How to Understand Math Intuitively? 

Probelm understanding, looking at problem with different perspectives, and different outcomes or combos!

Stop with this bs. Math problem *are* real life problems, anyone who believes the contrary is a damn engineer! Jk 😂

@@AliphbayIt’s like “Math ‘Street Fighter II’”!!

What I had in my curriculum was just mugging up formulas, without understanding why's and what's and the significance of the concept. I'd just mug up formulas and patterns of questions and apply the methods for such questions. Now I'm 24 and want to learn how to learn math intuitively. And my question is how'd you start from scratch?

@@gokulsrinath7714did you find the answer?

Very true, arigato

Step-4 Bread

i think you are missing the bigger picture here !

@@hunterxhunter9493 We have to go beyond the three steps to learn anything.

@@julians9070 sometimes,it is not necessary to follow these steps , but understanding the general rules ,the patterns of things around you is better , may be you have a big ammount of knowledge , but you are not that wise . that what i'm trying to say, yes you learned things you are conscious about them , but how does this affect your thinking, how can you find similar patterns in other domains etc , this is why we say a good mathematician is a good physician is because of his general understanding of things .

Absolutely !

Best of luck

Me too. No math since college and starting again. Struggling with order of operation. My tool box is a fisher price one.

Heyyy, thanks for your comment! I’m very happy to hear that you like these references 😁 yeah, I wish I had known about them already before high school. But it’s never too late. Best of luck to you!!!

U bought them all they super expensive:/

Indeed I did buy them all. It can be expensive, especially if you don't live in the US, but if you like math, and want to get really good at it, I can highly recommend.

@@W1llbam@W1llbam Are you referring to the art of problem-solving curriculum?

@@ElijahHemingway I was referring to their books specifically such as: Introduction to geometry, introduction to algebra, introduction to number theory, and so forth and then their intermediate books as well, plus NT course. All materials are really good.

I'm so proud of you

@@chugwhanio9538 Hahhahah I'm not sure what point you're getting at? :)

Very difficult with ADHD and mindset😐😐

Use the scale given on the map to measure the separare lengths by a ruler or a measuring instrument. Do the conversion for each and total the individual results for each to determine the actual length of the tube.

Hey Rafaela, thanks so much for your really kind comment! I wish you best of luck with all your mathematical endeavors 🙂

its sorta obvious' isnt it? solving probs is always the harder part

Compliments of probability?? I totally freak out if probability. Please guide me here

@@rapoliithe didn’t mean a compliment in the sense of niceties towards probability. He meant a COMPLEMENT, which means the complementing probability to something. You will understand immediately: If the probability to win in your favorite game is 10%. Then the complementing probability is 90% which is the probability of loss.

Very nice solution! 🙂

This is a basic Fermi Estimation question. I had fun making random questions up and then solving them.

Overestimated at 100,000 kms. I've got to downsize my estimate of city sizes.

Hello. I understand why a horizontal is 20 miles. It is an estimation, a proposition. I also understand why a vertical is 16 miles. Since a horizontal is 1.25 the lenght of a vertical, the vertical is 16 miles. Nevertheless, I do not understand why you have picked up 5 and 7 for the number of horizontals and verticals, respectively. Could explain it to me?

@@wilhelmjosephus4830 i suppose its just random numbers, youd get a similar no. anyway

Good for you. But fair warning, the math books from “The Art of Problem Solving” are geared to train future professional mathematicians. Meaning, they attempt to teach you to think in terms of doing mathematical proofs for everything. That is fantastic but I don’t think that will make you proficient quickly enough, because it will take a couple years for you to transform your brain to think in that way. Instead I would recommend you get two books from Saxon Publishers, by John Saxon, along with the solutions manual for each: Advanced Mathematics, and Saxon Calculus. I’ll explain why I think these books will prepare you better. Just as this video explained, normal math books teach you a new concept in one lesson (he called them “tools” in the video) and then move on to a new topic in the next lesson so you never really get to truly master the previous “tool”. The Saxon books are different than any other books I know of because the homework problems for any given lesson includes problems from the previous lessons. In other words, the problems for lesson 56 will contain problems from at least 15 previous lessons, and they will be a little harder than they were in the lesson you first learned those concepts. By the time you get to the end of the book you will be doing trig, algebra 2, pre-calculus, and calculus problems almost as automatically as if someone asked you “what’s three times two?” The secret is to do every problem. If you get stuck, don’t waste a ton of time going trying to figure it out. Instead go to the solution manual, study the step by step solution, close it, then do the problem again, remembering each step and duplicating the solution. That method will train you in both problem solving techniques and how to think mathematically. And while you’re at it, since you are going for engineering, get Saxon Physics too. I promise you if you can do every problem in all three books you will crush every subject in your engineering studies.

Put you behind? Take some damn responsibly and stop being a victim.

@@ccriztoff Dude. What’s wrong with you? He’s here making conversation with the video creator, thanking him, and he’s trying to learn math on his own. But you gotta butt in to dis him - a complete stranger minding his own business? Instead of lecturing him maybe you need to take responsibility where your parents obviously didn’t: teaching what constitutes appropriate manners and conduct.

M graduating class was 35, and teachers were hired off of how well known they were in the community rather than teaching skills

@@ATFofficial. Your class should have buck broke that teacher then 🥵

Hi can you update the url because the old one is expired.

do you know anywhere I can get the doctored pdf for the introduction to algebra (solution manual) PLEASE

yeah . . . . FUCK math!

Link expired :(

Invalid Invite Link 🙃

@@user-mm1yr8rx2l Thanks for your comment! 🙂 I understand that this must be quite disappointing. All I can say is: keep practicing, until you become better at it. It's surely not the last exam you will take, so just make sure you do better next time. I wish you best of luck!!! 🙂

Funny story. One day while at school I was thinking “When will I ever use math?” I ended up looking up a video on what the quadratic formula is used for (for simulated light rendering in that case). And I couldn’t really understand why they don’t just tell us that in school. I ended up going down on a rabbit hole, looking for different ways math is used and started realizing that the math problems they teach in school is a tiny fraction of the math that could be used and that there was so much more math out there that could be learned. I looked around for books and watched many videos that would talk about other kinds of math and understanding math at an intuitive level (3b1b, numberphile, veritasium etc) and I started realizing how rich the stuff was. One day at school I stumbled across a book on Mandelbrot’s discovery of fractals and on the inside of the book cover was a dedication from Princeton University to a student named Richard Ruyszick. I didn’t know who Richard Ruyszick but he went to my school and later went to Princeton so I decided to look him. Apparently he is the same guy who Created AoPS. Small world.

The key is to be humble and not be scared to afmit that you've missed a few steps during school. Start again so that you're ready to face greater challeges with the assurance that you have the necessary tools now.

​@@whannabi Yea, I'm good now in terms of confidence and humility. The humility has always been with me, to harmful degrees. I had to build up the ego and integrate it to some degree to come out on the other side. I had to be forced to come up with solutions on my own without me seeking direct guidance. I had to see that when I come up with solutions on my own, they are often praised by others who I regard much higher than myself. I just wonder how many people are running into psychological software limits while thinking they are running into hardware limits. Sometimes I think that psychology and philosophy should be major parts of the curriculum in any technical field of study. Not only for personal self-development, but to provide much deeper insights into the study than can be provided by technical analysis alone.

@@LCC2731 When I'm feeling overwhelmed by a technical explanation, I make sure to stop what I'm doing, and then really focus on trying to understand the simple underlying concepts. I try to build an intuition for the underlying concepts by searching for explanations written in plain English or with simple diagrams (minimal math, minimal symbols, minimal technical jargon). There's so many great/learner-friendly sources of information out there these days. And I might look at multiple sources to confirm that I am indeed on the right track in understanding the concept. Once I've gained a decent intuition, I use it as a roadmap to guide my interpretation of the mathematical formalisms that were originally overwhelming me. Knowing that I have to take the time to do this ultimately adjusts my expectations of myself to a more realistic level. A lot of feelings of frustration, intimidation, imposter syndrome are due to starting off with unrealistic expectations of yourself. And it's easy to get frustrated if you're comparing yourself with others and how quickly they appear to "get it". But you have little idea of how much time/energy they've previously spent learning the concepts or related concepts. So even if you're exposed to the same information at the same time, those who know how to learn and those who have already put in some time to understanding will appear to be much smarter/quicker than you. I think a lot of that is an illusion. To use an engineering analogy, each of our brains have very different initial conditions (which are hidden). Also, spend extra time on understanding fundamentals. For example, since I'm in electrical engineering, I spent some extra time understanding Maxwell's equations (there's only 4 of them) and surface charge distributions in the context of electrical circuits. Knowing how electrons fundamentally behave on a conductor, behave within electrical components, and behave with different excitations (high frequency vs. DC) allowed me to build a decent intuition that can be applied to any situation. You begin to realize most of the complexity and the conceptual breadth of your field of study comes simply from different configurations of a handful of fundamental concepts. But when starting off in a field of study, you may not understand this and it will look like everything is conceptually disconnected from itself. That's how it was for me in college. It just felt like everything was disconnected, loosely related, and as a result, very little of it stuck with me in a meaningful way. At the time, I simply learned what I needed to pass tests. I think a lot of people experience this. Hopefully that wasn't too jumbled of a stream-of-consciousness. Just pick and choose the things that resonate with you. Sometimes it's just one or two sentences in a wall of text that really resonate with your current state of mind. Hopefully some of it helps.

Your series of comments are amazing. I'm an undergraduate math student and sometimes I feel like I've hit my limit with my grades. I need to build up my ego, understand the basic stuff and force myself to stop memorising Definitions and formulae.

You’re very welcome :)

Thank you so much 😀

Heyyy Lucas, thanks for the comment and sorry for my slow reply! Yes, I would he happy to make such a video. Probably this summer already :) I just have to find a way of making it fit the main topic of this channel, given that I’m not running a fitness channel 😉

Super dass du es wie ich gemacht hast. Ja, Übung macht wirklich den Meister. Und, wie du gesagt hast, Mathe kann wirklich Spaß machen :)

Hätte mal eine Frage: Seitdem ich anfing mit dem Informatik Studium wollte ich meine Mathe Skills verbessern. Desto mehr ich über Mathe herausfand desto mehr wurde mir bewusst, wie wenig ich eigentlich kenne und die Themen scheinen endlos zu sein. Kommt man später an einen Punkt wo man sagen kann, nun verstehe ich alle Mathematischen Gebiete? Fühle mich so überrollt.

@@UberBossPure Eigentlich nicht ganz, aber es fällt dir viel leichter in andere Gebiete einzulesen. Die Grundmodule in der Mathematik sind wichtig und dann versteht man die fortgeschrittenen Themen besser. Vor allem in Physik, Informatik wo gewisse Ideen Konzepte uns mathematisch irgendwie bekannt vorkommen. Pure Mathe geht schon über viele Aspekte vorbei. Da gibt es mathematische Gebiete, wo sich nur ein handvoll Leute aus der ganzen Welt sich auskennen. Aber du musst für dich wissen (als Informatikstudent), was willst du später machen und welche Mathematik würde man brauchen. Für Physik viele Differentialgleichungen, riemannsche Geometrie für Allgemeine Relativitätstheorie, oder vllt Topologie für Kondensierte Materie Physik. Du musst schauen, was du brauchst und dann verwendet man die Konzepte für deine Forschung und deinen Erfolg.

@@nosferatu5500 so, master basics and you master everything?

​@@WhiteStripesStripiestFan Master basics and you'll surpass those who merely memorize without understanding the advanced.

😊

You’re very welcome 😊

Glad to hear that! 🙂

Suuuuuuuperrrrr!!!!!! 😁💪💪💪💪 Viel Erfolg wünsche ich dir dabei!

Beautiful insight, and I totally agree with you @David Walker 👏🏾 I think math, like a lot of other subjects, can be cultivating as a natural part of understanding the world - it’s even, in fact, touted as the first natural science. As intuitive and creative as we can be, I think everyone has the potential and capacity to do math - math, as universal as it is, is not reserved for some exclusive club. I enjoy fostering the math intuition within every student I work with as a math tutor. ☺️

Thank you 🙂

A physics version will come soon 😊

Thank you Arno! yeah, the intro is new 🙂

Thank you Arno! yeah, the intro is new 🙂

You’re very welcome 😉

You're very welcome 🙂

Hey Leo, thanks for the comment and sorry for my slow reply! Yes, I completely agree! It’s so easy for a teacher to discourage someone from going in a certain direction, but so difficult to motivate people to do so. Especially the example you just mentioned Also, good job with the estimation process - I really like your answer! 🙂

If a teacher tells girls that maths is for boys. Tell the teacher to look up the name Ada Lovelace then ask if they still think their statement holds true!

About the math isn't for girls , girls are all on onlyfans nowadays...

My dad and I would try to learn math when I was little and it never worked out. I guess I always wanted to do something else or he wasn’t great at teaching it. So basically throughout my schooling and as I was going into collage he would say but you can’t do the math and I decided against classes with math or even trying a class with math. Now I graduated and thinking back that I could have done the math and not I’m stuck doing something I’m not that interested in. I had one small math class when I was at college and came out top of the class of 14 students. Not huge it was simple math but I realized I could do math.

Thank you :)

That’s great!!!

The Khan Academy is a great resource!

@@Shannon_Robbie Thanks!

>That’s kind of what this video describes, it just says “learn it and then you’ll know it” circumventing the fact that plenty of people are just genuinely bad at math, because they have a low aptitude for abstract reasoning in general. Hello, Rigo. Yes, the video in question did not address the lack of abstraction. However, it is important to note that abstract reasoning is a skill (not a fixed genetic trait) that can be trained, developed, and refined. The key to achieving success in mathematics lies in identifying the right approach. Therefore, the use of the term "aptitude" is not entirely appropriate in this context. In reality, almost anyone can acquire proficiency in mathematics, and topics such as Linear Algebra and Analysis are often considered basic, trivial and straightforward within the higher-level mathematical community (no offense). With that said, the reasons for individuals failing to achieve success in these subjects are typically related to other factors such as laziness, lack of discipline, inadequate education systems, cognitive biases including anxiety, imposter syndrome, constant comparison to others, or aversion, insufficient motivation (worth noting that there are relevant studies on the subject, such as hypofrontality, ADHD, behavioral addictions (which are increasingly prevalent in our modern society), etc.), flawed learning methodologies, knowledge gaps, lack of interest, unrealistic expectations, or an unsupportive environment. Additionally, in a world where computers and machines can automatically solve problems, there is a tendency towards complacency among many individuals, leading to a reluctance to engage with concepts that machines can easily process in a fraction of a second. Studies have been conducted on this phenomenon, known as "Google Syndrome," which can also be applied to other fields. Neurology did also shed light on these issues. _Perhaps I’m being too negative but this approach hand waves away the biggest problem of natural ability, and people struggle with math despite spending time to learn the concepts because they don’t have that aptitude_ Your statement may create the impression that this issue applies to the majority of people, while this is not the case at all. There are certainly rare cases of individuals with severe mental handicaps who may face significant challenges in mastering mathematical concepts. However, these individuals often struggle with communication, basic conceptual understanding, and other fundamental skills that can make learning mathematics difficult. In any case, it is unlikely that such individuals would pursue higher education, so they do not represent the "average" student population. The video in question is addressing the average Joe or Jane, and it is reasonable to suggest that such individuals are certainly capable of grasping concepts in linear algebra and analysis, provided that they approach the material correctly. _[...]and people struggle with math despite spending time to learn the concepts because they don’t have that aptitude_ Once again, this assertion of yours is, for the most people, incorrect. In fact, aptitude is often the least consequential factor, the least of the problems in explaining why individuals struggle in mathematics. Rather, the primary factors that account for failures in these subjects are often more complex, including behavioral addictions, such as those involving pornography, social media, or being chronically online in general, and the tendency towards instant gratification that characterizes a major part of our modern society. These behavioral addictions have been shown to contribute to hypofrontality, which can have profound effects on the brain's reward system, and thus, necessarily on the ability to learn, to concentrate, to abstract reasoning, etc. People didn't believe this at first, and since it is so prevalent nowadays, because people are "used to it", they don't even notice the consequences arising from this. In the past year, these concerns have received official recognition, with the International Classification of Diseases, Eleventh Revision (ICD-11) formally recognizing compulsive sexual behavior disorder (CSBD) (Remember: People laughed about this, admittingly because certain religious sects use the empirically verified unhealthy nature of pron addiction to further their cause of religious living). The negative impact of such behavioral addictions on learning is then often compounded by the pace of modern life, which emphasizes speed and immediacy over deliberation and depth of understanding. In the past (say the 50s), individuals who wished to research a question or topic were required to visit a library and sift through multiple books, which encouraged thorough reflection and deeper engagement with the material or think about it entirely themselves without assistance. By contrast, in today's world, people are often satisfied with minimalist, on-point and immediate answers that require little effort or time investment (just look at this video or other video essays and for the comments that "summarized" this long video and then see how this summary often misses the in-depth idea of what the video is trying to say, but people still being satisfied and thanking them "for saving them 30 minutes" => Necessity for fast answers, symptomes of impatience, etc), which can lead to factual knowledge without a more profound understanding (which is why I have a personal dislike of "fact checkers" that even more so encourage this behavior). This minimalist attitude and lack of attention span can also be observed in the popularity of platforms like "RU-vid Shorts" and "TikTok" compared to longer video content that is often skipped. Admittingly, the average person has less free time to devote to a particular subject nowadays, making it more challenging to engage with complex material fully in length. In other words, this modern trend towards instant gratification has become so pervasive that individuals who reject this approach are still often forced to conform to it. People *still* (despite the vast amount of research in neurology and existing brain scans) underestimate how severely the age of the Internet influenced and still influences our brains. Therefore, average individuals who might have succeeded in the 60s with linear algebra and analysis may struggle in the present due to the constraints imposed by modern society, Internet and educational systems. The introduction of the Bologna Process and the adoption of bachelor's and master's degrees in Europe and Germany have resulted in the boiling down and confinement of knowledge into shorter time frames, making it more challenging for average individuals to engage with material comprehensively, therefore ultimately elitizing academies for the few talents who can keep up with the modern pace. In contrast, in the past, diploma programs allowed for more extended periods of learning, which facilitated deeper engagement with the material, allowing even the average Joe to confidently get a degree without being a fast-learning genius.

@@Kaje_ Are there amazing books, spectacular tutoring programs and other wonderful resources you'd like to recommend that can enable one to master the fundamentals of math?

Heyyyyoooo Nosferatu, danke für den Kommentar 😉 jaaa, Feynmantechnik ist super. Leider musste ich auch so viel auswendig lernen im Studium. I’m gymnasium war es viel besser

Thank you so much Zenin 🙂

Hahhahah that's 🙂

hey what do you mean by cutting it into arithmetic series of circles?? I don't really understand the wording lol......

Hahahhah thanks you 😊

You’re very welcome 😉 I’m glad you enjoyed it!

Danke für den netten Kommentar! Kommt mehr 🙂

Thank you Sergio :)

Hey Osman, great proposed solution! And very detailed! Let's hope this channel blows up and we get the required 100k views on this video 🙂

@@SamuelBoschMIT Danke mein Freund! Hoffe ich auch für dich. Großartige Einstellung die du hast. Vielleicht läuft man sich ja irgendwann über den Weg. Bewege mich auch in der Finanzbranche

I enjoyed reading this very much. I’m a programmer that at age 40 nearly just got into math (because of computer science stuff and general interest) You have any books you’d recommend for folks wanting to start over and build up

Hahhahah that’s what I’ve been asking myself too 😂 Thank you!

Hahahha thanks Jan 😁

Thank you Cristian! I really appreciate it 🙂

You’re very welcome 😉

You should! 🙂

Thank you! I’m glad you liked it! 🙂

Excellent! Thank you Sheldon! 🙂

Hey Franz, thank you for this really nice comment! 🙂 It really depends on what you want to do. One hour per day is surely not bad at all. But if you want to go to IMO and IPhO, you might have to step up your game even more. I think most people who went to these competitions did between 2 and 8 hours of practice per day (it varies significantly) for quite some time. If you wanna go to the IPhO, you should probably also focus a bit more on physics competition problems. They are similar to math (=knowing one will help you with the other), but they are also different in some ways. Being so good in physics helped me get into the top 10 in math on national level, but obviously wasn't enough to get me to the IMO. I wish you best of luck with your IPhO preparations. Maybe I'll make a video about this topic at some point 🙂

Hey Jun, thanks for your comment! This week’s podcast episode might be super interesting for you - we talk about computer science competitions :) It will be released on Sunday, so stay tuned :)

Thank you - I’m glad you enjoyed the video! 🙂

Very nice 😊

Freut mich Finn :)

Haahhahah Wow, this was really fast 😁

Thank you Jay! I’m glad you enjoyed the video 🙂

Very interesting solution! Yeah, I never even thought about this. But of course, the tunnels should be structured in a certain (logical) way, so this is a very smart estimate 🙂