How to Understand Math Intuitively?
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- Опубліковано 12 чер 2024
- How to prepare for math competitions? How to understand math intuitively? How to learn math? How to practice your math skills? These are all questions I address in this video!
▬ In this video ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
I explain why most people struggle with mathematics, despite not lacking intelligence or knowledge. This video is not limited to students in school or university, but rather applies to everyone interested in using the full potential of math intuition. I created this video together with my friends at MIT, who all attended and won different math competitions/olympiads during high school.
▬ Resources from the video ▬▬▬▬▬▬▬▬▬▬▬▬▬
Solution to the London underground tunnels problem: tvCd3PKt9A...
For all ages and educational backgrounds:
- Art of Problem Solving (artofproblemsolving.com/store)
Intro level resources:
- A Concise Introduction to Pure Mathematics by Martin Liebeck (www.amazon.com/Concise-Introd...)
- Mathe ist Cool in German (matheistcool.de/)
More advanced high school and above:
- Problem-Solving Strategies by Arthur Engel (www.amazon.com/Problem-Solvin...)
- Love and Math: The Heart of Hidden Reality by Edward Frenkel (www.amazon.com/Love-Math-Hear...)
▬ Follow me for more tips ▬▬▬▬▬▬▬▬▬▬▬▬▬▬
Instagram: / samuel.bosch
LinkedIn: / samuel-bosch
TikTok: / boschsamuel
▬ Credits ▬▬▬▬▬▬▬▬▬
Special thanks to my friend and former classmate, Jonas Wolter, for providing me many of the insightful resources from this video! You can find him on LinkedIn: / jonas-wolter-40b277147
Also, thanks to my friend and colleague, Emily Mu, for helping me with with this video!
If you don’t have Instagram, feel free to watch the solution to the London underground tunnel problem here: Solution to London subway problem from Instagram
• Solution to London sub...
▬ Timestamps ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
0:00 - Intro
0:53 - Why most people don't get math?
2:17 - How to learn math intuitively?
4:38 - Best math resources and literature
6:30 - Practice problem
7:57 - Outro
My mom has a master's in math, and she can solve complex math problems, but later, I learned how difficult it was for her to apply this knowledge to solve real-life problems. After I became an engineer, I realized how difficult it was to figure out what mathematical approach you can use to solve problems. That is when math becomes exciting and fun.
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?
"The key to the learn anything"
Step-1 Understanding
Step-2 Remembering
Step-3 Practice
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 .
Math for me is the greatest endeavor of human kind. Pencil, paper, handwork, life commitment and the greatest adventure on the search of beauty.
Absolutely !
Absolutely fantastic recommendations, literally the best, I am in fact now going through these books and it's incredible how much it differs from the standard education I had
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.
As 73 year old studying math for the first time and having a mechanical background your analogy is crystal clear and very logical. Thanks for taking the time to put your thoughts on line. With the tube question I would telephone the org’s engineering section and ask them for the total length.
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.
I definitely agree that learning mathematics needs a proper mindset. One just needs to understand the concepts when it comes to solving problems.
Math is actually at least two skills or disciplines. One is understanding the concepts. The other is being able to solve problems involving them. I would argue that the former is not quite as difficult in terms of sheer work required to develop as the latter. A lesson I learned much later than I would have liked to, owing to my personal history. Learned from bitter experience. Which also shows that luck counts, too. Hard work and luck are not totally independent variables. They can influence each other.
its sorta obvious' isnt it? solving probs is always the harder part
Check out my new video about math skills you ACTUALLY need in life 🙂 ua-cam.com/video/A5CMAcyI42o/v-deo.html
@@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.
Thank you for explaining in a calm and friendly way. It really makes difference in how we can view mathematics as something helpful and not just bunches of information to be decorated. Hope you can keep spreading acknowledge and helping many people like me who is trying to relearn this amazing topic. Take care x
Hey Rafaela, thanks so much for your really kind comment! I wish you best of luck with all your mathematical endeavors 🙂
Join my Discord community for more content: discord.gg/m79PRZ4tCe 🙂
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 🙃
For the London Underground problem I would first of all aim for a rough ball-park figure. I'm also too lazy to take any train speeds as inputs. Looking at the Underground map, I'd like to simplify it further by imagining the network as a set of pure horizontals and verticals. The total length is then just the sum of H horizontals plus V verticals. Let's also consider the map's rectangular shape, so a horizontal is e.g. 1.25 the length of a vertical. How wide is London? Let's say 20 miles? So a horizontal is 20 miles, a vertical is 16 miles. For our "raster" Underground, let's pick 5 for the number of horizontals, and 7 for the number of verticals. Total? 5 x 20 + 7 x 16 = 212 miles. I still don't know what the true figure is, but I expect the real length to be closer to 212 than 21,2 or 2120 miles. Quick Google check.. 249 miles.
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
I think intuitively understanding probability is the most useful specifically, understanding the compliments of probabilities, which is super easy honestly. Understanding the compliment of a probability will completely change your mindset when coming across a statistic.
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.
Thank you brother. I went to a rural highschool with teachers who were poor at their jobs, and therefore put me behind. Im re teaching myself math from algebra all the way up through calculus currently so i can hopefully get into school for engineering next fall. This helped tremendously
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 🥵
I am 71 and starting the furtherance of my math journey.
I'm so proud of you
Check out my video on problem-solving strategies used by the greatest scientists of all times 💡
ua-cam.com/video/yoFBiyg_ulg/v-deo.html
Check out my solution to the London subway tunnels problem: instagram.com/tv/Cd3PKt9At0A/?igshid=MDJmNzVkMjY=
@@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.
Yep, this resonated with me. I did pretty average/crappy in my engineering undergrad because I didn't know how to learn. Like you said, it just felt like memorizing procedures and very little of it stuck with me in a way that could be extrapolated to novel situations. Too many concepts immediately obscured by dozens of symbols and pages of rearranging equations. That was intimidating and prevented any confidence in thinking that I was grasping any of it at all. In retrospect, after figuring out how to learn new things, the underlying concepts of the stuff that used to terrify me are embarrassingly simple to understand. I think a lot of professors are super smart in their field of study, but very little of their brain power goes to thinking about teaching and how people actually learn things. I can say that because I also had a few professors that were really good teachers and I remember everything I learned in those classes.
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.
I've recently got interested in math again, and I find it amazing how it appears in many areas. There's also many solution to one problem, but I like to think with simplicity. If not only I can solve a problem, but explain it to others, then it's a simple solution. I feel like if math is taught correctly, people can see a form of beautiful art in it too. The best if you can find answers logically and intuitively as well.
Amazing video, another one. So I think it's not my discipline but I'm highly impressed by this skills. Great work.
I see forward to a video about your fitness routine! Keep going, it's great!
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 😉
Your brilliance, dedication and enthusiasm are shown in your work. Keep on doing great jobs for everyone, you're a legend
Thank you so much 😀
Thank you for your inspirational message and friendly presentation. I'm already in my 40's but I hope by picking up the textbook you recommended, and doing some of the problems in the book and online, I can refresh some knowledge and learn new ideas!
For the tube problem, i took a fairly simple approach. I once took the track from Heathrow to downtown which took around 1 hour (on average it seems airports take a rough 1 hour by transit). Assuming an average speed of 40km/hour this would be around 40km of track. I know that line extends beyond my stop so just guessing that that line is 60km long. Starting from there, I remember 8-10 colors on the subway map which indicates around 10 lines. Now I’m guessing heathrow line is one of the longer ones so let’s say the average line is around 40km, times 10 lines gives us 400km of total line. Now tunnels are generally underground in the city and overground in the suburbs, I’m guessing central London is about 40km in diameter while the total tube coverage area stretches 80km in diameter. Thus Longer lines are primarily overground but more sparse while shorter lines are primarily underground but more frequent. These effects should mostly cancel each other out so around 40-50% of the total line should be tunnelled which gives a range between 160-200km!
Ich war auch nicht gut in Mathe, aber tada, bin ein theoretischer Physiker geworden. Ich habe erst ab der 8.Klasse gemerkt, dass wenn man sich hinsetzt und übt und übt, dann merkt man in der Klasse wie weit fortgeschritten man ist und wie sich das bemerkbar macht in Form von Lob eines Lehrers. Erst im Studium merkte ich, je mehr Mathe ich mache, desto mehr Spaß macht es. Das war davor für mich nicht so denkbar, dass das Freude in mir auslöst :).
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.
Your analogy on the tooltips was really interesting and helpful, thanks a lot!
You’re very welcome :)
I’ve always been amazing at math. I’ve always understood it intuitively. Its why I love it.
My solution to the tube problem:
The tube trains average speed is 33Km/hr. Find the total time it takes for each train to travel it’s entire line and add them together, formatted as a decimal of hours. Then multiply that value by the average speed (33) to get an average of the total distance of the tube line.
I think that we gain intuition (a feeling how to solve the problem) when we do some memorable examples (about some physics; a different way of doing the geometry) and reconnect the idea using space repetition (explaining to a 5yo).
And not doing a bunch of exercises in a day; maybe that after you have understanding and test your new tool set.
Interconnecting ideas related to different subject with wide scope (hate when say I leave that to your physics teacher or that is part you will see in probability), not isolated pure math.
I kind get some of that on edX.
A concise introduction to pure mathematics
Great Video! I love the idea of gaining an intuitive understanding of math.
For the tunnel problem solving I will check the radius of London. I will calculate the length of a line using the assumption that it crosses the whole city usually lines make , then I will count number of lines and I'll multiple the length of a line with the number of lines ,and if I have a map of the tunnels maybe will give them some 5 -10 % bonus because metro trains don't go always in a straight line , that's my 4 grade math solution 🤣
You give me hope for myself. I’m always down on my ability. Thank you so much
😊
Dankeschön! I’m currently pursuing an EE degree and definitely need to improve my math skills. I appreciate your advice.
This is absolutely terrific. I believe that this could crack the code with public schooling in general. Couldn't it be explained that the height of education is intuition? Intuition being the accumulation of knowledge, wisdom and creativity. I'd love to have a real discussion on this because we must know how to develop our intuition if we hope to progress as a species. Unfortunately, the modern school system has failed us.. math was never explained to me in this way and I struggled.... But now, all of a sudden, math is being explained in an intuitive sense and it's clicking with me. Now let's try and translate intuition to the other academic pillars. This is just so incredible!
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. ☺️
Great video, Samuel! As someone who studies math, I totally agree with the mindset-part. But a lot of time it's not even a person's fault but the fault of their teachers/parents etc. who tell them that "math just isn't for them" really early on (or even worse, there are teachers saying "math isn't for girls" which just excludes half of the population).. One of my friends at uni told me that some of his teachers in elementary school said the same thing to him but he was lucky to have a physics-teacher in 6th grade who really encouraged him to understand physics and math and in the last few years of school he even went on to win some national & international math-competetions.
For the tube-problem my thought would have been the following: I know that Berlin has a diameter of roughly 40 km with a population of around 4 million. I also know that London's population is around 9-10 million (let's just assume 10) and that London's population density should be a bit higher (I would have guessed by 20%) since Berlin is really spaced out. So London_Pop/Berlin_Pop = 2.5 and London_Dens/Berlin_Dens = 1.2 which means that London_Area/Berlin_Area = 25/12; therefore Londons diameter should be sqrt(25/12) which is close to sqrt(2) which is close to 1.4 times the diameter of Berlin, so around 56km. I would also expect that the tube does not go to the edges of the city as a tube system is usually built for the parts closer to the city center so I would subtract 10 km from the city's diameter to get the tube-system's diameter. (This would be around 46km now) I also know that Berlin has 3 lines which, kind of, cross the whole city (so their length would be close to the diameter of the tube-system) and another 5 which only cross half the city. I would just double those amounts because, while London's population is 2.5 times the population of Berlin, I also think that London's tubes have a slightly higher frequency than Berlin's so that makes up for that. With all these assumptions I would therefore arrive at 6*46 + 10*23 = 506km. This certainly isn't the best guess but it works without knowledge of any city's rail-length.
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.
This video is such a gold nugget in the whole "math" youtube content! Maybe it's not a unique idea for the video but with such a elegant execution!
Also sorry for my English, overall very positive about this type of content!
The most important mathematical tool I've _ever_ come across, is the lowly percentage calculation. Armed with it, you'll gain an entirely new and useful understanding of the world. But... By gaining that knowledge, and using it in practise, I sadly also found out how few others employ it daily, such as professed "experts" even within medicine and economics.
Quick calc for London underground tunnels is, 10 lines (I think there are more), times 25km per line (~15k radius out from City Centre sound reasonable?), 33% tunnels (most of Zones 1/2 (there are like 6+) are underground then other tunnels); so a third of 250km is about 63 km (its probably more). I read its 200-250 miles somewhere, so about 400+ km
My lecturer of physics used to tell us that there are 3 stage of learning a certain topic in this ascending order.
1. Understanding
2. Telling
3. Writing
Hey Samuel! Great video! What do you recommend about how long we should practice per day? Right now I am solving problems for 1hr per day, is that enough? And you said you were solving math problems every day and then you participated at IPHO. Do math and physics problems the same effect or why didn’t you focus on physics problems. I also want to participate at IPHO and would be glad about your response.
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 🙂
It's surprising how powerful ratio is, I could solve most of my high school level problems with it.
For the benefit of others interested, let me share two obvious titles as a first read.
1. A Primer for Mathematics Competitions by Zawaira & Hitchcock
(A recent Googling pops up a couple of similar titles but I am unfamiliar with those.)
2. The Art and Craft of Problem Solving by Paul Zeitz
Excellent! Thank you Sheldon! 🙂
I love the series "Everything you need to learn in a simgle fatbook" and the series "Fundamentos da Matemática Elementar". The FME can teach you from the basics in math (from probability) to some things more complex (like introduction to calculus). The geometry books, for example, have 1400 geometry problems (really). It has 11 books, but only in portuguese language
Damn! Worked like a charm! Thank you soooo much!
Glad to hear that! 🙂
Great information. I am very happy with these tips
This is a total life changer to my point of view in mathematics, hope you make more of these spectacular videos
A physics version will come soon 😊
Thanks for this video, Samuel. Your new intro looks very good too!
Thank you Arno! yeah, the intro is new 🙂
Thank you Arno! yeah, the intro is new 🙂
A typical city maybe has about 8 subway lines, each one of those taking about an hour if you ride them from one end to the other. I'd say the average speed of a subway train is 40 km/h, so that gives you roughly 320 km in total.
Thank you very much for this video sir Samuel!
You’re very welcome 😊
Danke für das Video, hole mir nun auch die Bücher! Hab geliked und kommentiert damit der Algorithmus dein Video hoch pushed
Suuuuuuuperrrrr!!!!!! 😁💪💪💪💪 Viel Erfolg wünsche ich dir dabei!
For context I do not know anything from the London subway system and I’m at 9th grade level mathematics.
I would look at it from a top down aproach in a 2D space because even if subways change elevation the values in any meaningful way. And I would add all of the values together. But to be more precise with this I would find the area of intersections if there is any. And I would subtract it from the final value.
The Tube system in London is build in a very simple fashion. There’s lines which go horizontal (east to west vice versa), lines that go vertical (north to south). All the lines kind of meet in the middle of the city (downtown). So it kind of resembles a circle. The really roughest way would be to take the area of London and pretend that it is a circle. Then calculate the diameter of that circle and multiply it by the number of tube lines there is. Kind of creating a wheel with the spines being the lines that go through the city and the outer ring of the wheel being the borders of the city.
The way I’d go about it is creating an estimating models. I’d take the width of London (roughly east acton to Stratford) and height (roughly Barnet to Mordon).
From here on out it depends on how sophisticated you want to model this problem. The simplest way would be to count the number of tube lines London has and multiply this number by the average length and width of the city.
If you want it more sophisticated you could count how many horizontal and vertical tube lines there is and multiple these amounts with the width respect length of the city.
You could also go ahead and get more sophisticated by iteratively aiming towards a more precise recreation of reality. You could model out the circle line as a the circumference of a certain diameter within this model.
To consider things like curvature of the tube lines I would just estimate a 20-30% add on top of the length and width basis we used
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
It's pretty ez as the tube is most likely to be used by Subway train and u can calculate the length by knowing the time taken to reach from first to last stop then tweaking it a little to adjust for stopage time and using the avg speed to find the time
In my secondary and high school I absolutely hated mathematics but recently when I understood the reasons behind the formula and standard forms, my interest in mathematics bloomed! Can someone help me find the resource to understand mathematics better? Thank you mate for the resources and video!!
The Khan Academy is a great resource!
@@Shannon_Robbie Thanks!
About the problem you mentioned, we can just calculate the length of all subway trips in one direction
Thanks man I'm improving slowly and slowly I hope I'll countinue like this your really helped me thanks sir
You're very welcome 🙂
Mr.Samuel, I'd really like to offer my gratitude to you for the content you provide. This is so helpful.
Thank you so much Zenin 🙂
take hillingdon as "x" and havering as "y"
Take into consideration of all the routes from x to y from the tunnels london's public transport system.
find the length of each route.
and lastly add them all up to find the total length of all the tunnels.
thank you
I have naturally learned these skills over the past year or so in my persuit of a maths degree. Although I personally found it much easier to find these skills by basically being talented at mathematics, lowering the time it took for me to completely understand and gain new tools. Like a 60 unit hour course in linear algebra was completely sufficient for me to understand most all of the material, from the most basic applications of matrices to vector spaces and eigenvalues. All the fun stuff which would normally be too quick for most people to pick up, I simply applied myself for hours on end and learned the material effectively. 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. 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.
>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 "UA-cam 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?
For the tunnels I think you need to have central point to start the tunel like the beginning of the city .....then this will be going in all directions south north west east and back to the central point with sub roots or additional lines .....so you need to know how far of the places that can be reached which can equal the length of the tunels like for if from the beginning of the city to the north was 154kmeter the length of the 154000m ....
Ich habe für mich gemerkt, dass das Anwenden das Beste ist. Wenn man viele Aufgaben löst dann prägt sich dass besser ein als das Script mit den Beweisen ins Hirn zu stopfen. In den Aufgaben hat man auch viele Fallbeispiele und wenn man etwas ähnliches kennt, weiß man vielleicht wie man das lösen muss. Außerdem ist die Feynmanntechnik auch sehr empfehlenswert.
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
Sweat the small stuff, and never be afraid to go back to basics.
As Tobias Dantzig wrote in his book 'Number : The Language of Science ( which I highly recommend ) : ' Why is it that only mathematical processes can lend to observation, experiment, and speculation that precision, that conciseness, that solid certainty which the exact sciences demand ? When we analyze these mathematical processes we find that they rest on the two concepts : number and function ; that function itself can in the ultimate be reduced to number ; that the general concept of number rests in turn on the properties we ascribe to the natural sequence : one, two, three'.
the way i would estimate the length is first thinking about the size of London. London as a circle should roughly have a diameter of 40 km. Since the density of the tunnels should be higher in the center of the city i thought of a tunnel strucuture similar to a star (first draw two straight tunnels across london that look like crosshairs then add another two tunnels splitting every quarter of the circle(London) so that london looks like a pizza, and then another four tunnels that split every eighth of London.) That makes 8 tunnels with a length of the diameter of London which is 40 km which results to a total length of 8 * 40km = 320km
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 🙂
Thank you Sir.U have inspired a lot.May God bless you 😊
Thanks for the book recommendations. Would you say that learning math through a book is more useful than practicing the topics from youtube tutorials?
The problem with School and college tuition is that its a factory. You are sorted into your supposed intelligence then you get trapped there. Depending on what conveyer you are on you will only be given the course material to satisfy a pass for your stream. Everything is pace and grades, the actual knowledge gained doesn't really matter. The big problem is that textbooks also follow this as well, the books just target the exams or levels to be studied. Continuity which is so important doesn't seem to exist. This of course is partly due to the physical size of a book. In the UK it was very hard to learn advanced shool maths as they all jumped in at more difficult areas without the all important lead up. I found American text books so much better and I only wish I could have obtained them when I was younger, (pre Internet).
huge respect and good wishes to you.
Thank you 🙂
thank you for your advice. your tips will be very helpful to my students.
Really a great video!! Kindly make more videos on Maths and share some more resources like this...Once again thanks a lot!!
Bro, this guy did all the side-quests in life
to solve the 'tube' problem i would first know the total height of the tunnels and the length and the would use the formula of curved surface area of a cylinder by breaking it down and then would add all of it to get the area the reason why I chose curved surface area is because he asked the area through which the train goes but not how much can he put into the tunnel.If i am still wrong correct because i tried my best :))
I would use a square or rectangle and estimate length and width, such as 20 by 30. I would then assume there is some pattern to the tube system for it to make sense to riders, such as an array/grid and thus estimate somewhere around 400 to 500 miles, or 640 to 800 km. The problem is I have no reference for how big London actually is but I know Toronto is about 45 km across so would make an assumption that since Engand is a smaller country, London might also be smaller. Therefore I might say that a reasonable answer is 640 to 700 km.
The radius of greater London is about 45 miles. This makes the circumference about 270 miles. Imagine each tube running from the center out 45 miles to the urban perimeter. You need enough tubes so that one need not walk too far to get to a tube station, so guess about 4.5 miles between end stations on these radially arranged tubes. Each tube is thus 9 miles apart at the perimeter. But nearer the center you don’t need as many, so the number is cut in half every 9 miles. So there are 9x30 + 9x15 + 9x7.5 + 9x3.75 + 9x2 or rounding to 9 x 57 = 513 miles of tubes.
This was a fun problem, I've been seriously studying mathematical fundamentals since the start of the year.
I imagined London as a circle with radius 10mi. I cut the circle into 8ths with 4 lines, and I cut it again into an arithmetic series of circles from 2 miles to 10 miles. Then you just sum up everything, 60•pi mi + 80mi = 268.4 mi. I looked up the actual number and saw 250mi online.
Anyways I hope my math checks out, but I was happy to get the right solution. A few months ago I wouldn't know where to start :).
hey what do you mean by cutting it into arithmetic series of circles?? I don't really understand the wording lol......
I just thought another solution for "The Tube" problem. I'll just look for 50-100 cities which resemble the population density of London and then take out the mean of the underground tunnels. Obviously I consider that the length of underground tunnels in these cities are known to me. And geographic factors will too deviate the mean length but still it will be a good estimate. The more cities I consider, more will be the accuracy...
My answer is to figure out how fast the train goes which should be common knowledge of the train company (mph if your in America like me, or km per hour) then ride the train to see how long it takes to go from station to station. Don’t count time between stations as they are not part of the tunnel system anyway. I’m not sure how the stations are set up, but you should be able to use all of this information to calculate roughly the length of the entire tunnel system.
i live in paris so im gonna base my answer on the paris subway : i know there are 15 lines, each line has approximately 17 stops and it takes approximately 1min 30 to get to one stop. so in total there are approximately 1245 subway stops in paris, the subway moves at approximately 60km/h so it means appropriately 1,5km per station so 1245 x 1,5 = 1873km
Great video! Thanks sir
Good resources. You know your stuff! 🖖
400 miles without looking anything up. But I am a Londoner. So I know there are twelve or so tube lines. Diameter of London about 20 miles. 12 times 20 is 240. But some of the lines run above ground. So call it 200 miles. It is 400 miles if you mean the total length of tunnels in both directions. (When I looked it up, I found that 55% of the system runs above ground, so that was the major error I made.)
For the tube problem I just used a series of concentric circles with varying degrees of tunnel density (inside being the most dense). I assumed that the distance between tunnels would be around 1/5th a mile apart for the innermost circle and gradually increases the further you are from the center. I ended up getting 115 miles of tube which was a lot closer than I thought I'd get.
That’s a nice and creative solution 🙂
i have always had good grades all throughout school,i could solve problems on paper with understanding & some meorization too. but i have realized i always lacked application of math which is why i i didnt develop love for it . Time to give math another chance & enough time
I know London is approximately 20miles Nth-Sth by 30miles East-West. I know the main lines and their very approximate layout e.g. Central Line runs East-West: therefore ~30miles. I add up all the lines and this gives me a total length of ~150+ miles. I did this mentally in ~5mins and used my fingers to add the lines (in multiples of 10miles).
Other than some consumer math, I occasionally try to solve other types of problems. My latest attempt was to estimate the speed of an F-18 buzzing an aircraft carrier. I wasn't sure where to start. I looked up the average length of a carrier, the Mach speed at sea level and estimated the distance traveled in one second. I figured the F-18 had to be going near the speed of sound because mist was forming at the leading edge of the wing (Compressibility). Not bad I guess. I think I came within less than 50 NMPH. Of course you have to have a reason to do this stuff. Mine is to use it with C# and Unity game engine.
well by reading the train route provided by the station and carefully calculating and measuring it by kilometers,
Great info; thanks!
You’re very welcome 😉
Hey ! I've just landed on your channel and it's absolutely brillant. I fully agree with you on the way you should APPropriate ourselves the concepts we learn so that we can re-use it in our daily life. By the way I took a guess on your subway problem, here is my reasoning. Actually I'm French and I live in Paris, so I may not no the London Underground, however I know the Paris one pretty well. We have about 20 different lines that go all across the city. From one point to another I know that it takes me roughly 2h by foot. A person walks 5km per hour on average. So we have 20 lines × 2h × 5 km we got 200km for the Paris underground (which is correct!!). Then I know London is 2 to 3 three time bigger so my guess was a few couple of hundreds of km, and I said 500km. The right answer being a bit less... so I'm quite far and quite close but quite happy to see that I can find a quite reasonably close number without knowing almost nothing about London and with no calculator.!!
Stephane Mallat, one of the founders of wavelet theory along with Abel Prize winner Yves Meter and Jean Morlet, states in his book " A Wavelet tour of signal processing: the sparse view": "The Fourier transform is everywhere in physics and mathematics because it diagonalizes time-invariant convolution operators". Please explain the "intuition" behind this comment given the Lagrange and Poisson had doubts about the "intuition" behind Fourier's original heat conduction work.
We make a train goes from the start of the tunnel to the last point with a defined speed and then multiplying that speed by the time it takes to arrive at the last time , and then we will find the distance.
Great Video! It would be interesting if you can also make similar videos (tips && tricks) for CS, physics, and other STEM fields.
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 :)
I think the average speed can be used to calculate the distance covered by the metro during the full journey to which we add the length of the metro
The question asked about the tunnel, i would try to just get an idea how many station it connects and then try to calculate the total distance that might be the length of tunnel
Great video Samuel! More of this :)
Thank you :)
I have to admit that the exercises in the book from martin liebeck are already advanced. Do you have more recommendations for german math books?
I took advantage and I am taking advantage of doing well with your books thank you
That’s great!!!
I see London as a circle with diameter of 25 miles. I take the center as hotspot which most of the railways crosses. Then I take 8 lines through the center point (like a pizza with 8 pieces) since I would build a railway in a city this way since the density of the traffic is and should be more center focused. Additionally I put 1/4 (7,5 miles) of the diameter in the middle of every piece on the outside since in the outer regions are connected to weak since with further distance from the center railway connection gets separated to far with too big radius. So I come to a result of 8x 25 miles + 8x 5 miles = 240 miles
Great solution!
I would have nailed that interview question 😂 I just saw your instagram video and I was luckily correct even though my guess was very simple. I just estimated that the main part of london is within a 20x20km square and I wanted to calculate the length of the lines of a grid within that square. So I thought a tunnel every 1km would be too close so I went with every 2km. Therefore I summed up 10 vertical tunnels with 20km length with 10 horizontal tunnels with 20km length and thats exactly 400km. I was very surprised when I saw the true answer 😂
Very nice 😊
I don’t know if this is the okay way to do it and I think I’m late for this but personally I would get a map of the London tube. Let’s say a 1:1000 scale. I would then at the center of every track, put a line from the beginning to the end, then measure it up and do the conversion back
There is time where the tube goes right or left and not straight, that’s why I would put the line in the center, in order to get more straight line sas possible. Otherwise I should measure the angles.
Wow this really is another way for looking at math thanks brk
I realised that the resources you mentioned (art of problem solving)were ones I had already come across a while back when I was looking to get a better grasp of Mathematics but they were too expensive. Just one book is like $60 so a whole course would cost hundreds. Bit of a shame but the site has free videos going over the topics which are super useful !
Hmmm yeah, that is kinda annoying. If I had known that people would really care about this video, I would have tried to get some discount code from them. Or found a different resource instead. Are there really no free (pirated) versions anywhere? Not encouraging it, but …
@@SamuelBoschMIT There are pirated versions of the books but not of most of the solution manuals which makes going through them harder.
i dont think people thoroughly understand math, they just do what they know as right. for example, i didnt understand how fractions with different denominators were added, i just did what i knew as right. now i know why it works.