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Speaking frankly, it's ambitious to complete even one of these courses in a single summer. Most students would be taking one or two of these in 12 - 16 weeks wh

Speaking frankly, it's ambitious to complete even one of these courses in a single summer. Most students would be taking one or two of these in 12 - 16 weeks while doing nothing else but being a student. Accomplishing the same in about eight weeks from video lectures will be more difficult. I would strongly urge the author to slow down, choose a single course they have the background for and work from there. I can't tell if the author has done a real analysis course before, but if they haven't that's the one they should choose next. If they already have that under their belt they should go to probability or complex variables. I don't see the utility of re-doing calculus or linear algebra if the author is already strong in both. I appreciate the enthusiasm for math that's evident here, that's great! But if you're learning math the "right" way - by actively engaging with the material - there's only so much of it you can learn at once.

hsikka 7y ago
Hey, author here, I really appreciate the feedback. Are there any specific prerequisites to Real Analysis?
- throwawaymath 7y ago
  Technically speaking, no. Real analysis is pretty self-contained. You basically start out by constructing the reals from scratch and deducing continuity as a consequence of the completeness of the real field. You use a tiny bit of set theory to establish notation and define bounds, and then from there you go into limits, derivatives, integrals and maybe the Lebesgue measure. I wouldn't expect a real analysis course targeted at applied math majors to do much other than that. The reason real analysis is useful is because it's (loosely) a deeper calculus course with proofs. Since probability theory becomes more proof-based (and ventures into measures), real analysis is good preparation for it.
  - hsikka 7y ago
    Fantastic, thank you for the tip! I'll probably prepend Real Analysis to the list!
  - umanwizard 7y ago
    > You basically start out by constructing the reals from scratch Not necessarily -- a lot of books just take the existence of a unique set with certain properties as an axiom and call it R. The main topics of basic real analysis IMO are differentiation, integration, (uniform) continuity, compactness, convergence, etc.; how to construct the reals from the rationals is a side point at most. This could be personal bias as I just don't personally think that the exercise of constructing the real numbers is very interesting.
- ghufran_syed 7y ago
  As someone who also went back to study math after not doing it for a long time, I would recommend doing a book that focusses on teaching how to prove things, using the math you already know. Most upper div math courses either assume you know how to prove things, or try and teach it in passing while also teaching the topic itself (classically linear algebra and calc 3 were taught this way, though now many places now have separate "proof" classes instead) I would strongly recommend Chartrand and Zhang's "Intro to mathematical proofs", once you know how to prove things, every other more advanced math class you take will be much easier than it would have been otherwise. [Edit] In particular, do the book, including the chapters on "proofs in calculus" and "proofs with real and complex numbers" before doing real analysis - you'll enjoy it much more that way! [/Edit] I went through most of this book a few years ago, now about to finish my MS in math :)
  - dorchadas 7y ago
    Agreed. I'm trying to move from being a secondary mathe teacher to actually learning pure math, and proofs are the killer. I feel like I understand topics, but creating proofs kills me.
- chestervonwinch 7y ago
  Analysis requires a certain level of mathematical maturity to motivate, follow, understand, and appreciate. This level is roughly acquired after having taken undergraduate coursework in ODEs, linear algebra, or abstract algebra.
behnamoh 7y ago
This guy did something similar: completing a 4-year MIT undergrad CS degree in just one year! https://www.scotthyoung.com/blog/myprojects/mit-challenge-2/ https://www.scotthyoung.com/blog/myprojects/mit-challenge-2/
- usgroup 7y ago
  It'd be more believable if he did it in 3 years :)
- HAL9000Ti 7y ago
  Scott Young is a publicist - his goal is to sell, sell, sell. Completing a world-renowned 4-year curriculum in 1 years time is impossible. (He also admitted to reading the solutions instead of doing the problem sets)
  - barry-cotter 7y ago
    This is akin to saying that Math 55 is impossible and neglects that he didn’t claim to complete the requirements for an MIT degree but for a Computer Science Major. That’s two years of classes, not four, and if your aim is to pass the exams for them rather than learn everything to mastery doing it in one year seems possible if still incredibly hard. https://en.wikipedia.org/wiki/Math_55 https://en.wikipedia.org/wiki/Math_55
nsfmc 7y ago
> I can't tell if the author has done a real analysis course before, but if they haven't that's the one they should choose next... I don't see the utility of re-doing calculus or linear algebra if the author is already strong in both. having (somehow) completed many of these requirements for my 18c degree, i would say that analysis is not necessary if your interest is actually applied math. There's a great line in rudin's preface that says that his approach is ~"pedagogically sound at the expense of being logically incorrect," and recommending analysis for somebody that's not looking to mainline a pure math degree to me feels "pedagogically unsound (but logically correct)"[0]. I took analysis and i appreciated it, but i really loved the applied classes in my degree: 18.310, 18.311, 18.781 (theory of numbers) along with algebra 18.701/702. If you haven't taken a higher-level algebra class, it will let you know if analysis is right for you because you'll brush up against the edges of it without (what i consider to be, at least) its hallmark punishing density. There are other great electives in math at mit, shop around the 18.4* classes and dial in by interest, most of them only require a prereq of 18.02/18.03/18.06 and you can sort of figure the rest out along the way. Something to be aware of is that for a while 18.310 didn't have a dedicated instructor, so it really was all over the place. 18.311 was also somewhat hastily structured the semester i took it, but it is actually pretty good material. You may find that after you've done all these classes that you are actually interested in pure math and at that point i would suggest looking at 18.100b (analysis), 18.700 (linear algebra), 18.100c(real analysis), 18.901(topology) and the rest of the "hard math classes," but i really do think that you'll find that the rationale for those classes doesn't click if you haven't taken a few classes like 310 or 701 first. just my two cents! good luck, have fun! [0]: this is the actual quote, it's in the preface rudin's principles of mathematical analysis 3rd ed. which is the 'textbook' for 100b.
- jacobolus 7y ago
  > i would say that analysis is not necessary if your interest is actually applied math If you start reading research papers in applied math, there’s a ton of measure theory and functional analysis there. More generally, both introductory real analysis and introductory complex analysis are assumed basic foundational background for pretty much any kind of research mathematics, applied or otherwise. I’m also not sure I would recommend trying to self-study them though. Some expert guidance/feedback is pretty helpful for someone starting out.
jpmattia 7y ago
> Most students would be taking one or two of these in 12 - 16 weeks while doing nothing else but being a student. From my experience (at MIT in the 80s and 90s): The usual student takes 3 technical courses and one humanities course per term. From my own history: I took 18.03 in one term with two EECS courses and a humanities course, and then 18.04 the following term, also with two EECS courses and a humanities course. I would guess that 3 math courses at one time (and taking nothing else seriously technical) would constitute a pretty full load for most people. Much beyond that probably impacts how much is being truly absorbed for the long term.
- pdonis 7y ago
  > From my experience (at MIT in the 80s and 90s): The usual student takes 3 technical courses and one humanities course per term. My experience (in the 80s) was the same. There were a couple of semesters where I had to take 4 technical courses; I had no desire to repeat the experience. :-)
usgroup 7y ago
I agree with this comment and would supplement as follows: whatever you intend to learn, consider how you're going to retain it. I'm quite sure that I've forgotten a lot more maths than I know. I'm not sure about the utility of forgotten maths. You may think it rehydrates well, but it doesn't. It won't be as hard to relearn but it'll be hard. So consider, for whatever you learn, are you going to be using it? Are you going to be building upon it? Is it just a tourist expedition? These are all fine reasons but they each have different implications.
- c0vfefe 7y ago
  I think most of us are highly invested in the value of easily-rehydratable knowledge, because an incredible amount of the first third of our lives is spent learning facts that we'll forget. If we admitted there might be a better way, we'd have to do the hard and thankless work of institutional overhaul in our schools.
  - barry-cotter 7y ago
    This mistakes the purpose of school. It’s less learning than childcare for primary school, socialisation and teaching children to do apparently boring and meaningless work as directed by superiors, and ranking and proof of intelligence and conscientiousness at higher levels. We’ve known most people forget almost everything they learn at school for a very long time and spaced repetition and the forgetting curve date from Ebbinghaus (1885) while mastery learning is probably older than writing though the name dates from 1968 and the earliest unambiguous instance of the concept dates from the 1920s. https://en.wikipedia.org/wiki/Forgetting_curve https://en.wikipedia.org/wiki/Forgetting_curve https://en.wikipedia.org/wiki/Mastery_learning https://en.wikipedia.org/wiki/Mastery_learning
    - c0vfefe 7y ago
      I suppose I was taking more of an idealistic view.

hsikka 7y ago

Hey, author here, I really appreciate the feedback. Are there any specific prerequisites to Real Analysis?

throwawaymath 7y ago
Technically speaking, no. Real analysis is pretty self-contained. You basically start out by constructing the reals from scratch and deducing continuity as a consequence of the completeness of the real field. You use a tiny bit of set theory to establish notation and define bounds, and then from there you go into limits, derivatives, integrals and maybe the Lebesgue measure. I wouldn't expect a real analysis course targeted at applied math majors to do much other than that. The reason real analysis is useful is because it's (loosely) a deeper calculus course with proofs. Since probability theory becomes more proof-based (and ventures into measures), real analysis is good preparation for it.
- hsikka 7y ago
  Fantastic, thank you for the tip! I'll probably prepend Real Analysis to the list!
- umanwizard 7y ago
  > You basically start out by constructing the reals from scratch Not necessarily -- a lot of books just take the existence of a unique set with certain properties as an axiom and call it R. The main topics of basic real analysis IMO are differentiation, integration, (uniform) continuity, compactness, convergence, etc.; how to construct the reals from the rationals is a side point at most. This could be personal bias as I just don't personally think that the exercise of constructing the real numbers is very interesting.
ghufran_syed 7y ago
As someone who also went back to study math after not doing it for a long time, I would recommend doing a book that focusses on teaching how to prove things, using the math you already know. Most upper div math courses either assume you know how to prove things, or try and teach it in passing while also teaching the topic itself (classically linear algebra and calc 3 were taught this way, though now many places now have separate "proof" classes instead) I would strongly recommend Chartrand and Zhang's "Intro to mathematical proofs", once you know how to prove things, every other more advanced math class you take will be much easier than it would have been otherwise. [Edit] In particular, do the book, including the chapters on "proofs in calculus" and "proofs with real and complex numbers" before doing real analysis - you'll enjoy it much more that way! [/Edit] I went through most of this book a few years ago, now about to finish my MS in math :)
- dorchadas 7y ago
  Agreed. I'm trying to move from being a secondary mathe teacher to actually learning pure math, and proofs are the killer. I feel like I understand topics, but creating proofs kills me.
chestervonwinch 7y ago
Analysis requires a certain level of mathematical maturity to motivate, follow, understand, and appreciate. This level is roughly acquired after having taken undergraduate coursework in ODEs, linear algebra, or abstract algebra.

behnamoh 7y ago

This guy did something similar: completing a 4-year MIT undergrad CS degree in just one year! https://www.scotthyoung.com/blog/myprojects/mit-challenge-2/ https://www.scotthyoung.com/blog/myprojects/mit-challenge-2/

usgroup 7y ago
It'd be more believable if he did it in 3 years :)
HAL9000Ti 7y ago
Scott Young is a publicist - his goal is to sell, sell, sell. Completing a world-renowned 4-year curriculum in 1 years time is impossible. (He also admitted to reading the solutions instead of doing the problem sets)
- barry-cotter 7y ago
  This is akin to saying that Math 55 is impossible and neglects that he didn’t claim to complete the requirements for an MIT degree but for a Computer Science Major. That’s two years of classes, not four, and if your aim is to pass the exams for them rather than learn everything to mastery doing it in one year seems possible if still incredibly hard. https://en.wikipedia.org/wiki/Math_55 https://en.wikipedia.org/wiki/Math_55

nsfmc 7y ago

> I can't tell if the author has done a real analysis course before, but if they haven't that's the one they should choose next... I don't see the utility of re-doing calculus or linear algebra if the author is already strong in both. having (somehow) completed many of these requirements for my 18c degree, i would say that analysis is not necessary if your interest is actually applied math. There's a great line in rudin's preface that says that his approach is ~"pedagogically sound at the expense of being logically incorrect," and recommending analysis for somebody that's not looking to mainline a pure math degree to me feels "pedagogically unsound (but logically correct)"[0]. I took analysis and i appreciated it, but i really loved the applied classes in my degree: 18.310, 18.311, 18.781 (theory of numbers) along with algebra 18.701/702. If you haven't taken a higher-level algebra class, it will let you know if analysis is right for you because you'll brush up against the edges of it without (what i consider to be, at least) its hallmark punishing density. There are other great electives in math at mit, shop around the 18.4* classes and dial in by interest, most of them only require a prereq of 18.02/18.03/18.06 and you can sort of figure the rest out along the way. Something to be aware of is that for a while 18.310 didn't have a dedicated instructor, so it really was all over the place. 18.311 was also somewhat hastily structured the semester i took it, but it is actually pretty good material. You may find that after you've done all these classes that you are actually interested in pure math and at that point i would suggest looking at 18.100b (analysis), 18.700 (linear algebra), 18.100c(real analysis), 18.901(topology) and the rest of the "hard math classes," but i really do think that you'll find that the rationale for those classes doesn't click if you haven't taken a few classes like 310 or 701 first. just my two cents! good luck, have fun! [0]: this is the actual quote, it's in the preface rudin's principles of mathematical analysis 3rd ed. which is the 'textbook' for 100b.

jacobolus 7y ago
> i would say that analysis is not necessary if your interest is actually applied math If you start reading research papers in applied math, there’s a ton of measure theory and functional analysis there. More generally, both introductory real analysis and introductory complex analysis are assumed basic foundational background for pretty much any kind of research mathematics, applied or otherwise. I’m also not sure I would recommend trying to self-study them though. Some expert guidance/feedback is pretty helpful for someone starting out.

jpmattia 7y ago

> Most students would be taking one or two of these in 12 - 16 weeks while doing nothing else but being a student. From my experience (at MIT in the 80s and 90s): The usual student takes 3 technical courses and one humanities course per term. From my own history: I took 18.03 in one term with two EECS courses and a humanities course, and then 18.04 the following term, also with two EECS courses and a humanities course. I would guess that 3 math courses at one time (and taking nothing else seriously technical) would constitute a pretty full load for most people. Much beyond that probably impacts how much is being truly absorbed for the long term.

pdonis 7y ago
> From my experience (at MIT in the 80s and 90s): The usual student takes 3 technical courses and one humanities course per term. My experience (in the 80s) was the same. There were a couple of semesters where I had to take 4 technical courses; I had no desire to repeat the experience. :-)

usgroup 7y ago

I agree with this comment and would supplement as follows: whatever you intend to learn, consider how you're going to retain it. I'm quite sure that I've forgotten a lot more maths than I know. I'm not sure about the utility of forgotten maths. You may think it rehydrates well, but it doesn't. It won't be as hard to relearn but it'll be hard. So consider, for whatever you learn, are you going to be using it? Are you going to be building upon it? Is it just a tourist expedition? These are all fine reasons but they each have different implications.

c0vfefe 7y ago
I think most of us are highly invested in the value of easily-rehydratable knowledge, because an incredible amount of the first third of our lives is spent learning facts that we'll forget. If we admitted there might be a better way, we'd have to do the hard and thankless work of institutional overhaul in our schools.
- barry-cotter 7y ago
  This mistakes the purpose of school. It’s less learning than childcare for primary school, socialisation and teaching children to do apparently boring and meaningless work as directed by superiors, and ranking and proof of intelligence and conscientiousness at higher levels. We’ve known most people forget almost everything they learn at school for a very long time and spaced repetition and the forgetting curve date from Ebbinghaus (1885) while mastery learning is probably older than writing though the name dates from 1968 and the earliest unambiguous instance of the concept dates from the 1920s. https://en.wikipedia.org/wiki/Forgetting_curve https://en.wikipedia.org/wiki/Forgetting_curve https://en.wikipedia.org/wiki/Mastery_learning https://en.wikipedia.org/wiki/Mastery_learning
  - c0vfefe 7y ago
    I suppose I was taking more of an idealistic view.