I started my studies in mathematics by taking a group theory course. Yeah, crazy, huh? It was the only class that met late in the day and I was working full time and I could get to the class after work. I loved it immediately. Learning symbolic logic, set theory, all of that, was to me like the most fascinating thing I had ever come across in my life. Difficult, yeah, but a complete joy to think about something for days and days and then suddenly have the Ahah! moment. I still have the Herstein book from that course. I went in to topology and absolutely loved that as well. By this time I was doing directed study because there weren’t a whole,p lot of students interested in taking these courses. And here’s the thing - I find calculus so much harder than abstract algebra! But I force myself to do problems every morning just to keep my brain sharp. (I’m 78).
Hey Math Sorcerer, how about you come up with a proof writing series where you take randomly some problems from those books you just mentioned and break them down in a "explain like I'm five" manner so you can fill that hole on RUclips? It would be so so great especially with you explaining the problems! Think about it and leave a comment on how this sounds. As always keep up the great videos! Love you and your work!
Proofs are often hard to follow and understand, and, of course, difficult to write yourself. But definitely rewarding when you "get it." So satisfying. A+ video sorcerer.
Before I took a class on on writing proofs, I took a class specifically on logic. I feel that proofs benefitted immensely due to that decision. I loved both classes, but where I excelled in writing proofs was in Abstract Algebra and Real Analysis. It was there I learned more than ever how careful organization is an absolute necessity. I took care to make a Word document with all the definitions as they were introduced and a separate doc containing each theorem, corollary, lemma, etc. in their chronological order. Then when presented with a proof, I would copy and paste only those parts of the documents that were relevant to the task. With less clutter, it was easier to organize a plan of proof. And knowing that the odds of everything needed to write a proper proof was right there helped focus. This is how I got past these classes successfully. As a math tutor, this skill has helped me develop more insightful approaches to any given problem students encounter. A good tutor must not only know how to solve a problem: he must be able to apply logic in such a way as-to break it down to elementary terms for most students. Proofs open one’s mind to this type of thinking. I’ll cherish it for as long as I am capable.
I like to think of it this way: solving equations in computational math classes is proof writing with a really narrow scope and tons of supporting assumptions given to you for free. Venturing into proof-based math is learning to widen that scope and remove things given to you for free by building things from scratch yourself.
I love u Math Sorcerer !! Because of you i started realearning Math for !Fun! in my Summerholidays. Right now im at Algebra I-II and I aspire to get to Precalculus in 3-4 weeks. Thank you
I discovered you at just the right time! Thank you, thank you. I'm struggling in graduate level Abstract Algebra because of the proof writing. I downloaded the Hammack book and will be practicing. I must get a handle on proof writing to be successful in my program. Again, thank you!
Honestly for me, my introduction to proofs class was when I fully fell in love with my major. Don't get me wrong, I'm not great at it, but despite the difficulty and even perhaps because of it, I can't see myself doing anything else. That class was kind of like true love's punch in the mouth. Side note: I taught myself Boolean Algebra after learning to write proofs, and it was comically easy for me after being introduced to basic set theory.
I did my PhD at Charles University in Prague, now I am a postdoc. It is interesting that our curriculum was kind of different from the one in USA. In particular, our first year at university everything was already with proofs. The first year involved analysis (1 and 2), linear algebra (1 and 2), discrete mathematics and everything with proofs. So we were really forced to learn how to write proofs in the first year and it was very tough in the beginning, but it paid off. Also during the high school the university organized a distance seminar in mathematics for high school students. The organizers would send out (not only) olympiad-style problems, but we would have roughly one or two months to solve one set of problems, so it was less stressful and one could think about it at home for a long time. This was basically about writing proofs. Also, some problem sets had accompanying text which would introduce a university-level topic (group theory, graph theory, probability, combinatorial geometry, etc).
Proof writing is one of the most difficult yet rewarding things, every problem feels like a rubix cube you have no idea how to solve but after playing around for a while you can get it, Just proved the average of two numbers is between those two numbers or, prove a
I got my Bachelor's in Engineering, and went to the US for my Masters. There my uni basically allowed us to take as many courses from any school as possible a semester so long as we hit 4 courses from the School of Engineering. We had to take 2 courses in a minor. I took a course and then another and then another. Soon I was taking 7 courses a semester, in math and Engineering combined. Real and functional analysis, topology, the works. Brought down my aggregate GPA, but I had an absolute blast. Not to mention, I became known as the weirdo who did 2 Masters degrees in STEM in 2 years and wrote a thesis and survived to tell the tale. Algebraic topology without a shadow of doubt was the hardest course I did. Analysis you basically go backward from conclusion to assumptions. Algebra is pretty straightforward. PDEs and ODEs are fun and very applied. Ditto for numerical stuff. But Hatcher broke me. I never got an intuition about what I was trying to do.
I'm taking discrete mathematics next semester and I'm looking forward to it. My friend told me it's the hardest math class he's taken but he actually really enjoyed it. I know it's entry level proofs, but I wouldn't mind taking more advanced proof based math courses in the future!
"Try to apply that careful thinking to your life situations." Believe it or not, this is good advice. Obviously, don't start throwing QEDs around in conversations with the chef, they're probably not gonna know or care, but I remember trying to employ this careful approach to problem-solving with things outside of math specifically and it was very helpful. The logical thinking process, anyway.
I took a proofs course and we used Hammack's Book of Proof. It's a great book with great explanations and exercises. It was a really challenging course but now I see those skills have been indispensable for all the math I've done since.
“How to prove it” and “A transition to Advanced Mathematics” are my favorite. Even a dumb dumb like me was able to get through them both during undergrad. 15 years later, and they are the few books I made sure to keep when moving from place to place.
Schools in the US usually don't teach proofs until college, except for simple two-column concurrent proofs in high school geometry classes. That is why a lot of people feel overwhelmed when they first encounter proofs in college. I am learning proofs for fun and might teach them to my kids as well.
That's the same in the UK too. I'm a 2nd year math major and I taught myself how to write proofs at 16-17 because i knew i would need to be at least competent in them going foreword. It was crazy to see how many guys in first year had no exposure to basic proofs (direct, contrapositive, induction, etc)..
I learned a few special situations earlier in life. I had an introduction to symbolic logic when I was in 6th grade in elementary school, and I also learned proof by mathematical induction when I was in junior high school. But those are nothing like trying to do a proof in abstract algebra.
I've heard one of the hardest Maths out there are Calculus and Trigonometry. I've had friends who had to take both subjects at once. They're both aspiring to be medical professionals, and so Maths for them is an essential.
@@vicepedro Large parts of abstract algebra and number theory ARE discrete math. Which field is the hardest is definitely subjective, as a computer science student I find discrete stuff much easier than continuous stuff. To me all the theory surrounding the Riemann hypothesis (the hypothesis is kind of a connection between discrete math and analysis) seems to be absurdly complicated
calculus is primarily taught to 16-18 yr olds in the UK. We cover it all in first year of university too, to make sure everyone is at the same level. I am a 2nd year math major and the classes I have had to take this year are abstract algebra, stats 2, linear algebra, vector calculus, real analysis, Riemann integration and lebesgue measure, analytical mechanics. Next year I will be taking group theory (galois theory etc), number theory, commutative algebra, graph theory, financial derivatives, complex analysis, combinatorial optimisation and another module I haven't decided on yet 😛.. But no, calculus (generally) I would not say is particularly tough. It is just a well known math subject to "non-math people".
Dear Math Sorcerer, I love this topic and your positive energy. I am in my mid-sixties and trying to learn some of these things. However, I do want to point out that there is a tremendous, by that I mean order of magnitude price difference between the same book that is sold in USA and third world countries like India. Yes! Mr. Jay Cummings book on proofs costs $ 18.35 + shipping in the USA, but in India it is nearly $68 + shipping. I can understand the higher shipping charge, but four times the price is daylight robbery. However, Mr. Jay Cummings' book on Real Analysis is reasonably priced in both countries. Pricing is such a puzzle!
I've been seeing a lot of comments that talk about getting over the creativity hurdle. Proof-writing really started opening up to my mind when I finally got over the psychological milestone of believing that I did, in fact, have the creative capacity to devise my own solutions to new problems. Receiving my first nonzero score on the Putnam at university was another contributing step, and over the next year I practiced and had gained more confidence. I did end up scoring again the following year, and I hope to score again this year.
I feel so much better after watching this. I’ve never taken an advanced mathematics course, but I have taken introduction to logic. (The proofs made this class feel impossible). After learning how to properly cite (the hardest part about logical proofs in my opinion) I felt pretty accomplished.
I have always been "good" at math, from primary school to secondary school (British system), and I was never worried about not getting a good grade in math. When I entered college, I wanted to do theoretical physics. One semester in, I realized that my thinking is more of a mathematician than a physicist, so I switched major. I spent most of the time writing proofs, honing and honing along the way. End up hurting my GPA because I did not balance the theoretical side of math and the calculational side of math, which is crucial when taking courses. For students, do keep that aspect of doing math in mind also.
I quite agree with everything you have said. Proof writing is really hard to learn (and would also hard to teach). Writing proofs consider a dense amount of knowledge imo. It considers the way one understand the statement, what statements have been proven already, what’s obvious or needs to be shown first. By Van Hiele’s paper, rigor is the highest level of math knowledge. Despite of all of these struggles, it’s very fulfilling to write proofs that are correct and clear, at least for me. Taking a pure math course change the way one thinks and reasons.
I'm going through "Linear Algebra Done Right" and I did not realize it's a book centered around proofs before I bought it. I find it really tough since I don't have background in doing proofs other than Discrete Math and Algorithms/Graph Theory courses. I've never really read a book on proof writing but I find that if I can work through the solutions one-by-one, I can recognize some patterns and solve a few on my own. It's definitely been quite difficult 😅
Thanks..as maths are ever growing materials, we need to compact it again to make easier for human beings to perceive it in a life span... modern computers helped us in bearing the part of the task,(of practice and testing), I call it task, because we have other things to do in Life.. Here we come back to numbers, arithmetics and theory...here I found you talking about proofs...I considere it the first step for maths compaction...thanks again
I firmly believe that none will ever truly master the art of proving stuff. Each time you have to prove some new, non-trivial result you kind of have to invent something. Sometimes chaining some known techniques/results in a clever way is sufficient, sometimes you have to come up with completely new observations and techniques. Proving stuff often requires CREATIVITY
I'm so happy with what you're doing!! I follow you frequently especially in the times I have challenging learning goals to achieve. I'm A software Engineer, Functional programmer, distributed Software architect, but my first passion is Physics that I studied at university before my engineering studies first in electronics and Automation and then moved to Software Engineering field in which I've been working for years now in my US company. At secondary school I was not showing too much interest in mathematics, I was seing them not concrete and not so difficult and It made me not give too much efforts in Mathematics. I was not assisting to Maths classes but at exams I was always above the average. I was and I I'm still very passionated by Physics And was passing all my time studying Physics. My love for mathematics came during my undergraduate Physics studies at university specially when I discover the speculative and abstract part of Quantum Physics, General relativity, particle physics that are parts of Physics deeply rooted in theoretical and abstract concepts but that have deeply impacted the science and the technology and our life. I started giving more efforts to Maths And I as a consequence having great Scores. Actually my next field of studies is going to be Artificial intelligence starting soon in a graduate study program leading to a Phd specially with a lot of mathematics. I'v been following you for months and I've finally decided to also go back to mathematics and study them. I I've already bought on Amazon the book. "Everything You Need to Ace Pre-Algebra and Algebra I in One Big Fat Notebook (Big Fat Notebooks) Kindle Edition" that you recommended recently It's a good book to go back in math. I read a lot, I pass my time studying, I love the struggle of learning difficult things and my brain like complex things and abstract. I should get to learn the proofs and move further. You inspire me so much. I hope I'll may be soon or in the coming months share my experience and my passion with your public. Thanks for what you do, for inspiring us so much! I love what you do and the way you do it with so much dedication and passion.
I once told a teacher in my high school I wanted to study theoretical mathematics at a university, to which she responded with something along the lines of "I had a student, she was interested in mathematics, she went to uni and gave up. You will hate proofs, you will come back in a year and tell me exactly what she had told me." So, me being me, I said to her "Thats exactly why I want to study mathematics, I want to learn about proofs and theoretical, abstract mathematics. In fact I can promise I will come back to you in not one but six years to show my degree." At that moment i felt pride. I was so pride I stuck with my choice and what i love. Right now im selfstudying beginning abstract mathematics and in two months I start my first year of uni. Even if I don't like it, i think abstract mathematics is one of the most beautiful things in the universe and I am proud of myself for at least trying. I wish myself luck and happiness, as well as to other people who want to learn more mathematics. I may do an update in some time Edit: I also forgot to mention that after reading the first few chapters of Book of Proof it changed how i view the world, how i look at arguments of other people, i can more easily identify flaws in logic, and i think thats very beautiful.
It is the most difficult but also the most beautiful and important topic! And do not be afraid of basic topology. The beginning is not hard. The proofs are really straight foreward mostly really short and a little bit talented student can even do them by him/herself. Fun fact: there is a famous topology book without proofs, where ALL proofs are left as an exercise for the reader.
I just finished my first semester of Pure Math. I had a proofs course, which I passed. However, I did not get the results I wanted. I'd been feeling nervous for the continuity of my career, considering I will take Calculus I next semester, and after seeing some of the assignments of previous years' courses I thought, for a brief moment, I might not make it through. But I found your video just in time. Thank you for giving me the confidence boost I needed in order to keep studying proofs during vacation. For the course I used Ethan Bloch's "Proofs and Fundamentals" (great addition to your collection, if you do not already own it); I will continue with the exercises from "Book of Proof".
I am undergraduate student of Mathematics and Physics and every single math subject from first to fourth year has both writing and speaking part. We need first to pass writing part where do problems and next week go to speaking part where we get theory questions and present them in front of teacher. This theory always includes motivation for studying topic, definitions, theorems, lemmas, but everything with proofs that we need to know. It is actualy necessary for passing subject. These proofs are always big deal for students and sometimes are really hard, some of them appears also on writing parts. But I am thankful because we are faced with them and now, in my third year I can figure out many proofs on my own, especialy in topology. I forgot to say, my faculty is part of University of East Sarajevo, Bosnia and Herzegovina.
For me the hardest math that I have encountered is 3 dimensional vector calculus taught to me during my engineering degree...imagining problems in 3 dimensional space and then applying vector calculus to solve them was so challenging...a lot of this was being used for my subject Electromagnetic Fields and Waves...boy was it tough...now imagine moving the same math to higher dimensional problems...one can at least vsiaulize for 3 dimensional space...doing the same for higher dimensions is just something human brains are not cut out for...
I had a good professor for my sets and logic class, which was my introduction to proofs. It was very down-to-earth, I loved the class. Dude had the heir of “I would really like this proof to be an elegant 1 or 2 line- oh doh, it’s gonna take more, *sigh.*” (Maybe not exactly like this). He also liked talking about the history of math, which frankly, really helps make the field more alive. There’s a lot of interesting history, it makes these topics feel more alive. There’s a lot of intro to proofs books, but even an online lecture that’s engaging is really the sort of thing that I’d want to suggest.
I recommend also doing keyboarding to help out solving hard math problems because It really helps out the visual and imagination aspect of writing or solving proofs. I just started implementing this skill and it really helps.
Imagine taking a basic undergraduate course in Geometry... to find out it is about how to build all the geometry from scratch, only with few axioms and no coordinates. Through functions like the metrics and Euclid axioms, reformulated by Hilbert, step by step, what is a point, a line, a plane (sets and sets...), an angle (using points, lines, distances), a triangle, isometries, equivalences,... polygons,... circles... painfully, proof by proof, given to you and asked to you as well. It seems one century ago it was the fashion to use exactly this course to train future Mathematicians in how to do rigorous Mathematics. Half a century later, it went off fashion and many places dropped geometry. They moved this proof training to Advanced Calculus (Analysis 1), and it is still the situation today in many countries (at least in Europe). How I can found a place where they still teach this thing? Well, some places have a very long tradition of not hurrying up because educational trends 😀
If there is any advice I would give my younger self to help improve my proof writing is to memorize and understand the definitions and theorems. You can not write efficient proofs if you don’t understand and remember the definitions and theorems surrounding the topic you are proving.
you saying that you personally feel algebraic topology is way harder than functional analysis is unironically motivating me to get my real analysis and linear algebra in order so that I can finally be in a really good spot to learn functional analysis (maybe I should just keep kreyszig's book open too in the meantime)
Great vid as always and btw nice new beard look! I really loved Richard Hammock Book of Proof which you have recommended in other vids. I printed it myself at work, bound the pages together and studied every single day. Loved it. Another great book on proof writing is Transition to Higher Mathematics by Eggen and Nielsen.
You are right, but write proof is not all the game. Another part of the game is develop good theories and guess or develop important theorems in those theories.
This kind of video would have been helpful back when I was in university. Nobody then had told me that math was about proofs. We didn't even have a proofs class and were expected to either already know it or pick it up on the fly while jumping into abstract algebra and real analysis. No wonder half of the students dropped those classes and the major.
It’s really sad that there are people out there who are willing to take a giant dump on people who have hopes, dreams and goals. I had a teacher do the same thing to me in high school, and it killed my confidence, and I never pursued math as a career, and I really regret that today.
Been collecting math and physics books for over 30 years. Basement is stacked. Since expanding interest to philosophy and literature, had to go the climatized public storage route.
This was what "the New Math" back in the 60's tried to address. There is no reason why the basics of Set Theory and Formal Logic can't be taught in high school--or earlier!
Hello Sorcerer, great video. Also, would like to see more book review videos like the ones you used to do a lot till a month or two ago. That way we get to know about more of the great books in your collection. One book recommendation you made earlier that has proven itself was of the 'Wizard Book' on the Structure and Interpretation of Computer Programs. Great book.
Taking intro to math reasoning(ie intro to proofs) in the fall and real analysis in the spring. I’m both scared and excited! I enjoyed doing the few basic proofs I saw in probability theory and intro linear algebra, but I’ve never taken a class that’s all or mostly proof-based, so I’m kinda anxious.
When you said hardest math, I noticed you were talking about graduate level courses; but I immediately remembered things like the Putnam exam ~ As an undergraduate, what would you need to do to be able to get every single question on the Putnam exam correct? By the time you get yours Masters in Pure Math would you be able to solve 'em all? If you could recommend a collection of books/textbooks etc to a mature undergraduate specifically oriented to scoring well on the Putnam exam, what would that list look like ~ here I'm expecting a set of books where the enthused-student would be able to do every single problem in preparation ~ Assume this hypothetical student has unlimited time to study; and by unlimited lets say at most a decade ~ Thanks for any input! Love your videos!!!!
This video has resonance for me. The problem with proof writing is mathematicians of old thought in sentences, where as modern algebraists (the erstwhile math novitiate) think in terms of WFF's -- well formed formulae -- thank goodness proof writing is becoming more like computer programming and less like writing a medieval dissertation.
I have directed my own studies from high school onward through Calculus, ODE's, Linear Algebra, and Real Analysis. No proofs yet. Proofs next, following this.
I wish I had understood this when I was in school. I was a CS major so I took a lot of math classes. I considered a minor or double major in math but the proof heavy content courses were not attractive. I wish there had been a course dedicated to how to do proofs. Closest I found was a symbolic logic course in the philosophy department of all places. It was an excellent foundational course.
I’m sorry to ask this question, but I watched a video you posted two years ago and one of the two books you recommended people start with who want to learn math from Zero was “An Introduction to Abstract Mathematics.” There you say this is a great way to start (proof writing). Now you’re saying it is the hardest math to learn. I bought the book and it arrived today, and I’m already intimidated by the idea about relearning math (even if I was decent at it back in the day it’s been 15 years). I’m confused as to why if it is the “most difficult” type of math, why do you start someone off wanting to learn math with the absolute hardest?
Never learned to write proofs. I couldn't even memorize basic proofs. Had to miss a couple of questions in my high school because I couldn't memorize them. I have done algebra, calculus, complex numbers and other branches of math and passed my exams, but never came close to this proving stuff.
I was suggested symbolic logic by my counselor and her aides since I wasn't really struggling with computer programming, it's just they assumed I was, it was just a tough grading teacher. It's a bit more comprehensive than what is needed to learn the logic taught in mathematical logic, but hey it helps, just as much as learning greek can be to get an indepth understanding of the symbols.
I actually really enjoy proof writing and I wish I was better at it. I got a good taste for proof writing when I took a class on complexity theory during my CS masters.
Yeah I'll be honest, get your numbers out of the way. Jk, I know there's some quantifying that needs to be done. There's a certain set of words and rules of writing proofs that teachers want to see, and sometimes you fall short of that complete proof cause you reused certain words too much or something didn't make sense to the teacher. I'm so glad I don't have to take a course in Real Analysis and am free to make mistakes I guess, cause I got a C in introduction to proofs. Makes me glad I did my grad in Economics, while you do see set theory used in things like Microeconomics, when you take tests, you're not expected to write complete proofs or write out sets depending on the problem, generally, you just have to understand how to derive the formulas, part of which is in functional form like U(x,y) = (x^p + y^p)^p that has to include the Utility function which is indirectly differentiable itself if the utility is the constraint when taking the Lagrange to 0. Which is just complicated algebra as we do in optimization problems. Same thing with econometrics, but in the homework we are asked how key equations we use is derived given so and so equation so we can understand where it comes from. Which is not the same as writing a complete proof. Deriving is only part of that equation to writing proofs.
Makes me less confident having taken many classes....I now question everything and try to deduce everything logically before I jump to conclusions.....always using deductive reasoning and it drives my wife crazy....she always says, you have a reason for everything.....I literally don't know how else to do life anymore!
I am a slow learner. Math is hard. Here's the hardest subject.... the one you aren't trained in or practiced in or are supplied with tools for. With practice and time, you gain all those things. So I left off one part in another video comment... is the hesitation that comes from painful results of doing bad at first and from fear of failure. Not just laziness, but from a painful experience. Time to do a come-back round like you used to see in 1980's martial arts movies. Where the underdog gets training to strengthen their weak points and learn those basics. The hardest course, is always the one you are least familiar with in techniques. Don't let doubt keep you from getting back in there and training.
man i'm doing an electric engineer undergrad and my calculus from 1 through multivariate have been more on the real analysis side because of an insane profesor, but goddamn i'm fucked by hating and sucking at the computation stuff i wanna say i have an incredible knack for proofing because i'm very logical but more on the declarative side of things and not so much algebraically where you need to be very organized and learn all the implicit algorithms to solve a problem it's been a weird 2 semesters and i kinda get the feeling i'd have a much better time with a math degree but i also just want to do robotics not academia
I feel like I i got really lucky in college with the professors I learned to write proofs from. I still struggled, and so did everyone else, but it never felt like the hurdle it seems to be for some folks.
The hardest math is Math word problems basic to advanced Or complex math word problems with real life practical application using all operations symbols or signs at once . Plus proof writing mathematical logic ! That's true too ! Sometimes or according to the statistics people had a hard time with math problems numbers and words like solving a riddle A key to advancing world whatever are disciplines are in science and mathematical fields. Myself Had a hard with math word problems , thats why I like math in a applied way, and still working at it, strive better at it on my own pace great comment what's hardest math Thank you
I don't think the transitions to advanced mathematics in the video is the same one linked to amazon. The book on amazon is by Eggen, Smith and St. Andre, and the book shown in the video is by Chartrand, Polimeni, and Zhang.
I still struggle just comprehending the proofs in the books I read that contain them. As a programmer, I should be used to following and applying logic and tracking what a variable refers to, but I think math proofs are written in a deliberately very obscure way. No wonder people find it the most difficult part of mathematics.
I think that quote by von Neumann somehow often misunderstood even among mathematicians. It's actually quite hopeful quote instead of a hopeless cynical one.
Hello Math Sorcerer! Sorry for an unrelated (albeit not completely as proofs are integral to the following) comment, but have you had the chance to look at Michael Sipser's "Introduction to the Theory of Computation"? I think it would be a worthwhile addition to your CS books list.
I started my studies in mathematics by taking a group theory course. Yeah, crazy, huh? It was the only class that met late in the day and I was working full time and I could get to the class after work. I loved it immediately. Learning symbolic logic, set theory, all of that, was to me like the most fascinating thing I had ever come across in my life. Difficult, yeah, but a complete joy to think about something for days and days and then suddenly have the Ahah! moment. I still have the Herstein book from that course. I went in to topology and absolutely loved that as well. By this time I was doing directed study because there weren’t a whole,p lot of students interested in taking these courses. And here’s the thing - I find calculus so much harder than abstract algebra! But I force myself to do problems every morning just to keep my brain sharp. (I’m 78).
you are AWESOME
Is it possible for me to be this cool?!?
@@artophile7777Definitely, you just gotta BELIEVE...
@@artophile7777and study.
Awesome
in my student life, the hardest math actually had hidden hints which led to easy solutions. just that hints were hidden.
Hi thats arcane difficulty right?
The hardest part for me was becoming comfortable with using the freedom and creativity necessary for writing your own proofs.
Hey Math Sorcerer, how about you come up with a proof writing series where you take randomly some problems from those books you just mentioned and break them down in a "explain like I'm five" manner so you can fill that hole on RUclips? It would be so so great especially with you explaining the problems! Think about it and leave a comment on how this sounds. As always keep up the great videos! Love you and your work!
Yeah I should make more of those. I have a few playlists with these 🔥
This!!^^^
@@TheMathSorcereryou already have those? Which playlist you mean?
That would be awesome!!
Where do I sign up?
What is the prerequisite? How far down your math journey do you have to be for proofs?
Basic math? Pre-algebra? Algebra? Calculus?
Proofs are often hard to follow and understand, and, of course, difficult to write yourself. But definitely rewarding when you "get it." So satisfying. A+ video sorcerer.
Absolutely!
Before I took a class on on writing proofs, I took a class specifically on logic. I feel that proofs benefitted immensely due to that decision.
I loved both classes, but where I excelled in writing proofs was in Abstract Algebra and Real Analysis. It was there I learned more than ever how careful organization is an absolute necessity.
I took care to make a Word document with all the definitions as they were introduced and a separate doc containing each theorem, corollary, lemma, etc. in their chronological order.
Then when presented with a proof, I would copy and paste only those parts of the documents that were relevant to the task. With less clutter, it was easier to organize a plan of proof. And knowing that the odds of everything needed to write a proper proof was right there helped focus. This is how I got past these classes successfully.
As a math tutor, this skill has helped me develop more insightful approaches to any given problem students encounter. A good tutor must not only know how to solve a problem: he must be able to apply logic in such a way as-to break it down to elementary terms for most students.
Proofs open one’s mind to this type of thinking. I’ll cherish it for as long as I am capable.
I like your approach and organization of the subject.
Honestly I need to thank you. Firstly, I’m just 15 and I’ve always loved math. Your book recommendations are just superb!
I like to think of it this way: solving equations in computational math classes is proof writing with a really narrow scope and tons of supporting assumptions given to you for free. Venturing into proof-based math is learning to widen that scope and remove things given to you for free by building things from scratch yourself.
Kind of like making a game in Unity or Unreal engine vs making the game engine yourself.
I love u Math Sorcerer !! Because of you i started realearning Math for !Fun! in my Summerholidays. Right now im at Algebra I-II and I aspire to get to Precalculus in 3-4 weeks. Thank you
That is awesome!
I discovered you at just the right time! Thank you, thank you. I'm struggling in graduate level Abstract Algebra because of the proof writing. I downloaded the Hammack book and will be practicing. I must get a handle on proof writing to be successful in my program. Again, thank you!
Honestly for me, my introduction to proofs class was when I fully fell in love with my major. Don't get me wrong, I'm not great at it, but despite the difficulty and even perhaps because of it, I can't see myself doing anything else. That class was kind of like true love's punch in the mouth. Side note: I taught myself Boolean Algebra after learning to write proofs, and it was comically easy for me after being introduced to basic set theory.
This is so true. I did terrible in intro to proofs and barely understood it but I still love it.
I did my PhD at Charles University in Prague, now I am a postdoc. It is interesting that our curriculum was kind of different from the one in USA. In particular, our first year at university everything was already with proofs. The first year involved analysis (1 and 2), linear algebra (1 and 2), discrete mathematics and everything with proofs. So we were really forced to learn how to write proofs in the first year and it was very tough in the beginning, but it paid off. Also during the high school the university organized a distance seminar in mathematics for high school students. The organizers would send out (not only) olympiad-style problems, but we would have roughly one or two months to solve one set of problems, so it was less stressful and one could think about it at home for a long time. This was basically about writing proofs. Also, some problem sets had accompanying text which would introduce a university-level topic (group theory, graph theory, probability, combinatorial geometry, etc).
Oh wow so cool, thanks for sharing 🔥
Thank you so much for the Book of Proof recommendation; just downloaded it!. And great video as well 👍
Proof writing is one of the most difficult yet rewarding things, every problem feels like a rubix cube you have no idea how to solve but after playing around for a while you can get it,
Just proved the average of two numbers is between those two numbers or, prove a
I got my Bachelor's in Engineering, and went to the US for my Masters. There my uni basically allowed us to take as many courses from any school as possible a semester so long as we hit 4 courses from the School of Engineering. We had to take 2 courses in a minor. I took a course and then another and then another. Soon I was taking 7 courses a semester, in math and Engineering combined. Real and functional analysis, topology, the works. Brought down my aggregate GPA, but I had an absolute blast. Not to mention, I became known as the weirdo who did 2 Masters degrees in STEM in 2 years and wrote a thesis and survived to tell the tale. Algebraic topology without a shadow of doubt was the hardest course I did. Analysis you basically go backward from conclusion to assumptions. Algebra is pretty straightforward. PDEs and ODEs are fun and very applied. Ditto for numerical stuff. But Hatcher broke me. I never got an intuition about what I was trying to do.
I'm taking discrete mathematics next semester and I'm looking forward to it. My friend told me it's the hardest math class he's taken but he actually really enjoyed it. I know it's entry level proofs, but I wouldn't mind taking more advanced proof based math courses in the future!
"Try to apply that careful thinking to your life situations."
Believe it or not, this is good advice. Obviously, don't start throwing QEDs around in conversations with the chef, they're probably not gonna know or care, but I remember trying to employ this careful approach to problem-solving with things outside of math specifically and it was very helpful. The logical thinking process, anyway.
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I took a proofs course and we used Hammack's Book of Proof. It's a great book with great explanations and exercises. It was a really challenging course but now I see those skills have been indispensable for all the math I've done since.
One of my all time favorites
In computer science classes, before algorithm analysis i took a class on proof writing and it helped so much!
“How to prove it” and “A transition to Advanced Mathematics” are my favorite. Even a dumb dumb like me was able to get through them both during undergrad. 15 years later, and they are the few books I made sure to keep when moving from place to place.
Schools in the US usually don't teach proofs until college, except for simple two-column concurrent proofs in high school geometry classes. That is why a lot of people feel overwhelmed when they first encounter proofs in college. I am learning proofs for fun and might teach them to my kids as well.
That's the same in the UK too. I'm a 2nd year math major and I taught myself how to write proofs at 16-17 because i knew i would need to be at least competent in them going foreword. It was crazy to see how many guys in first year had no exposure to basic proofs (direct, contrapositive, induction, etc)..
I learned a few special situations earlier in life. I had an introduction to symbolic logic when I was in 6th grade in elementary school, and I also learned proof by mathematical induction when I was in junior high school. But those are nothing like trying to do a proof in abstract algebra.
I've heard one of the hardest Maths out there are Calculus and Trigonometry. I've had friends who had to take both subjects at once. They're both aspiring to be medical professionals, and so Maths for them is an essential.
Nah bro calculus isn’t even near what the hardest math offers. A tiny example is abstract algebra, real, complex and convex analysis, etc…
Medical professionals dont need maths at all. I am one and maths is my hobby but most physicians need little more than arithmetic
@@John-ru4gz not even near, check out abstract algebra, all the analysis number theory…
@@vicepedro Large parts of abstract algebra and number theory ARE discrete math. Which field is the hardest is definitely subjective, as a computer science student I find discrete stuff much easier than continuous stuff. To me all the theory surrounding the Riemann hypothesis (the hypothesis is kind of a connection between discrete math and analysis) seems to be absurdly complicated
calculus is primarily taught to 16-18 yr olds in the UK. We cover it all in first year of university too, to make sure everyone is at the same level.
I am a 2nd year math major and the classes I have had to take this year are abstract algebra, stats 2, linear algebra, vector calculus, real analysis, Riemann integration and lebesgue measure, analytical mechanics.
Next year I will be taking group theory (galois theory etc), number theory, commutative algebra, graph theory, financial derivatives, complex analysis, combinatorial optimisation and another module I haven't decided on yet 😛..
But no, calculus (generally) I would not say is particularly tough. It is just a well known math subject to "non-math people".
Dear Math Sorcerer, I love this topic and your positive energy. I am in my mid-sixties and trying to learn some of these things. However, I do want to point out that there is a tremendous, by that I mean order of magnitude price difference between the same book that is sold in USA and third world countries like India. Yes! Mr. Jay Cummings book on proofs costs $ 18.35 + shipping in the USA, but in India it is nearly $68 + shipping. I can understand the higher shipping charge, but four times the price is daylight robbery. However, Mr. Jay Cummings' book on Real Analysis is reasonably priced in both countries. Pricing is such a puzzle!
I've been seeing a lot of comments that talk about getting over the creativity hurdle. Proof-writing really started opening up to my mind when I finally got over the psychological milestone of believing that I did, in fact, have the creative capacity to devise my own solutions to new problems. Receiving my first nonzero score on the Putnam at university was another contributing step, and over the next year I practiced and had gained more confidence. I did end up scoring again the following year, and I hope to score again this year.
Fantastic content as always, mate! Grounded, realistic, and STILL inspiring.
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I feel so much better after watching this. I’ve never taken an advanced mathematics course, but I have taken introduction to logic. (The proofs made this class feel impossible). After learning how to properly cite (the hardest part about logical proofs in my opinion) I felt pretty accomplished.
I have always been "good" at math, from primary school to secondary school (British system), and I was never worried about not getting a good grade in math. When I entered college, I wanted to do theoretical physics. One semester in, I realized that my thinking is more of a mathematician than a physicist, so I switched major. I spent most of the time writing proofs, honing and honing along the way. End up hurting my GPA because I did not balance the theoretical side of math and the calculational side of math, which is crucial when taking courses. For students, do keep that aspect of doing math in mind also.
I quite agree with everything you have said. Proof writing is really hard to learn (and would also hard to teach). Writing proofs consider a dense amount of knowledge imo. It considers the way one understand the statement, what statements have been proven already, what’s obvious or needs to be shown first. By Van Hiele’s paper, rigor is the highest level of math knowledge. Despite of all of these struggles, it’s very fulfilling to write proofs that are correct and clear, at least for me. Taking a pure math course change the way one thinks and reasons.
I'm going through "Linear Algebra Done Right" and I did not realize it's a book centered around proofs before I bought it. I find it really tough since I don't have background in doing proofs other than Discrete Math and Algorithms/Graph Theory courses. I've never really read a book on proof writing but I find that if I can work through the solutions one-by-one, I can recognize some patterns and solve a few on my own. It's definitely been quite difficult 😅
Thanks..as maths are ever growing materials, we need to compact it again to make easier for human beings to perceive it in a life span... modern computers helped us in bearing the part of the task,(of practice and testing), I call it task, because we have other things to do in Life..
Here we come back to numbers, arithmetics and theory...here I found you talking about proofs...I considere it the first step for maths compaction...thanks again
Great advice! You can accomplish so much more in your mathematical journey once you master proof writing. Thank you Math Sorcerer.
You're very welcome!
I firmly believe that none will ever truly master the art of proving stuff. Each time you have to prove some new, non-trivial result you kind of have to invent something. Sometimes chaining some known techniques/results in a clever way is sufficient, sometimes you have to come up with completely new observations and techniques. Proving stuff often requires CREATIVITY
I'm so happy with what you're doing!! I follow you frequently especially in the times I have challenging learning goals to achieve. I'm A software Engineer, Functional programmer, distributed Software architect, but my first passion is Physics that I studied at university before my engineering studies first in electronics and Automation and then moved to Software Engineering field in which I've been working for years now in my US company. At secondary school I was not showing too much interest in mathematics, I was seing them not concrete and not so difficult and It made me not give too much efforts in Mathematics. I was not assisting to Maths classes but at exams I was always above the average. I was and I I'm still very passionated by Physics And was passing all my time studying Physics. My love for mathematics came during my undergraduate Physics studies at university specially when I discover the speculative and abstract part of Quantum Physics, General relativity, particle physics that are parts of Physics deeply rooted in theoretical and abstract concepts but that have deeply impacted the science and the technology and our life. I started giving more efforts to Maths And I as a consequence having great Scores. Actually my next field of studies is going to be Artificial intelligence starting soon in a graduate study program leading to a Phd specially with a lot of mathematics. I'v been following you for months and I've finally decided to also go back to mathematics and study them. I I've already bought on Amazon the book. "Everything You Need to Ace Pre-Algebra and Algebra I in One Big Fat Notebook (Big Fat Notebooks) Kindle Edition" that you recommended recently It's a good book to go back in math. I read a lot, I pass my time studying, I love the struggle of learning difficult things and my brain like complex things and abstract. I should get to learn the proofs and move further. You inspire me so much. I hope I'll may be soon or in the coming months share my experience and my passion with your public. Thanks for what you do, for inspiring us so much! I love what you do and the way you do it with so much dedication and passion.
I once told a teacher in my high school I wanted to study theoretical mathematics at a university, to which she responded with something along the lines of "I had a student, she was interested in mathematics, she went to uni and gave up. You will hate proofs, you will come back in a year and tell me exactly what she had told me." So, me being me, I said to her "Thats exactly why I want to study mathematics, I want to learn about proofs and theoretical, abstract mathematics. In fact I can promise I will come back to you in not one but six years to show my degree." At that moment i felt pride. I was so pride I stuck with my choice and what i love. Right now im selfstudying beginning abstract mathematics and in two months I start my first year of uni. Even if I don't like it, i think abstract mathematics is one of the most beautiful things in the universe and I am proud of myself for at least trying. I wish myself luck and happiness, as well as to other people who want to learn more mathematics. I may do an update in some time
Edit: I also forgot to mention that after reading the first few chapters of Book of Proof it changed how i view the world, how i look at arguments of other people, i can more easily identify flaws in logic, and i think thats very beautiful.
It is the most difficult but also the most beautiful and important topic!
And do not be afraid of basic topology. The beginning is not hard. The proofs are really straight foreward mostly really short and a little bit talented student can even do them by him/herself. Fun fact: there is a famous topology book without proofs, where ALL proofs are left as an exercise for the reader.
Nah, so far, listening to negative reviews has never been a mistake for me. Helped a lot.
Manifolds, Abstract Algebra, Analysis.
I just finished my first semester of Pure Math. I had a proofs course, which I passed. However, I did not get the results I wanted. I'd been feeling nervous for the continuity of my career, considering I will take Calculus I next semester, and after seeing some of the assignments of previous years' courses I thought, for a brief moment, I might not make it through. But I found your video just in time. Thank you for giving me the confidence boost I needed in order to keep studying proofs during vacation. For the course I used Ethan Bloch's "Proofs and Fundamentals" (great addition to your collection, if you do not already own it); I will continue with the exercises from "Book of Proof".
Thanks a lot for making such a helpful content
I am undergraduate student of Mathematics and Physics and every single math subject from first to fourth year has both writing and speaking part. We need first to pass writing part where do problems and next week go to speaking part where we get theory questions and present them in front of teacher. This theory always includes motivation for studying topic, definitions, theorems, lemmas, but everything with proofs that we need to know. It is actualy necessary for passing subject. These proofs are always big deal for students and sometimes are really hard, some of them appears also on writing parts. But I am thankful because we are faced with them and now, in my third year I can figure out many proofs on my own, especialy in topology. I forgot to say, my faculty is part of University of East Sarajevo, Bosnia and Herzegovina.
For me the hardest math that I have encountered is 3 dimensional vector calculus taught to me during my engineering degree...imagining problems in 3 dimensional space and then applying vector calculus to solve them was so challenging...a lot of this was being used for my subject Electromagnetic Fields and Waves...boy was it tough...now imagine moving the same math to higher dimensional problems...one can at least vsiaulize for 3 dimensional space...doing the same for higher dimensions is just something human brains are not cut out for...
For me the hardest course I took was called hyperbolic geometry. The content was fantastic but the assignments were hell on earth.
BTW completely proof based assignments and no exam
After that probably harmonic analysis
I had a good professor for my sets and logic class, which was my introduction to proofs.
It was very down-to-earth, I loved the class. Dude had the heir of “I would really like this proof to be an elegant 1 or 2 line- oh doh, it’s gonna take more, *sigh.*”
(Maybe not exactly like this).
He also liked talking about the history of math, which frankly, really helps make the field more alive. There’s a lot of interesting history, it makes these topics feel more alive.
There’s a lot of intro to proofs books, but even an online lecture that’s engaging is really the sort of thing that I’d want to suggest.
I recommend also doing keyboarding to help out solving hard math problems because It really helps out the visual and imagination aspect of writing or solving proofs. I just started implementing this skill and it really helps.
Imagine taking a basic undergraduate course in Geometry... to find out it is about how to build all the geometry from scratch, only with few axioms and no coordinates. Through functions like the metrics and Euclid axioms, reformulated by Hilbert, step by step, what is a point, a line, a plane (sets and sets...), an angle (using points, lines, distances), a triangle, isometries, equivalences,... polygons,... circles... painfully, proof by proof, given to you and asked to you as well. It seems one century ago it was the fashion to use exactly this course to train future Mathematicians in how to do rigorous Mathematics. Half a century later, it went off fashion and many places dropped geometry. They moved this proof training to Advanced Calculus (Analysis 1), and it is still the situation today in many countries (at least in Europe). How I can found a place where they still teach this thing? Well, some places have a very long tradition of not hurrying up because educational trends 😀
If there is any advice I would give my younger self to help improve my proof writing is to memorize and understand the definitions and theorems. You can not write efficient proofs if you don’t understand and remember the definitions and theorems surrounding the topic you are proving.
Hated school level maths (liked geometry though). Now doing CS theory full time. Proofs can be hard, but this is the only way I can do maths.
you saying that you personally feel algebraic topology is way harder than functional analysis is unironically motivating me to get my real analysis and linear algebra in order so that I can finally be in a really good spot to learn functional analysis (maybe I should just keep kreyszig's book open too in the meantime)
Always enjoy your vids. Thanks. Cool beard, math dude.
Thank you Sir! Love those kind of videos! I'd love to be your student....
Great vid as always and btw nice new beard look! I really loved Richard Hammock Book of Proof which you have recommended in other vids. I printed it myself at work, bound the pages together and studied every single day. Loved it. Another great book on proof writing is Transition to Higher Mathematics by Eggen and Nielsen.
I always liked proofs even when I didn't fully understand them. That's where all the good stuff is at in mathematics.
I'm from Vietnam, love your positive energy ❤.
You are right, but write proof is not all the game. Another part of the game is develop good theories and guess or develop important theorems in those theories.
I raised my daughter not to listen to other peoples negative comments.
This kind of video would have been helpful back when I was in university. Nobody then had told me that math was about proofs. We didn't even have a proofs class and were expected to either already know it or pick it up on the fly while jumping into abstract algebra and real analysis. No wonder half of the students dropped those classes and the major.
It’s really sad that there are people out there who are willing to take a giant dump on people who have hopes, dreams and goals. I had a teacher do the same thing to me in high school, and it killed my confidence, and I never pursued math as a career, and I really regret that today.
Been collecting math and physics books for over 30 years. Basement is stacked. Since expanding interest to philosophy and literature, had to go the climatized public storage route.
Hey Math Sorcerer, can you talk a bit about your CS career?
Love your content!!!
This was what "the New Math" back in the 60's tried to address. There is no reason why the basics of Set Theory and Formal Logic can't be taught in high school--or earlier!
Probably a scarcity of qualified teachers?
@@davidsabbagh6815 And/or a surplus of administrators
Completely agree. It was crazy to me the amount of people in first year of my math degree who had no prior exposure to proof writing.
I made sure to get the book of proof and the long form proof book! :D
Hello Sorcerer, great video. Also, would like to see more book review videos like the ones you used to do a lot till a month or two ago. That way we get to know about more of the great books in your collection. One book recommendation you made earlier that has proven itself was of the 'Wizard Book' on the Structure and Interpretation of Computer Programs. Great book.
Thank you for the valuable advices
Taking intro to math reasoning(ie intro to proofs) in the fall and real analysis in the spring. I’m both scared and excited! I enjoyed doing the few basic proofs I saw in probability theory and intro linear algebra, but I’ve never taken a class that’s all or mostly proof-based, so I’m kinda anxious.
Sorry to hear about your old friend. May he rest in peace
When you said hardest math, I noticed you were talking about graduate level courses; but I immediately remembered things like the Putnam exam ~ As an undergraduate, what would you need to do to be able to get every single question on the Putnam exam correct? By the time you get yours Masters in Pure Math would you be able to solve 'em all?
If you could recommend a collection of books/textbooks etc to a mature undergraduate specifically oriented to scoring well on the Putnam exam, what would that list look like ~ here I'm expecting a set of books where the enthused-student would be able to do every single problem in preparation ~ Assume this hypothetical student has unlimited time to study; and by unlimited lets say at most a decade ~ Thanks for any input! Love your videos!!!!
I think a course of book in abstract algebra often is a good introduction to abstract algebra
As soon as I clicked the link I thought Two words: Algebraic Topology- and Bingo great minds think alike
This video has resonance for me. The problem with proof writing is mathematicians of old thought in sentences, where as modern algebraists (the erstwhile math novitiate) think in terms of WFF's -- well formed formulae -- thank goodness proof writing is becoming more like computer programming and less like writing a medieval dissertation.
nice video. make me miss doing small proofs, algebra, functional analysis and set theory
I have directed my own studies from high school onward through Calculus, ODE's, Linear Algebra, and Real Analysis. No proofs yet. Proofs next, following this.
The Cummings books are great . . . except for when it comes to answers and hints: there are barely any, which can be frustrating for self-study.
I wish I had understood this when I was in school. I was a CS major so I took a lot of math classes. I considered a minor or double major in math but the proof heavy content courses were not attractive. I wish there had been a course dedicated to how to do proofs. Closest I found was a symbolic logic course in the philosophy department of all places. It was an excellent foundational course.
Books: 6:00
Great Neumann quote🙏
I am not good at grasping pure math proofs while I fare well in applied math. I think I am miles away from being actually good at math.
I’m sorry to ask this question, but I watched a video you posted two years ago and one of the two books you recommended people start with who want to learn math from Zero was “An Introduction to Abstract Mathematics.” There you say this is a great way to start (proof writing). Now you’re saying it is the hardest math to learn. I bought the book and it arrived today, and I’m already intimidated by the idea about relearning math (even if I was decent at it back in the day it’s been 15 years). I’m confused as to why if it is the “most difficult” type of math, why do you start someone off wanting to learn math with the absolute hardest?
Never learned to write proofs. I couldn't even memorize basic proofs. Had to miss a couple of questions in my high school because I couldn't memorize them. I have done algebra, calculus, complex numbers and other branches of math and passed my exams, but never came close to this proving stuff.
I was suggested symbolic logic by my counselor and her aides since I wasn't really struggling with computer programming, it's just they assumed I was, it was just a tough grading teacher. It's a bit more comprehensive than what is needed to learn the logic taught in mathematical logic, but hey it helps, just as much as learning greek can be to get an indepth understanding of the symbols.
This quote is force me to take seriously my math practice habits
I actually really enjoy proof writing and I wish I was better at it. I got a good taste for proof writing when I took a class on complexity theory during my CS masters.
Yeah I'll be honest, get your numbers out of the way. Jk, I know there's some quantifying that needs to be done. There's a certain set of words and rules of writing proofs that teachers want to see, and sometimes you fall short of that complete proof cause you reused certain words too much or something didn't make sense to the teacher. I'm so glad I don't have to take a course in Real Analysis and am free to make mistakes I guess, cause I got a C in introduction to proofs. Makes me glad I did my grad in Economics, while you do see set theory used in things like Microeconomics, when you take tests, you're not expected to write complete proofs or write out sets depending on the problem, generally, you just have to understand how to derive the formulas, part of which is in functional form like U(x,y) = (x^p + y^p)^p that has to include the Utility function which is indirectly differentiable itself if the utility is the constraint when taking the Lagrange to 0. Which is just complicated algebra as we do in optimization problems. Same thing with econometrics, but in the homework we are asked how key equations we use is derived given so and so equation so we can understand where it comes from. Which is not the same as writing a complete proof. Deriving is only part of that equation to writing proofs.
I don't do pure math but I love this channel
Thanks for the pep talk!!! It helped motivate me. So what are the prerequisites I should have mastered before trying to learn proofs??
Makes me less confident having taken many classes....I now question everything and try to deduce everything logically before I jump to conclusions.....always using deductive reasoning and it drives my wife crazy....she always says, you have a reason for everything.....I literally don't know how else to do life anymore!
I am a slow learner. Math is hard. Here's the hardest subject....
the one you aren't trained in or practiced in or are supplied with tools for. With practice and time, you gain all those things. So I left off one part in another video comment... is the hesitation that comes from painful results of doing bad at first and from fear of failure. Not just laziness, but from a painful experience.
Time to do a come-back round like you used to see in 1980's martial arts movies. Where the underdog gets training to strengthen their weak points and learn those basics.
The hardest course, is always the one you are least familiar with in techniques.
Don't let doubt keep you from getting back in there and training.
man i'm doing an electric engineer undergrad and my calculus from 1 through multivariate have been more on the real analysis side because of an insane profesor, but goddamn i'm fucked by hating and sucking at the computation stuff
i wanna say i have an incredible knack for proofing because i'm very logical but more on the declarative side of things and not so much algebraically where you need to be very organized and learn all the implicit algorithms to solve a problem
it's been a weird 2 semesters and i kinda get the feeling i'd have a much better time with a math degree but i also just want to do robotics not academia
His Universe that is the cumulative hierarchy is just simply amazing
I feel like I i got really lucky in college with the professors I learned to write proofs from. I still struggled, and so did everyone else, but it never felt like the hurdle it seems to be for some folks.
The hardest math is
Math word problems basic to advanced
Or complex math word problems with real life practical application using all operations symbols or signs at once .
Plus proof writing mathematical logic ! That's true too !
Sometimes or according to the statistics people had a hard time with math problems numbers and words like solving a riddle
A key to advancing world whatever are disciplines are in science and mathematical fields.
Myself Had a hard with math word problems , thats why I like math in a applied way, and still working at it, strive better at it on my own pace
great comment what's hardest math
Thank you
I don't think the transitions to advanced mathematics in the video is the same one linked to amazon. The book on amazon is by Eggen, Smith and St. Andre, and the book shown in the video is by Chartrand, Polimeni, and Zhang.
OMG! I’m friends with Jay Cummings! He teaches at the University that’s about 10 minutes from my home!
I still struggle just comprehending the proofs in the books I read that contain them. As a programmer, I should be used to following and applying logic and tracking what a variable refers to, but I think math proofs are written in a deliberately very obscure way. No wonder people find it the most difficult part of mathematics.
I think that quote by von Neumann somehow often misunderstood even among mathematicians. It's actually quite hopeful quote instead of a hopeless cynical one.
That von Neumann quote is so funny coming from a guy that started entire fields of study just by riffing on other researchers' work.
Hello Math Sorcerer! Sorry for an unrelated (albeit not completely as proofs are integral to the following) comment, but have you had the chance to look at Michael Sipser's "Introduction to the Theory of Computation"? I think it would be a worthwhile addition to your CS books list.
Hi Math Witch! Can you review AOPS(Art of problem solving) books? It would be really interesting.
When in doubt, just claim that the proof is trivial
Beard looks good brother!!! keep it!!!
Hello, my favourite math wizard. How useful would it be for me to learn proof writing in my first year of Electrical Engineering? (In my free time)