. They have discounts all the time, each course for $10-20. Never pay the full price. See the reviews, and you can always take a risk for that price, plus they have a 30 day money back guarantee.
* Algebra, Gelfand
* Geometry, Kiselev
Here's how it describes its content:
Part One: Size and Shape
In which we begin our investigation of abstract geometrical figures. Symmetrical tiling and angle measurement. Scaling and proportion. Length, area, and volume. The method of exhaustion and its consequences. Polygons and trigonometry. Conic sections and projective geometry. Mechanical curves.
Part Two: Time and Space
Containing some thoughts on mathematical motion. Coordinate systems and dimension. Motion as a numerical relationship. Vector representation and mechanical relativity. The measurement of velocity. The differential calculus and its myriad uses. Some final words of encouragement to the reader.
Here people are saying you need a solid pure math background. I skimmed through it and it gets complex very quickly. Around page 58 with the formulas I get lost.
Thanks though, I think I'll be ready for this after Gelfand and Kiselev. Can't wait.
For easing into higher maths I recommend this one on proofs: https://www.amazon.com/How-Read-Proofs-Introduction-Mathemat....
Also if you're concerned about quality the MAA reviews math books and publishes a basic library list for undergrad maths: https://www.maa.org/press/maa-reviews.