I've had this idea of starting back at basics and relearning math from the beginning since I never "really" learned it besides memorizing and skirting my way through it in school.
Do you know a good path or book that's suitable for that?
Do you know a good path or book that's suitable for that?
I bought myself a Remarkable 2 and signed up to Khan Academy. Now I'm revising algebra basics and I plan to go as advanced as Khan Academy lets me.
I was really bad at maths in school (UK A Levels). But I'm a successful software developer today. I felt like knowing more advanced maths could make me a better developer and not feel intimidated by a lot of the things I see.
I'm actually enjoying it as well. Maths isn't just something I have to do to get out of school, now it's something I want to do. And it gives me the same satisfaction as solving puzzles like sudoku.
I'd recommend it to anyone. The Remarkable 2 is actually really nice to write on too, since I want to store my notes digitally. And I make so many mistakes when writing, so undo is great.
also: the remarkable 2 is great, we have one, but the screen broke and the refurbished replacement arrived with a screen that's not functioning correctly at all, making it a unusable device. Good reminder to reach out to them again.
Thanks!
The Remarkable, at least for me, is good because I can organise my notebooks into folders by certain math lessons or concepts. And I can undo any mistakes, so my notes are clean. Even if I am quickly working something out I am clean it up and make it a good note for future me. The feel of writing on it is much nicer as well versus my laptop's pen or my iPad's pen.
Also, I like that it basically just does notes. There's no Android bullshit, it's just no nonsense note taking. Some competitor tablets have Android and all that baggage.
It can OCR you writing, but I don't know how good that would be for math.
The Remarkable isn't the only tablet that can do this, but it's the one I bought because I like the style and the simplicity of the software.
There are hundreds of videos, organised in playlists, from undergraduate lectures ( https://www.youtube.com/playlist?list=PL55C7C83781CF4316 ) and research seminars ( https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF ) all the way to basic fundamentals like how to think about counting (e.g. https://www.youtube.com/watch?v=Puk-ipOTiD4&list=PL5A714C94D... )
The reason I find them fascinating is that Wildberger doesn't agree with some of the conventional approaches, in particular with the use of infinity and taking limits. This leads him down interesting paths (e.g. Rational Trigonometry and Algebraic Calculus), which (a) show the process of mathematics (exploring, making definitions, building up in different directions, etc.), whilst (b) remaining mostly grounded and approachable (e.g. no appeals to inscrutable lemmas from abstract research areas).
For example, he's recently been making videos about "multisets" (computer scientists would call them Bags), their arithmetic (where "adding" is union, and "multiplying" is pairwise/cartesian product of the elements), and how this generalises: from an algebra containing only empty bags (trivial, but self-consistent; behaves like zero), to bags of zeros (behaves like natural number arithmetic), to bags of natural numbers (behaves like polynomial arithmetic), to bags of polynomials (behaves like polynomials in arbitrarily-many variables) https://www.youtube.com/watch?v=4xoF2SRp194
So no transfinite ordinal analysis or large cardinals? Hard to take him seriously.
It's similar to 'reverse mathematics' (trying to find the minimum set of assumptions required to prove a known result)
Yes, we can construct such a line segment; but line segments are not numbers.
We don't actually need "legs of length one" (which pre-supposes some system of units); all we need is the ratio of the lengths of the sides. However, finding lengths requires the ability to take square roots, which would either make this a circular definition (e.g. that √2 = √2 / 1), or requires the limit of an infinite process (like Newton's method, or equivalent).
Instead, it's much easier to count the areas of the squares on each leg (1 and 1), and add them together to get the area of the square on the hypotenuse (1 + 1 = 2). No need for lengths, so no need for square roots, so no need for √2.
Wildberger abbreviates 'area of the square on a segment/vector' as the 'quadrance' of that segment/vector (defined as the dot-product with itself). Likewise we can avoid angles by taking ratios of quadrances (e.g. 'spread' is defined via a right-triangle as the quadrance of the opposite side / quadrance of the hypotenuse); together this gives rise to a whole theory of Rational Trigonometry, which gives efficiently computable, exact answers; works in arbitrary fields (except for characteristic two), and with arbitrary dot-products/bilinear-forms (e.g. euclidean, relativistic, spherical, etc.). Here's Wildberger's textbook on the subject http://www.ms.lt/derlius/WildbergerDivineProportions.pdf
Infinities are very interesting but the non-infinite maths have kind of got neglected over the past 100 years. I had to memorize Laplace transforms in college but never heard of Fairey sequences until I watched his videos.
People get upset at him but he's basically just having fun seeing how far you can go in Math without infinity. It's quite interesting to a certain audience (like myself).
You could insist on sticking with the Axiom of Countable Choice if you wanted to avoid some of that.
I'd say it's pretty hard to avoid thinking about 'infinity' though.
https://www.amazon.co.uk/All-Math-You-Missed-Graduate/dp/100...
https://www.amazon.co.uk/Princeton-Companion-Mathematics-Tim...
Gives a bird's eye view of math very nicely. Even from a skimming it was very useful to help me understand the gaps I have, and the shape of those gaps, and partially filling them.
I don't know how good it is, but her earlier entries on Physics and Philosophy were well-received.
HN thread: https://news.ycombinator.com/item?id=30591177&p=2
And my impression was that Saxon Math was the worst. What I mean by worst is that it just make you practice an algorithm by doing lot of repetition but doesn't force you to have a deep understanding or problem solving skill.
My experience is that I didn’t really feel like I was memorizing an algorithm. Because the problem set includes assignments from all of the old sets, it is hard to memorize all of the algorithms. So you instead memorize the different moves that are allowed and have a general idea of what types of moves might be useful.
I dunno. I went on to do engineery stuff as an undergrad rather than pure math stuff, it seems like a good match because engineering problems are also often in the “no need to be super clever, just don’t mess up” vein, so it could be just a lucky match. This is what I mean by muscle memory — I’ll use the famous names theorems when necessary but sometimes you just need to bash the math until the thing you want is on that side of the equal sign and the other stuff is on the other side.
I think anything that results in
1) actually reading some textbook
2) actually working through problems for a couple hours a week
will compare well to the typical US math education pretty well anyway.
I like khan academy back in 2010 because all the videos were in one place and you could see everything right there in front of you
So, I decided I wanted to study maths for the maths. I was in the fortunate position of being able to self fund myself through the Open University (uk based) Maths and Statistics BSc. One module at a time I’m now on my last module. There many things I’d studied before (calculus, sequences) and many new to me (group theory, graph theory)
I've been using Professor Leonard's Youtube video series[1] mostly, along with some of those "workbook" type books by Chris McMullen, and a variety of books with titles like "1001 solved problems in $SUBJECT", "The Humongous Book of $SUBJECT problems", and the like. The nice thing about Professor Leonard is that he has videos on everything starting from pre-algebra, middle-school math, up through Differential Equations. Note that his diff-eq class isn't quite complete but he just announced he's about to start recording new videos to finish that, and he's also going to be starting a Linear Algebra sequence. And he's a great lecturer who does a really good job of explaining things and making them understandable.
I also use Khan Academy sometimes, and stuff on Youtube from The Math Sorcerer[2]. Oh, and of course there is 3blue1brown[3], whose videos are also useful. And for Linear Algebra I've been using Gilbert Strang's OCW videos[4] on Youtube.
FWIW, I've evolved the way I study math, and what I do now works for me, even though it's 100% not the way you'd ordinarily see suggested. That is, I watch math videos fairly passively and don't work problems at the same time and treat it like being in a class per-se. I used to do the thing of treating it like a class, pausing the video to work examples, and what-not, and that does work. But it's very slow and tedious.
Now, I just watch the videos, acknowledging that I won't absorb everything and that I also need to work problems for long-term retention. So now what I do is watch passively to a certain point (which I determine fairly subjectively) then I stop with the videos for a while, pick up a textbook or one of those "workbook" type books I mentioned earlier, and work problems for a while. Then I review the parts that I find myself struggling with. I'm also just now starting to add "creating Anki cards" as something I do during that second pass.
Once I start getting a decent Anki deck built up, I'll be reviewing that regularly as well to help build retention. I only create cards for things that seem amenable to rote memorization, and TBH, I'm still working on figuring out what things are best to include, and how to structure those cards. What I don't intend to do is include specific problems where all I'd be doing is memorizing the answer to a problem. So far it's just formulas and things are are very obvious candidates to be memorized, and "algorithm" things like the "chain rule" from calculus, and similar.
[1]: https://www.youtube.com/@ProfessorLeonard
[2]: https://www.youtube.com/@TheMathSorcerer
[3]: https://www.youtube.com/c/3blue1brown
[4]: https://www.youtube.com/playlist?list=PLE7DDD91010BC51F8