Teaching myself an upper level pure math course (we almost died)

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00:00 Intro
2:41 What is real analysis?
5:30 How long did the book take me?
6:18 How to approach practice problems
8:08 Did I like the course?
8:42 Quick example
10:53 Advice for self teaching
15:38 Textbook I used
17:50 Ending/Sponsorship

►My Setup:

[zachstar]

Then something I forgot to mention for this textbook/class specifically, the pre-req's would be single variable calculus, proofs (induction, direct, indirect, contrapositive, etc), basic set theory, logic, and functions/relations. Last time I made one of these videos it was about going through the book 'how to prove it' and that covers everything you would need to take go through this real analysis book (besides the calculus 1 and 2 concepts).

[connordavis]

As a freshly minted PhD in math, I assure you the word "obviously" in a math textbook actually just means "there was something about this list of expressions that was annoying to LaTeX."

[cf]

A mathematician calls a problem "hard" as long as he hasn't fully understood it yet.
When he finally does, he calls it "trivial".

[daniellucas]

"They use the word 'obviously' way too loosely in upper level math courses." Yes. In upper level math textbooks, the word obviously gains an additional meaning. It can either have its original meaning, or it can mean "proving this fact is too inconvenient for me to bother with right now, so I'm not gonna."

[standardtrickyness]

Flammable maths: "Today we are going to do proofs"
Zach Star: "what was that noise?"
Flammable maths: You mean proof?
Zach Star: There it is again.

[Fujibayashi]

"...staring at a problem and having no clue how to start it."
Welcome to the life of a math major

[JasonOvalles]

Real Analysis is where you talk about all the times your Calculus professor told you not to worry about it.

[joemiller]

I learned my favorite proof strategy from my Linear Algebra professor, “Proof by Force.” Where you state “clearly this is true” and then no one will have the confidence to call you out for being wrong.

[ruferd]

"Well of course it's YOU REALIZE IT!"
Truer words have never been spoken. It sounds like a contradiction, but so many times in my Math Major as an undergrad, I would go through exactly that. I'd read something that claimed to be obvious 15 times over for an hour straight before giving up. Then the next day I woke up and I just understood exactly what it was saying. Like, of course it's true. Why wouldn't it be true?

[AndrewDotsonvideos]

More like teaching yourself how to be such a g-dang beautiful person

[martinshoosterman]

The "it is obvious, once you realise it" part reminds me of a funny story:

There's a professor in a math class, during the lecture he declares a theorem and says that the proof is trivial, then moves on.

After class a student comes up to him and asks him about the proof that the professor claimed was trivial. The student says he doesn't see how you would do it, and it doesn't seem trivial to him.
The professor then looks at the problem and thinks about it. He realises that he doesn't actually immediately know how to prove it. He tells the student to talk to him the next day. That night the professor looks at the problem again and spends all night figuring out how to prove it. By the morning he's figured it out, and is able to prove it.
The next day the same student comes up to him and asks about the problem.
The professor says: ah yes, I thought about that problem some more, and I can confirm that yes, it is indeed trivial.

[DaveScottAggie]

"The proof is left to the student as an exercise."

[swozzlesticks]

Undergraduate Analysis: "Ugh... I can't believe I stared at it for an entire hour not knowing what to do"
Graduate Analysis: "Holy shit dude I only spent 2 days staring at it not knowing what to do."

[TyronTention]

My current Real Analysis Part 2 professor had Rudin as an advisor and I feel extremely honored.

[RCSmiths]

As someone who is interested in Pure math, Physics, and Engineering, to be able to appreciate them all together is what makes them engaging in any shape or form.

[jonathan.gasser]

Hahaha man I've never felt so dumb as when I went through a serious maths textbook. That thing about "just give me one I can solve" really hit home.

[xephyr]

Dude this is seriously so validating. I've tried to read through upper level texts a few times but I always get stuck on something or other and I get really discouraged. I feel like I'm going too slowly. Then I just kinda give up.

Hearing that you go through the same thing really helps. It's encouraging. Thanks!

[taproot]

I loved this class. Partly because the teacher was great, and partly because if you are teaching people this class, they are going to be a bunch of fast-witted, snarky, sarcastic dicks. Because these classes teach you to find loopholes better than a lawyer. And we had a lot of fun because of that.

But the best part of the class was the 2nd test. I was given the question "prove or disprove the validity of this statement: there exists a rational number R and an irrational number I such that R•I equals a rational number."

And I looked at that question for about 10 seconds, and wrote down "R=0" and moved on.

When he graded it, he wrote "I technically have to give you credit"

One of my friends got the question wrong, and when I told him R=0, he turned bright red and about died in embarrassment because, " I forgot zero existed".

But the best part is that on the next test, he asked the same question, but with the added "for R=/=0" in it.

It was great.

[bharatkrishna]

Me: thinking about studying math at an advanced level.
My brain: Softly 'dont'

[ZeroT]

A physics student walks up to his Prof and asks him whether a certain statement can be considered trivial. The Prof, deeply reflects on the statement, but cannot answer the student right away. The next day, the Prof happily concludes to the student, "This statement is indeed trivial."