Ask HN: Best resources to gain math intuition?

https://mitpress.mit.edu/books/art-insight-science-and-engin... . (free - look for pdf download link )

https://mitpress.mit.edu/books/street-fighting-mathematics (free - look for pdf download link)

https://www.betterworldbooks.com/product/detail/How-to-Solve...

Pure Mathematics for Beginners by Steve Warner runs through the (very) introductory elements of major branches of math [0]. If you want even more leisurely intro to math intuition without worrying about elements of topology/abstract algebra or whatever, Book of Proof by Hammack is great and free [1].

[0] https://www.amazon.com/Pure-Mathematics-Beginners-Rigorous-I...

[1] https://www.people.vcu.edu/~rhammack/BookOfProof/

Plug, but I have a blog for exactly this. As an EE/physics student you may appreciate this one on Euler's Formula:

https://betterexplained.com/articles/intuitive-understanding...

3Blue1Brown videos are awesome:

https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw

Also, take a look at A Programmer's Introduction to Mathematics by Jeremy Kun [0].

[0] https://www.amazon.com/Programmers-Introduction-Mathematics-...

Polya How to Solve it

If you are doing signal processing (Laplace and Fourier transforms especially) and want to gain an intuitive sense of Complex Analysis, I recommend the book Visual Complex Analysis by Tristan Needham.

If you mean "gaps in my education" or "basic things that I don't quite understand", you could try studying some high-quality texts. Look for books written by extremely smart people who are trying to explain the ideas rather than taking you through the standard topics. Hamming's books on probability and signal processing and Strang's books on linear algebra and applied math come to mind.

Alternatively if you're really interested in intuition, you could also look at the Math Olympiads. Pick a problem, beat your head on it, finally look at the solution, repeat. There are web sites and prep books.

3B1B is the best place to start.

http://immersivemath.com/ila/index.html

Learn linear algebra properly. Like many folks on HN, I really enjoyed Axler's Linear Algebra Done Right, but ymmv. I got a ton out of going through each of the proofs + practice problems and really taking the time to work through the solutions. It wasn't enough for me to just read the text.

There's a linear algebra lurking everywhere in the realm of applied math (some people like to joke that machine learning is really just linear algebra) so it really is worth your time to have a firm understanding of it.

The 3 wonderful Roger Nelsen books Proofs Without Words entirely consist of equations & inequalities from various mathematical fields 'proved' in intuitive pictorial form.

http://gen.lib.rus.ec/search.php?req=proofs+without+words

I used to love the abstract symbolic manipulation of pure math, but have not worked with it in many years and find it unintuitive now. I have a harder time trusting the abstractions. I wonder how to get my old sense back - it never, ever came from practice problems for example in the past. I miss proofs

It's probably too hard to answer the general question about "how to gain math intuition". Mathematics is just too vast.

However, you have written

> I think when I took physics it really brought out these flaws and lack of intuition

which suggests you have some good practical experience with physics problem solving which has precipitated a certain feeling that you need to learn more about some kind of math. I would advise that you try to exploit this. In the same breath I want to recognize (as someone who did their BS and MS in physics) that physicists are not always so careful or explicit in how they are doing their mathematics. So learning means eventually going beyond physics sources and into a much wider world of mathematical thought. The particular things that mathematicians care about may or may not be relevant to the problem you are trying to solve in physics, and a good part of developing that intuition is to figure out which particular caveats that a mathematician expounds upon (more often than not, some esoterica about the space(s) that they are working in or the class of isomorphisms under which their results are invariant) matter physically. As you develop and intuition about these things a bonus is that you will be able to skim through mathematics resources much faster.

this might be useful to read for some general high level ideas on how to proceed: http://m.nautil.us/issue/17/big-bangs/how-i-rewired-my-brain...

My suggestion would be this. Assume that you have your intuition, already, and it is good.

May be it is distinct from others, may be you will not be 'narrowing' to the right answer within seconds -- like many folks who do Olympiads...

Just do basic things, but every other day. Get books/materials that have solutions (not just problems). use those, compare your results, and then try again.

If you feel like you do not understand 'why', you will need a particular subject area. Switch to read about applications of that area, how historically it came about and so.

And then back to problem solving, proofs, and reading other people's papers (when you can..).

It is hard work, but over time you will build up your version of the so called intuition, it will be powerful, you will be able to apply it all over the place.

Also there are a number of math forums where you can reach out, if you are really stuck and cannot figure out how a particular proof, or solution was obtained.

A lot of intuition in math is from geometry. High school plane geometry is a good start, but high school solid geometry, where get good tools and intuition for seeing things in 3D instead of just the 2D of plane geometry, is quite a bit better.

Another part of intuition is from Max Zorn (from Zorn's lemma statement of the axiom of choice):

"Be wise, generalize."

E.g., for the set of real numbers R and a positive integer n, a lot that goes on in the n-dimensional vector space R^n is a generalization of what can see in 3D, e.g., from solid geometry.

E.g., in both cases, a biggie is a perpendicular (orthogonal) projection and, again, the Pythagorean theorem. E.g., regression in statistics is a perpendicular projection.

Perpendicular (orthogonality) is a biggie and is a major part of, say, Fourier series. I.e., each of the sine/cosine waves used is an orthogonal axis, and to find the corresponding Fourier series coefficient just project onto that axis. The projection is an integral of a product, and that is commonly an inner product which close to just a cosine of an angle as in plane and solid geometry and a perpendicular projection and close to correlation in statistics, etc.

E.g., a huge fraction of applied math is from analysis in pure math, and from G. F. Simmons the two pillars of analysis are "continuity and linearity". Linearity generalizes enormously: The quantum mechanics super position is essentially linearity. Under meager assumptions, differentiation and integration in calculus are linear operators. In probability theory, expectation is a linear operator. The wave equation is a linear partial differential equation. Linear programming works on linear equations. Of course, in linear algebra, matrix multiplication is a linear operator. When something is not linear, it may be locally linear which can be enough to get useful results.

For more, a good lesson is to approximate: Commonly we can't get just what we want in just one step but can iterate and approximate as closely as we please. So, can use simple things, sine waves, polynomials, continuous functions, and more, as means of approximation. Such approximation gets us close to more in continuity and, in particular, completeness -- the real numbers are complete and the rational numbers are not but via iteration can approximate the reals as closely as we please. Then this generalizes: The big point about Hilbert space (as mathematicians but not always physicists define it) is completeness. A joke, partly correct, is that "calculus is the elementary consequences of the completeness property of the real number system". E.g., the integral in calculus (and its better version in measure theory) is defined in terms of an iterative approximation. So, if you are good with sine waves, polynomials, continuous functions, wavelets, and more, then you can iterate and approximate a lot, in many cases, everything there is in that case.

I'm going to answer this assuming you mean gaining number sense, which is something I didn't realize I was missing until I gained it.

I got my undergrad in math and physics. I was good at math. It wasn't until I had been teaching high school for 3-4 years when some gave me a copy of Shortcut Math by Gerard Kelly. After reading it and practicing the techniques, arithmetic made so much sense. I was able to easily add, subtract, and multiply larger numbers in my head.

Interestingly enough, many of the techniques taught in this book are also part of the common core math curriculum. It's a way to help students gain number sense.

What exactly is your goal?

I think that the way to get math intuition is to learn a mathematical language, like Julia, and play around with it. Plot things. Change parameters.

Also learn a theorem prover. Maybe Agda or Coq or Lean.

For a small buffet of abstract math topics (with lots of exposition putting ideas into historical context and an emphasis on building geometric intuition) try Courant & Robbins "What is Mathematics?". Covers number theory, number systems, geometry, topology, optimization, algebra, and knots, among others. You could also play around with branches of math like Knot and Braid Theory, which are interesting in their own right, give insight into lots of different branches of math, and many interesting problems are still accessible to lay people. Number theory is also like this

I think it can also be helpful to learn some things about the history of math and the historical context that different ideas came from. Here's a nice example covering complex numbers https://www.youtube.com/watch?v=T647CGsuOVU

Maybe tangential to your ends, but the Crest of the Peacock is a nice book on non-European mathematical traditions, which provides some insight into how the process of establishing and validating mathematical knowledge works in other cultures.

You might try books written by physicists or that are about mathematical physics (an author to look out for depending on your level is VI Arnold), since arguments will be of a more geometric or physical nature and appeal more to intuition. Stillwell is another author (not a physicist) that tends to write books that give context and geometric intuition

You might like playing around with Pinter & Humphreys for Algebra, or Jänich for Topology (fantastic book for building intuition around topology).

Math is big. A good place to start building your intuition may be by learning proofs. I‘m currently revisiting math by working through Chartrand [0]. As a non-mathematician, I would recommend this text as a foundation, and a possible bridge to advanced subjects.

[0] Mathematical Proofs: A Transition to Advanced Mathematics https://g.co/kgs/stSmxJ

I'm interested in building up math intuition too. I'm engaging with math very slowly and just as a hobby, but part of what got me interested was encountering resources that are unreasonably effective at building intuition.

A common theme I have noticed is whatever it is, getting a geometric understanding of it aids intuition significantly. Others have mentioned the 3Blue1Brown videos. They are an excellent example of this.

These HN threads always bear out great resources and I've made note of (and acquired) a few of these, so I'll list them here.

Burn Maths Class - https://www.goodreads.com/book/show/26195956-burn-math-class

Book of Proof - https://www.people.vcu.edu/~rhammack/BookOfProof/

The Topology Of Numbers (number theory)- http://pi.math.cornell.edu/~hatcher/TN/TNpage.html

The Evolution of Trust (game theory) - https://ncase.me/trust/

Visual Information Theory - https://colah.github.io/posts/2015-09-Visual-Information/

Information Theory For Smarties - http://tuvalu.santafe.edu/~simon/it.pdf

Abstract Algebra - https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR...

Algebra Cheatsheet - https://argumatronic.com/posts/2019-06-21-algebra-cheatsheet...

Control Theory Basics - https://www.youtube.com/user/ControlLectures/playlists

Related question, anyone know any good resources for learning and developing intuition in Abstract Algebra? Especially ones that explain it well, like Calculus Made Easy did for Calculus. AA is a prereq for cryptography, category theory and and other areas of interest for CompSci.

I asked this a few weeks ago but at an off peak time and not many folks saw the question. But I know HN probably has good recommendations so trying again here. Should still be relevant and helpful to OP.

https://news.ycombinator.com/item?id=20733422

I run a 1-on-1 remote mentoring/teaching business where I teach the intuition and reasoning (including some history) behind mathematics, physics, and computer science. If you are interested in a full level overview into higher level mathematics, starting with set theory, reach out to me. We can organize an arrangement that suits your needs, desires, and schedule.

Shoot an email over to codertutor@gmail.com or text or call +1-718-360-3176 if you are interested or if you have any questions.

This is my print flyer for reference: https://i.imgur.com/Gdwa7m9.png

Looking forward to hearing from you.

Try finding a cool research paper that contains a lot of math you don’t understand. Then spend a few months learning the math in that one paper until you get it. That is a highly motivating and enjoyable way to learn. Far better than textbooks. I did this to learn algebraic topology.

This isn't an answer, but an observation. I taught an EE course many years ago -- the second semester of electrodynamics. This was after being a college math major, and getting a physics degree.

It doesn't shock me that there are holes. I noticed that some math topics are very important to engineering and physics coursework, but given short shrift in the math department. Examples are the way that complex numbers are used, and specific kinds of differential equations such as the general harmonic oscillator.

My college physics coursework actually had its own "math methods" class, intended to fill some of those gaps, and to get us prepared for the higher level physics courses.