The description of how the two fields of Mathematics get from point A to point B is so spot on haha! Especially for Analysis when you have to do an epsilon-delta proof. I always started at point B and tried to work backwards. I remember thinking it kinda felt like I was "cheating" but one professor I had laughed at me when I told him that and said it's just how it's done.
When I was studying aboard in New Zealand, I noticed that my classmates in the equivalent undergrad level were leagues ahead of me in our Analysis Course, but I seemed to have a bit more experience in our Algebra based courses. Talking to the students, grads, and professors there, I got the feeling that topics in analysis are explored and ingrained in a lot more in lower education than in the US while the US has a higher emphasis on the algebra side of things. I also noticed that the recommended order of courses for math is different as well. So I think there's some merit to that observation.
My proofs professor always said that Analysis proofs move in a U-shape, because you have to start at both sides, meet in the middle, and then write the real proof. Basically another graphical approach to your description of analysis proofs :)
I have been really struggling in my first analysis class. I feel like this perspective will be useful for me going forward, cannot thank you enough! loving these videos
I see algebra as the study of composition. It gives a framework for taking raw sets and giving us one or more ways to compose elements from that set to get another element from that same set. The ruleset for those compositions, both independently and how they interact, defines the algebraic structure. By studying all the ways elements of sets can compose, we end up with a very general theory that covers things like symmetry and patterns, as well as the mathematics objects we're typically familiar with beforehand, like numbers, polynomials, and functions. On the other hand, I see analysis as the study of measurement. The two main things we want to measure are change and size, which correspond to the derivative and the integral of elementary calculus. In order to study change, we need some way of "smoothly moving through a set", which is what topology affords us. In order to study size, we need some kind of "meter stick" to measure more general spaces against, which is what a measure space affords us.
Wow how you describe solving analysis problem is so on spot. I never really thought about styles of solving math problems, but this term I'm doing stochastic process, and the way to prove convergence is exactly like what you described, which is hilarious and sad at the same time (cuz I hate the backtracking approach).
As a structural engineer, I am so glad I went straight into the field than get a Phd in math. I know many of my colleagues who did take that path and this only looks insane. When it comes to my field, they really want an MSc. Degree. So, I can see how that happens, but by far the job market was not nearly this complicated to deal with.
Nice stuff. Im finished with my lower level undergraduate level classes I think. Finishing my GE's at CC, can't wait to transfer and get into the "meat". The epsilon-delta proof makes more sense though, pretty cool.
From the examples you gave, I found the Analysis one to be a bit more easier, but I guess the algebra one was hard to follow because I am not familiar with its jargon.
Well,, I'm not pursuing a Ph.D., so I can't answer the question about "What field do you want to get your Ph.D. in?" But I can say that my (primary) interest in maths is in service of my interests in AI and Cognitive Science and to a lesser degree, Physics. I was very inspired by the book "The Universe Speaks in Numbers" when I read it last year, and decided to really "lean in" to re-learning the maths I've forgotten, and then pushing forward into some material that I never studied back in the day (I was a C.S. major). Right now I'm still grinding through approximately the equivalent of what an undergrad math major would study early in their journey. Calc I, Calc II, Calc III, Differential Equations, Discrete Math, Linear Algebra, Statistics, Probability, etc. But I plan to keep pushing forward, using books, RUclips videos, web resources, etc. and probably get into some of the graduate level stuff. What you said about Algebra being all about patterns really resonated with me, as AI is also largely about patterns, albeit possibly at a different level of abstraction. But that brings me to my point - abstraction and the morphisms between objects are different levels of abstraction, are largely where my interests lie. So to the extent that some of this maths might be useful to me from an AI viewpoint, I suppose you can say that I have more interest in Algebra than Analysis. But then again, AI involves many probabilistic elements, which means Probability Theory, which in turn means Measure Theory, which means Analysis. And then I also have reason to be interested in Graph Theory, Combinatorics, and some more esoteric fields like Catastrophe Theory, and Pattern Theory, so... I guess I'm kinda in the mold of "I want to know all of it". :-) In the end, I'll follow Bruce Lee's advice and "be like water" and just flow where the research takes me.
Man, you are really reading my mind. I am math freshman for now, but I'm also into AI and ML and going to study math and CS fields needed for it. Also had exactly same sentence in my mind when he said about patterns in algebra.
Hi, you are very talented in mathematics and im not sure if you know this but is it possible to major mathematics as a premed and still be ready for the MCAT. Keep up the good work!
You can do both. The MCAT takes months to prepare for, and I don’t think it matters too much what your major is. The STEM fields probably will give you a slight advantage on the test.
Yes you can do both. The only problem you should be wary of, is that most medical schools require certain undergrad classes to be taken. If you can fit both those classes and your mathematics major requirements in your schedule, then you’re fine. Talk to your advisor about it. Typically for the mcat, youre studying information you’ve never learned, and its more of a test on your studying ability.
I'm undegrade in my first year, so I took Lineal Algebra 2 and this was a quite challenging class for me, and I like Analysis more, but I like Lineal Algebra because it's challenging. :) Thank you for your great videos.
I think you are really underselling the applicability of algebra. There are many applications, admittedly less than analysis but particle physics, chemistry and computer science are at least three major areas that are full of abstract algebra
I think the stereotype about domestic vs international student holds quite true for Asian international students! I think this is because in Asia (I know in East Asia this is true but I think other Asian countries also have heavy Calculus teaching system), a lot of the university entrance exam is heavy on Calculus but no so much on algebraic thinking so when these students study pure math in Uni, they tend to be more comfortable with Analysis than Algebra. A saw a lot of Asian international students being shocked by the abstractness of first course in Algebra but not so much in first course in Analysis.
This is awesome! For context, I am currently studying Calculus, and have just learned about the Chain Rule(which is surprisingly easy so far and awesome!). So I definitely have quite a ways to go in order to understand a lot of this. That said, I agree with what you've said regarding understanding the terminology(speaking as someone who's only learned up to this point, and could be incorrect). From what I've seen in the past, whether it be an integral, a derivative, a series, or heck even the notation for a set, it all looks like witchcraft until you understand what it's talking about. :) Anyways, I have been struggling with writing Proofs for the longest time. I always suspect that I'm leaving something out, or doing something incorrectly. While I didn't quite understand everything you just wrote down(because I have yet to learn what everything means as you said), it did at least shine a little bit of a light on how to go about it. It also makes way more sense how some solutions for differing problems that I've seen may have appeared(not trying to imply that's the case for all of them). They didn't appear out of thin air, but were worked partway through, only to go the end and work backwards, and use that to finish the proof. Really cool stuff! :) So yea, I suppose this is a long winded comment to say that I'm liking what I'm seeing, and to keep up the good work! :)
@@ClumpypooCP Honestly, it's still too early to say. Perhaps one day I will be, but there are still plenty of other things in life that could happen. For instance: Maybe I will end up becoming a Physicist instead, or maybe I get into GameDev, or maybe there's some other field of study I like that much more and don't even know about yet. Ultimately I have no idea! Whatever I end up doing though, I highly doubt that I will ever cease to use Mathematics. This is quite simply because I love it! :)
Just because there are some important inequalities in Analysis, or some proofs require them, doesn’t mean the whole field is about them. Would strongly disagree.
Algebra has more to do with isomorphisms, canonical or otherwise, and structure preserving maps. Analysis is more approximations with precise bounds. Strict equality in both cases is something that's overcome using various techniques.
@@Kraft2001inequality is important to embed items into a desired space which is at the heart of a loy analysis problems. There are subfields that deals with inequalities for example Convex Geometry.
Im German highschool the maintopic in math ist analysis and a bit vektor algebra and a little bit probability stuff, but some schools even skip all the probability stuff
Measure theory was so painful.... I approached the topic as if I was just writing, almost like ChatGPT. Of course, I struggled a ton and had to retake it. I agree with you, though, I like analysis so much more...
i can give you a prospective about the state of maths in morocco here we have different branches in hs and for the mathematics branch we studied derivatives and integrals as our main focus with limites in Analysis as for algebra we had in our last year of hs Groups/rings/fields though the problems and theorems were more introductory material and also had arithmetic in Zn obviously almost everyone struggled with the material 😅 and graduated after working hard
4:02 I’m only a 2nd year undergrad and I’m not a math major or an English major so forgive me if I’m wrong but I think the word you’re looking for would be abstraction
I think it might be just the matter of how one people tend to approach one problem？But as an international student from China, I think we do have a a lot of proof like the sketch you draw for analysis in our high-school study.
I think its also worth noting that students from the east side of iron curtain (aka communist countries in Europe before the fall of soviet union) were essentially trained to become an analyst from a very young age. I think the soviet union once had one of the best analysis program in the world, more so than their western counterparts. This was during the cold war when the U.S and USSR wanted to assert their dominance: aka space race. So, it makes sense for countries like USSR to allocate more resources into training their best and brightest students to study real analysis which has a direct application in aerospace, physics and numerical analysis etc. On the other hand, countries like France were more invested in Algebraic geometry and other algebra topics (Grothendirk and his colleagues are case in point. They were mostly citizens of western Europe). For real though, its quite interesting how culture and history often shapes the development of mathematics.
nice video. you write well, it was soothing to watch. i graduated in chem E 5 years ago, i would like to go for a PhD in math as a personal pursuit once i've "made it" and invest well
Hey man I really like your videos and the topics are always so Interesting, it’s a good break from all the other low level clean math shown on RUclips. However I really think you could improve your video quality so much by having a small tripod or something instead of the hand held camera. Hope to see more cheers.
For Analysis I think Ultrametricity or non-Archimedean metric spaces (i.e. Q_p the p-adic space etc...) is becoming a big enough part of analysis (literally the other half of the analysis universe) that it is a good mention.
Halfway through a math bachelor in Sweden. I can say that analysis was heavily prioritized over algebra in the education pre-uni, and to some extent also in the course recommendations at uni. Analysis is easier for me probably largely because of that. Although there is a lot to learn in real analysis, I feel definitions and theorems are rarely surprising and often similar to things you've seen before. Similarly with problems. I can quickly get a rough plan, and then it's "just" a matter of execution. In algebra on the other hand i often have no clue why a statement should be true after reading a theorem or a problem. Similarly with definitions. Their immediate consequences come as complete surprises. Makes it harder but also more interesting to me. Feels like there is more to discover.
deadass? I aint even know what Analysis was untill i searched it up lmfao. Im an undergrad at my place. Im doing my first year stuff but from what i see, aint analysis stuff like limits? What stuff in highschool did u learn for analysis? I feel like other countries learn way more in highschool than we do lol. We just learn the basics and rarely calculus unless we take that course specifically lol
About your remark why some outside students might do better in analysis than algebra, I can only speak from personal experience that in eastern / central Europe back in my days as a high school student in the +2000 years we did not studied calculus as it is studied in US, instead it was "approached" much more how you would approach real/complex analysis. P.S: Love your videos...keep going :)).
I think analysis textbooks, and other textbooks in general, should instead of just giving you the proofs and methods and stuff, they should show you a way you could get to them not even show problem solving strategies, but just show the problem being solved in a normal way cus I think that's not an obvious way to solve stuff at all if you're not used to it
"If you can't explain it to a six year old, you don't understand it yourself." I'm starting to think nobody understands graduate-level mathematics. Either that or I need to be six years old.
Yeah, because this statement is only partially true. A little improvement: "If you can't explain it to someone who speaks your language, you don't understand it yourself." 6 year olds and none-math people dont speak the language of mathematics.
The statement bothers me because the vast majority of things aren't simple enough to be understood by someone who wasn't aware of their own existence four years ago. How do you explain algebra to a kid who's just learning about addition? You went through school and college to get to those concepts.
I TA'd for discrete structures once as an undergrad. It is very similar to math proofs but under the context of computer science. ATM I have no video planned but I will consider the topic for future videos :)
Based on your assumptions, Algebra is a single path (abelian) connecting A to B, and potentially a multiple path going from A to B (non-abelian) ,so Jacobi doesn't shrug🤣🤓. With Analysis, a singularity on the real line forces us to make an inductive ansatz to connect A and B.
I would recommend some Analysis first, like Bartle and Sherbert's book. Believe it or not, Elementary Statistics and Probability theory might help you bridge the gap even better, because I believe measure theory was born from probability.
French mathematicians Henri Lebesgue and Emile Borel wanted a better version of integration (cause Riemann one is pretty weak when we're talking about sequence of functions) and so they developped Lebesgue integration which needed to devellop measure theory
Then, it appeared that probability could be rigorously defined as a brench of measure theory via the Kolmogorov's model ( a set + a sigma algebra + a probability measure)
If I have to explain what algebra and analysis are in 10 seconds to someone who doesn’t know: Algebra = Generalised arithmetics Analysis = Rigorous calculus (I know this is super inaccurate but I think it is intuitive😂)
Abduction. This is what one of my instructors called proofs like your analysis proof. It is not deduction, not induction, ergo abduction. You abduct the solution.
My university in Italy is extremely specialized in analysis but it's because of an almost 130 year-old tradition, probably you know that Italy isn't a developing country. In the XX century there were many great analysts, so other good analysts moved to work with them. So in the departament now like 70% of research is analysis focused. Maybe it's the same thing in the US for algebra, or it's because some education choice. Analysis might be a field in which US is relatively bad at, because of some reasons.
Algebra people are good at logic puzzles (and very smug about it) but terrible at visualization (and in total denial about it). The only physics they can handle is Chemistry. xSTx on the MBTI. Analysis people are the exact opposite. xNTx the MBTI.
Analysis is the logical foundations of calculus. Topology is the study of connectedness and continuity. Originally topology started with graph theory but there is significant overlap with analysis. Ideas that are vital in analysis like compactness are fundamentally topological in nature and all metric spaces have a topology induced by the metric. So if you can measure lengths or distances in a meaningful way you have a topology there too. Not all topologies are metrizable though so metric spaces are insufficient to classify topological ones.
Thanks man, I'm about to apply for college and I think I'm either going for an engineering degree or a math degree. I'm quite fond of and comfortable with calculus, linear algebra and math in general as I take advanced maths courses in my high school. But man, I'm still scared af that I wouldn't be smart enough for a math degree. But at least this video gives me a bit of information, so cheers!
In India students master single variable calculus in the high school itself. So your argument in favour of the utility of analysis skills is pretty solid ig.
This is a dumb comment from somebody that knows nothing about anything. Is proving trig identities, like what you do in high school, Analysis? I ask because that was always the area where you’d be told to work forwards and backwards from each instance of the identity
In University of Ljubljana if we studied only this, math would be a joke. In algebra year one we also do Permutations Vector spaces Matrecis Transformations Dot product (not just normal dot product as done when multiplying vectory) And some more stuff I don't know how to spell + ofcourse Monoids halfgrups grups rings fields and algebras. In analysis year 1 we do dedikinds axiom and stuff around that. Sequences, sums, differentiation, integration, reiman integrals, debroux integrals + metrics. Metric spaces or how do you spell it. Sorry for my bad english and not exactly knowing what things are called in English. I don't know if you are from USA, but this is literally a joke what you study.
Are you saying its easy or hard? Im in year 1 rn and we basically do the exact same stuff u do lol. tbf im retarded and still dont know the difference between algebra and analysis, but we basically did everything you listed up there. But I wouldnt say your wrong. I think we here in the states just dont do as much mathy stuff as people from other countries lol. I have no clue tho. Thats just what it seems, cus I was talking to a family friend in Srilanka, and the dude learnt all that shit in fucking HIGHSCHOOL bro. tbf he said it was cancerous and he prolly didnt fully understand it anyways, but still other countries are on a different tier. I think its maybe the mindset cus here we fuck around in highschool and then tryhard thru college, but maybe its different idk.
@@aviberezovskiy7633 that does not matter. If you consider the point itself or not does not change the limit. Since any sequence converging to that point must be considered. So if the function is defined there changes nothing. So your remark is of no use. Except when the function is not defined at that point of course
@@2funky4u88 You are wrong. Read about removable discontinuities. A classical example is f(x)=(x^2-9)/(x-3) for x=/=3 and f(3)=1. Avi Berezovskiy's point is right: when talking about limits we have to exclude the point to which x tends whether or not f is defined in that point. Just check the definition of the limits of a function if you don't believe.
The description of how the two fields of Mathematics get from point A to point B is so spot on haha! Especially for Analysis when you have to do an epsilon-delta proof. I always started at point B and tried to work backwards. I remember thinking it kinda felt like I was "cheating" but one professor I had laughed at me when I told him that and said it's just how it's done.
When I was studying aboard in New Zealand, I noticed that my classmates in the equivalent undergrad level were leagues ahead of me in our Analysis Course, but I seemed to have a bit more experience in our Algebra based courses. Talking to the students, grads, and professors there, I got the feeling that topics in analysis are explored and ingrained in a lot more in lower education than in the US while the US has a higher emphasis on the algebra side of things. I also noticed that the recommended order of courses for math is different as well. So I think there's some merit to that observation.
Enjoying these videos because you're giving us a nice "lay of the land" -- it's making higher math seem more approachable.
Man you are so damn right about the analysis part haha 😆😆😆
Glad to know it's a common thing. Backtracking is one good way to solve analysis problems.
Without question, that was the most accurate description of analysis I've ever heard 😂
My proofs professor always said that Analysis proofs move in a U-shape, because you have to start at both sides, meet in the middle, and then write the real proof. Basically another graphical approach to your description of analysis proofs :)
That is a brilliant way of describing analysis
I have been really struggling in my first analysis class.
I feel like this perspective will be useful for me going forward, cannot thank you enough! loving these videos
I see algebra as the study of composition. It gives a framework for taking raw sets and giving us one or more ways to compose elements from that set to get another element from that same set. The ruleset for those compositions, both independently and how they interact, defines the algebraic structure. By studying all the ways elements of sets can compose, we end up with a very general theory that covers things like symmetry and patterns, as well as the mathematics objects we're typically familiar with beforehand, like numbers, polynomials, and functions.
On the other hand, I see analysis as the study of measurement. The two main things we want to measure are change and size, which correspond to the derivative and the integral of elementary calculus. In order to study change, we need some way of "smoothly moving through a set", which is what topology affords us. In order to study size, we need some kind of "meter stick" to measure more general spaces against, which is what a measure space affords us.
This is dumb lmao. The study of composition is category theory, not algebra
Wow how you describe solving analysis problem is so on spot. I never really thought about styles of solving math problems, but this term I'm doing stochastic process, and the way to prove convergence is exactly like what you described, which is hilarious and sad at the same time (cuz I hate the backtracking approach).
As a structural engineer, I am so glad I went straight into the field than get a Phd in math. I know many of my colleagues who did take that path and this only looks insane.
When it comes to my field, they really want an MSc. Degree. So, I can see how that happens, but by far the job market was not nearly this complicated to deal with.
That Lara Alcock book about analysis one can see in the background is an absolute gem!
Nice stuff. Im finished with my lower level undergraduate level classes I think. Finishing my GE's at CC, can't wait to transfer and get into the "meat". The epsilon-delta proof makes more sense though, pretty cool.
From the examples you gave, I found the Analysis one to be a bit more easier, but I guess the algebra one was hard to follow because I am not familiar with its jargon.
Well,, I'm not pursuing a Ph.D., so I can't answer the question about "What field do you want to get your Ph.D. in?" But I can say that my (primary) interest in maths is in service of my interests in AI and Cognitive Science and to a lesser degree, Physics. I was very inspired by the book "The Universe Speaks in Numbers" when I read it last year, and decided to really "lean in" to re-learning the maths I've forgotten, and then pushing forward into some material that I never studied back in the day (I was a C.S. major).
Right now I'm still grinding through approximately the equivalent of what an undergrad math major would study early in their journey. Calc I, Calc II, Calc III, Differential Equations, Discrete Math, Linear Algebra, Statistics, Probability, etc. But I plan to keep pushing forward, using books, RUclips videos, web resources, etc. and probably get into some of the graduate level stuff.
What you said about Algebra being all about patterns really resonated with me, as AI is also largely about patterns, albeit possibly at a different level of abstraction. But that brings me to my point - abstraction and the morphisms between objects are different levels of abstraction, are largely where my interests lie. So to the extent that some of this maths might be useful to me from an AI viewpoint, I suppose you can say that I have more interest in Algebra than Analysis. But then again, AI involves many probabilistic elements, which means Probability Theory, which in turn means Measure Theory, which means Analysis. And then I also have reason to be interested in Graph Theory, Combinatorics, and some more esoteric fields like Catastrophe Theory, and Pattern Theory, so... I guess I'm kinda in the mold of "I want to know all of it". :-)
In the end, I'll follow Bruce Lee's advice and "be like water" and just flow where the research takes me.
Man, you are really reading my mind. I am math freshman for now, but I'm also into AI and ML and going to study math and CS fields needed for it.
Also had exactly same sentence in my mind when he said about patterns in algebra.
Oh my god that analysis “point A to point B” analogy was so true hahah
Hi, you are very talented in mathematics and im not sure if you know this but is it possible to major mathematics as a premed and still be ready for the MCAT. Keep up the good work!
Check out Elliot Nicholson on RUclips!
You can do both. The MCAT takes months to prepare for, and I don’t think it matters too much what your major is. The STEM fields probably will give you a slight advantage on the test.
Yes you can do both. The only problem you should be wary of, is that most medical schools require certain undergrad classes to be taken. If you can fit both those classes and your mathematics major requirements in your schedule, then you’re fine. Talk to your advisor about it. Typically for the mcat, youre studying information you’ve never learned, and its more of a test on your studying ability.
I'm undegrade in my first year, so I took Lineal Algebra 2 and this was a quite challenging class for me, and I like Analysis more, but I like Lineal Algebra because it's challenging. :)
Thank you for your great videos.
Linear*
@@ILoveMaths07 you didn't take lineal algebra?
@@crazybeatrice4555 Hahaha! You had me there!
I think you are really underselling the applicability of algebra. There are many applications, admittedly less than analysis but particle physics, chemistry and computer science are at least three major areas that are full of abstract algebra
„What am I trying to say…“ truly encapsulates analysis
Holy shit the part about the difference between proofs fits perfectly
Interesting that your description of the process of solving an algebra problem looks like a graph from graph theory.
I think the stereotype about domestic vs international student holds quite true for Asian international students! I think this is because in Asia (I know in East Asia this is true but I think other Asian countries also have heavy Calculus teaching system), a lot of the university entrance exam is heavy on Calculus but no so much on algebraic thinking so when these students study pure math in Uni, they tend to be more comfortable with Analysis than Algebra. A saw a lot of Asian international students being shocked by the abstractness of first course in Algebra but not so much in first course in Analysis.
been here since like 90 subs, in before this channel blows up
This is awesome! For context, I am currently studying Calculus, and have just learned about the Chain Rule(which is surprisingly easy so far and awesome!). So I definitely have quite a ways to go in order to understand a lot of this. That said, I agree with what you've said regarding understanding the terminology(speaking as someone who's only learned up to this point, and could be incorrect). From what I've seen in the past, whether it be an integral, a derivative, a series, or heck even the notation for a set, it all looks like witchcraft until you understand what it's talking about. :)
Anyways, I have been struggling with writing Proofs for the longest time. I always suspect that I'm leaving something out, or doing something incorrectly. While I didn't quite understand everything you just wrote down(because I have yet to learn what everything means as you said), it did at least shine a little bit of a light on how to go about it. It also makes way more sense how some solutions for differing problems that I've seen may have appeared(not trying to imply that's the case for all of them). They didn't appear out of thin air, but were worked partway through, only to go the end and work backwards, and use that to finish the proof. Really cool stuff! :)
So yea, I suppose this is a long winded comment to say that I'm liking what I'm seeing, and to keep up the good work! :)
you gonna be a mathematician?
@@ClumpypooCP Honestly, it's still too early to say. Perhaps one day I will be, but there are still plenty of other things in life that could happen. For instance: Maybe I will end up becoming a Physicist instead, or maybe I get into GameDev, or maybe there's some other field of study I like that much more and don't even know about yet. Ultimately I have no idea! Whatever I end up doing though, I highly doubt that I will ever cease to use Mathematics. This is quite simply because I love it! :)
cool! good luck with your future studies!@@superiontheknight963
As an international student I agree with everything you said
The visual analogies were very helpful.
Algebra is to do with equality and analysis is to do with inequality
That's a cool perspective...
Just because there are some important inequalities in Analysis, or some proofs require them, doesn’t mean the whole field is about them. Would strongly disagree.
Algebra has more to do with isomorphisms, canonical or otherwise, and structure preserving maps. Analysis is more approximations with precise bounds. Strict equality in both cases is something that's overcome using various techniques.
This is a very pathological summarization of both disciplines. Analysis have as much equality as algebra.
@@Kraft2001inequality is important to embed items into a desired space which is at the heart of a loy analysis problems. There are subfields that deals with inequalities for example Convex Geometry.
Im German highschool the maintopic in math ist analysis and a bit vektor algebra and a little bit probability stuff, but some schools even skip all the probability stuff
Measure theory was so painful.... I approached the topic as if I was just writing, almost like ChatGPT. Of course, I struggled a ton and had to retake it.
I agree with you, though, I like analysis so much more...
i can give you a prospective about the state of maths in morocco
here we have different branches in hs and for the mathematics branch we studied derivatives and integrals as our main focus with limites in Analysis
as for algebra we had in our last year of hs Groups/rings/fields though the problems and theorems were more introductory material and also had arithmetic in Zn
obviously almost everyone struggled with the material 😅 and graduated after working hard
“Purest form of mathematics”
Category Theory: hold my beer…
My copy of Algebra: Chapter 0 starts with category theory. The algebrists are coming for everything.
@@abebuckingham8198 i'm pretty sure category theory sprung out of algebraic topology. i think the algebrists can lay claim to it
analysis is about estimates, approximations, and infinitary processes. algebra is about structure, equivalence, and finitary processes.
4:02 I’m only a 2nd year undergrad and I’m not a math major or an English major so forgive me if I’m wrong but I think the word you’re looking for would be abstraction
The little amount of ink in that pen is impressive.
I loved your analysis problem ...analysis.
I think it might be just the matter of how one people tend to approach one problem？But as an international student from China, I think we do have a a lot of proof like the sketch you draw for analysis in our high-school study.
I find real analysis to be interesting. However, I prefer abstract algebra because in my opinion, it is easier to understand. I am from the Caribbean.
Analysis ftw 😁
I think its also worth noting that students from the east side of iron curtain (aka communist countries in Europe before the fall of soviet union) were essentially trained to become an analyst from a very young age. I think the soviet union once had one of the best analysis program in the world, more so than their western counterparts. This was during the cold war when the U.S and USSR wanted to assert their dominance: aka space race. So, it makes sense for countries like USSR to allocate more resources into training their best and brightest students to study real analysis which has a direct application in aerospace, physics and numerical analysis etc.
On the other hand, countries like France were more invested in Algebraic geometry and other algebra topics (Grothendirk and his colleagues are case in point. They were mostly citizens of western Europe).
For real though, its quite interesting how culture and history often shapes the development of mathematics.
Bro thats what I was thinking lmfao.
mf every analysis type book i see is written by some RUSSIAN dude lol.
nice video. you write well, it was soothing to watch. i graduated in chem E 5 years ago, i would like to go for a PhD in math as a personal pursuit once i've "made it" and invest well
Very interesting! Thanks.
Analysis: the study of metric spaces and maps between them.
Hey man I really like your videos and the topics are always so Interesting, it’s a good break from all the other low level clean math shown on RUclips. However I really think you could improve your video quality so much by having a small tripod or something instead of the hand held camera. Hope to see more cheers.
For Analysis I think Ultrametricity or non-Archimedean metric spaces (i.e. Q_p the p-adic space etc...) is becoming a big enough part of analysis (literally the other half of the analysis universe) that it is a good mention.
They both look interesting.
Surprisingly, both cryptography and cryptology mean the same thing.
What an interesting way to put it
Halfway through a math bachelor in Sweden. I can say that analysis was heavily prioritized over algebra in the education pre-uni, and to some extent also in the course recommendations at uni. Analysis is easier for me probably largely because of that. Although there is a lot to learn in real analysis, I feel definitions and theorems are rarely surprising and often similar to things you've seen before. Similarly with problems. I can quickly get a rough plan, and then it's "just" a matter of execution.
In algebra on the other hand i often have no clue why a statement should be true after reading a theorem or a problem. Similarly with definitions. Their immediate consequences come as complete surprises. Makes it harder but also more interesting to me. Feels like there is more to discover.
deadass? I aint even know what Analysis was untill i searched it up lmfao.
Im an undergrad at my place. Im doing my first year stuff but from what i see, aint analysis stuff like limits?
What stuff in highschool did u learn for analysis?
I feel like other countries learn way more in highschool than we do lol. We just learn the basics and rarely calculus unless we take that course specifically lol
@@honkhonk8009These guys are math majors, you don't take Analysis in high school.
@@NachoSchipsread avin's comment,je/she learned analysis in high school
About your remark why some outside students might do better in analysis than algebra, I can only speak from personal experience that in eastern / central Europe back in my days as a high school student in the +2000 years we did not studied calculus as it is studied in US, instead it was "approached" much more how you would approach real/complex analysis. P.S: Love your videos...keep going :)).
Difference between analysis and algebra
Analysis-a gem
Algebra-an annoying lot of theorems you learn that you most likely forget in 2 weeks time
Loved it
I think analysis textbooks, and other textbooks in general, should instead of just giving you the proofs and methods and stuff, they should show you a way you could get to them
not even show problem solving strategies, but just show the problem being solved in a normal way cus I think that's not an obvious way to solve stuff at all if you're not used to it
Cryptography is right.
"If you can't explain it to a six year old, you don't understand it yourself." I'm starting to think nobody understands graduate-level mathematics. Either that or I need to be six years old.
Yeah, because this statement is only partially true. A little improvement: "If you can't explain it to someone who speaks your language, you don't understand it yourself."
6 year olds and none-math people dont speak the language of mathematics.
The statement bothers me because the vast majority of things aren't simple enough to be understood by someone who wasn't aware of their own existence four years ago. How do you explain algebra to a kid who's just learning about addition? You went through school and college to get to those concepts.
@@caro8164 that or you’d need a really smart 6 year old to get it right away when you explain stuff conceptually
That only applies to an extent. It used to be cus you had retards making fancy words for random shit and acting hard for no reason.
how do you record your videos?
Did you study discrete mathematics? Is there a video planned about that topic?
I TA'd for discrete structures once as an undergrad. It is very similar to math proofs but under the context of computer science. ATM I have no video planned but I will consider the topic for future videos :)
I studied math and physics but I became a software engineer because I enjoy making money. My heart still belongs to math though
As a recent math undergrad. Algebra supremacy! Booo Analysis!
Sounds like someone know his epsilon delta proofs…
Based on your assumptions, Algebra is a single path (abelian) connecting A to B, and potentially a multiple path going from A to B (non-abelian) ,so Jacobi doesn't shrug🤣🤓. With Analysis, a singularity on the real line forces us to make an inductive ansatz to connect A and B.
So Analysis is just deep calculus 🤔
What would be the shortest path to get from calculus to measure theory ?
In theory, the first three chapters of Rudin would be mostly enough. In practice, at least the first seven chapters.
I would recommend some Analysis first, like Bartle and Sherbert's book. Believe it or not, Elementary Statistics and Probability theory might help you bridge the gap even better, because I believe measure theory was born from probability.
@@PhDVlog777 That's the other way around
French mathematicians Henri Lebesgue and Emile Borel wanted a better version of integration (cause Riemann one is pretty weak when we're talking about sequence of functions) and so they developped Lebesgue integration which needed to devellop measure theory
Then, it appeared that probability could be rigorously defined as a brench of measure theory via the Kolmogorov's model ( a set + a sigma algebra + a probability measure)
If I have to explain what algebra and analysis are in 10 seconds to someone who doesn’t know:
Algebra = Generalised arithmetics
Analysis = Rigorous calculus
(I know this is super inaccurate but I think it is intuitive😂)
Abduction.
This is what one of my instructors called proofs like your analysis proof. It is not deduction, not induction, ergo abduction. You abduct the solution.
That makes sense-it’s pulling the solution that’s already embedded within the question from the question as a whole.
My university in Italy is extremely specialized in analysis but it's because of an almost 130 year-old tradition, probably you know that Italy isn't a developing country.
In the XX century there were many great analysts, so other good analysts moved to work with them.
So in the departament now like 70% of research is analysis focused.
Maybe it's the same thing in the US for algebra, or it's because some education choice.
Analysis might be a field in which US is relatively bad at, because of some reasons.
I wouldn't say the US is relatively bad at analysis. Most of the well known analysis books I can think of were authored by Americans.
Ive heard from some people that a degree in math requires a lot of coding is that true?
Applied maths, yes
Pure maths, no
Some bachelor degrees are equipped with a computer science minor, so yeah... depends on the degree.
Algebra people are good at logic puzzles (and very smug about it) but terrible at visualization (and in total denial about it). The only physics they can handle is Chemistry. xSTx on the MBTI. Analysis people are the exact opposite. xNTx the MBTI.
2:14 there are several ways to write an ampersand, and that is NOT one of them, lmao
My degree is engineering, not mathematics. Am I correct to say the name of the 2 fields are abstract algebra and topology?
There is a nuanced difference between Analysis and Topology but they do have a significant overlap.
Analysis is the logical foundations of calculus. Topology is the study of connectedness and continuity. Originally topology started with graph theory but there is significant overlap with analysis. Ideas that are vital in analysis like compactness are fundamentally topological in nature and all metric spaces have a topology induced by the metric. So if you can measure lengths or distances in a meaningful way you have a topology there too. Not all topologies are metrizable though so metric spaces are insufficient to classify topological ones.
jni
What’s that funny sound at 14:44
I literally have no idea 😅
Thanks man, I'm about to apply for college and I think I'm either going for an engineering degree or a math degree.
I'm quite fond of and comfortable with calculus, linear algebra and math in general as I take advanced maths courses in my high school. But man, I'm still scared af that I wouldn't be smart enough for a math degree. But at least this video gives me a bit of information, so cheers!
Hey please make the video length 5 to 10 minutes
If you have lot to say break the video into two
No!
They are approx 10 minutes long, if you play them at x2.
Analysis is calculus on crack
Hahahaha! I like this quote!
😂😂😂
Analysis>>Algebra. It’s funny, all of my classmates prefer algebra but I abhor it.
Once you learn about Lie algebras you’ll come around.
@@AP0PT0SIS I already have, they aight
10:30 I’d disagree (I’m from Peru)
111
In India students master single variable calculus in the high school itself. So your argument in favour of the utility of analysis skills is pretty solid ig.
Nah this guy is trolling.
He doesn't know math
Huh
This is a dumb comment from somebody that knows nothing about anything. Is proving trig identities, like what you do in high school, Analysis?
I ask because that was always the area where you’d be told to work forwards and backwards from each instance of the identity
In University of Ljubljana if we studied only this, math would be a joke.
In algebra year one we also do Permutations
Vector spaces
Matrecis
Transformations
Dot product (not just normal dot product as done when multiplying vectory)
And some more stuff I don't know how to spell + ofcourse Monoids halfgrups grups rings fields and algebras.
In analysis year 1 we do dedikinds axiom and stuff around that. Sequences, sums, differentiation, integration, reiman integrals, debroux integrals + metrics. Metric spaces or how do you spell it.
Sorry for my bad english and not exactly knowing what things are called in English.
I don't know if you are from USA, but this is literally a joke what you study.
Are you saying its easy or hard?
Im in year 1 rn and we basically do the exact same stuff u do lol. tbf im retarded and still dont know the difference between algebra and analysis, but we basically did everything you listed up there.
But I wouldnt say your wrong. I think we here in the states just dont do as much mathy stuff as people from other countries lol.
I have no clue tho. Thats just what it seems, cus I was talking to a family friend in Srilanka, and the dude learnt all that shit in fucking HIGHSCHOOL bro.
tbf he said it was cancerous and he prolly didnt fully understand it anyways, but still other countries are on a different tier.
I think its maybe the mindset cus here we fuck around in highschool and then tryhard thru college, but maybe its different idk.
Hi, do you have LinkedIn? Would love to connect!
A great video!! But I would like to remind you that x is in R\{3} !!! 🥲
maybe if f would not be defined at that point, but since it is, x can be any real number
@@2funky4u88 no, since we are asked to find the limit at that point we must not consider the point itself.
@@aviberezovskiy7633 that does not matter. If you consider the point itself or not does not change the limit. Since any sequence converging to that point must be considered. So if the function is defined there changes nothing. So your remark is of no use. Except when the function is not defined at that point of course
@@2funky4u88 You are wrong. Read about removable discontinuities. A classical example is f(x)=(x^2-9)/(x-3) for x=/=3 and f(3)=1. Avi Berezovskiy's point is right: when talking about limits we have to exclude the point to which x tends whether or not f is defined in that point. Just check the definition of the limits of a function if you don't believe.
@@filipmunteanu2211 in the general case you want to exclude it, but here the result remains unchanged so it is overly pedantic