Real Math People Study the Appendix

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In this episode of Office Hours with The Math Sorcerer I answer a question I received from a viewer. He is trying to learn mathematics.
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Thank you:)

[@MikeSpinoza]

It's actually gastroenterologists who study the appendix.

[@GiovannaIwishyou]

That's a good one😂

[@KMMOS1]

MIchael is studying multiple appendices simultaneously. Gastroenterologists only study one at a time.

[@TheMathSorcerer]

LOL OMG!!!!!!!!!!!

[@assishratha2132]

You know I wonder why people study about appendix? It's nothing to with our body

[@jamieg2427]

learning tips: 1. review if you do a problem today, then don't touch it for five days, you may forget how to do the problem. for each book, keep a list of problems that exemplify each concept. if you find a juicy problem that bundles multiple concepts together, then you only need that one problem instead of several for each concept. 2. time difficult courses may take 10--15 hours (sometimes more) a week of study outside of class. don't overload yourself, so you have enough time for the class. 3. spread if possible, only do three to ten problems from a book section per day. once you hit your quota, move on. return tomorrow. this spreads out the learning, allowing you to absorb material slowly, while also moving quickly. 4. repetition when you finally get a tough problem, do it again once or twice. treat math like an instrument. repetition is helpful. 5. speed it is important to write out all your steps, but there are shortcuts too. even if you don't use these shortcuts on an exam, they can help you repeat a problem more quickly. here's an abbreviated difference quotient example from calculus, done the typical way: lim . . . /h = lim . . . /h (algebra) = lim . . . /h (algebra) = lim . . . /h (algebra) = answer that's a lot of repetition. instead of carrying the entire problem onto each line, use variables to name the pieces of your problem. this way, you can work on each piece, which is faster, then pull them back together at the end. L = lim Q (L for limit; Q for quotient) Q = . . . /h = . . . /h (algebra) = . . . /h (algebra) = . . . /h (algebra) L = answer 6. illusion of competence reading a book or your notes makes you FEEL like you know something, but it might be an illusion. you have to reproduce the knowledge with pencil and paper or voice. think of notes as a way to help you make quiz material for yourself: flash cards, problem descriptions, or similar things to QUIZ yourself with. you do not know something unless you ask yourself a question on it and give the correct answer without help. 7. solutions download the solutions manual. only use the solutions manual to nudge you in the right direction. you have to be able to reproduce the correct answer by yourself and understand why it works. 8. lectures lectures have been shown to give students false confidence and poor competence. a teacher only has a couple hours of class per week, so their treatment of the material will be superficial at best. you MUST fill in the rest of the details yourself outside of class with the textbook and other materials. this is not extra. 9. work linearly do a problem straight down a page. if you write a bunch of calculations on the side of your paper or jump around, it can make your logic hard to follow. in other words, keep your math solutions skinny. also, you can use indentation and headers to give you work some structure: slope? y = mx + b y - b = mx ■ m = (y - b)/x 10. books and notes extensive notes are often overkill. 1. look through a chapter's headings to see what's going on 2. read through and work the examples along with the text as if they were hookworm problems. definitions or equations you need for those examples can go on a separate sheet. this sheet will be notes for the chapter, and will contain only the most important ideas. 3. work problems. again, when you find you need an equation or definition, copy that equation to a separate sheet. after a bunch of problems, you can go back through the text to add any loose ends to your notes.

[@christressler3857]

When I started to review my calculus book by Larson, Hostetler (edition is old enough to include Edwards), I too started with the appendix, which is preliminary work. This book also has a chapter 0 that also includes pre-cal algebra material. When I got into Ueno's Introduction to Algebraic Geometry (NOT his other three volumes), I started with its appendix first also. Finally, I just started David Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, and again started with its appendix first. Lol but while it is useful to know before delving into the main material, it have a feeling it's going to be harder!

[@christopherallan4298]

The idea that a person can improve their proof writing by using proper "etiquette" resonates with me as someone who writes computer code for a living. Even if you work in a highly-technical field (as opposed to working in the arts or humanities), there's a big difference between building a thing that merely "works", and building something whose design reveals itself to an observer in a natural way (writing "self-documenting" code, or writing a proof or design specification with a high degree of simplicity and elegance), and that is easy (and maybe even pleasant) to understand, modify, or build upon.

[@TheMathSorcerer]

great comment:)

[@83jbbentley]

I just finished my first semester with a 4.0!! Let’s go to summer classes to grad faster!

[@TheMathSorcerer]

Wooo!!!!! GOOD WORK!!!!!!!

[@83jbbentley]

@@TheMathSorcerer it’s because of you and this community for the encouragement and motivation.

[@davidgrenier]

A few things: First, either have good handwriting or take the time to learn LaTeX. An overleaf subscription is expensive but worth it if you're in school. Second, when solving an equation, include every single step of the process. Only change one or two things per step. This will help identify where anything goes wrong. Thirdly, same with proofs. LEarning to write proofs is tough and a lot of math books have REALLY BAD proofs that just make a bunch of implicit assumptions. State everything explicitly and explain each logical step you make. I tend to break my proofs into a bunch of individual lemmas, each only stating one thing and each having its own proof. Then the final "main" proof is a lot of "By Lemma 1 we know X, By Lemma 2 we know Y, Therefore Z". It's similar to how I write code when I program. Every function should do one thing and be testable. Then I can write unit tests to make sure everything works and a main function that just calls the other functions in order.

[@OwenEkblad]

I have never found any reason to purchase an overleaf subscription (besides research collaboration) and have been using the platform for writing math for about 5 years.

[@xhivo97]

Huge congratulations to Michael! Good luck on your studies!!!

[@leelwx5694]

Memorization first and question later resonates deeply with my learning experience. I was one of those slowest to grasp new ideas. I thought I was dumb. Repetition helps me a lot to catch up.

[@helioliskfire5954]

I've never heard of "etiquette" in writing math solutions. That's interesting. Although, in a complex analysis class, I remember the professor striking out the "explanations" in my proof. Lolz. It was in such a way that I felt his annoyance if not rage at my submission. In a linear algebra class, where I used a theorem from the assigned reading but was not discussed in class, the professor refused to give me full points because I should have explained the theorem in my submission. So I guess, there's different kinds of professors out there.

[@KMMOS1]

First of all, good on you for seeking focus and clarity away from drugs. Second, Stewart and co-authors have a textbook about algebra and trigonometry that is much longer and better than the appendices in the calculus book. Get this book, work through the problems, and save your work in a three-ring binder as you progress so you can refer to it as you do more advanced math. Analogously, get Schaum's outline series books about your math subjects, and work through those problems, saving your work papers in three ring binders. This may seem like a lot of extra work, but it will instill personal discipline and form an excellent reference record for future math studies and homework.

[@TheMathSorcerer]

great advice, thank you my friend!

[@nuggamcnugget8335]

Michael been through some sh!t, damn. Coming up again and working your way to the top. Great! I been drinking and was worried I damaged my brain way too much in the process. I realized I lose noting on trying going for science courses. Good to hear someone have been going through the same thing I have been through. You are not alone :)

[@douglasstrother6584]

Look through the whole book cover-to-cover; a lot of interesting stuff can be packed into the appendices. "Oh, I wish I knew that while studying Chapter 5." Prayers to you to stay clean, and keep on keepin' on!

[@douglasstrother6584]

Take a stroll and get a lay of the land.

[@niteman555]

This is an idea that just occurred to me, but you could consider setting aside some of your proofs and come back to them days or weeks later to see if they still make sense to you once they're no longer fresh in your working memory. If you realize there are details that would be good to add, you can make a conscious effort on any proofs you're currently working on to add them in.

[@jsween]

I like what you said around the 8 minute mark. Memorise first, understand after. Sometimes people get stuck in the trap where they feel they need to understand something before learning it. If you just imbibe the information first, organise it somewhat in your head, and then apply it to routine problems in the textbook, the understanding will usually fall put naturally

[@Verbal27]

After self studying mathematics for a a couple years now, I naturally gravitated towards reading the appendix. Incredible video 🔥

[@suredeydo]

This is such a good KZbin channel 🥰 I'm so glad I found you.

[@nerd26373]

we appreciate your effort and hard work. God bless you.

[@stumerac]

Awesome story! I like watching a lot of videos or reading a lot of different books on the same marh concept to see how they all present their work, even if I have a strong understanding of the topic. There are so many ways to present a solution, and some techniques work better for certain problems . Finishing a beautiful presentation on an interesting topic is warm satisfaction.

[@budgarner3522]

Nice advice. I do this for all books that I read and study. It's a nice way to see the trend on what the author thought was important and how they organize their thoughts. You are more than just a Math Sorcerer. Keep it going.

[@martinhawrylkiewicz2025]

Love reading the appendix of the more advanced books.....Terence Tao's Analysis 1 has a great appendix on Logic (introduction). Love it.

[@bvds2007]

Really interesting question and answer. I did not expect the answer, as I feel math papers or even math in applied fields (e.g. financial engineering) are often very terse and very brief, with almost no wordy explanation. Great video.

[@TheMathSorcerer]

thank you:)

[@ianmorgan7906]

If your studying for electrical engineering, I would also look at probability and statistics with calculus. Seeing as a lot signals work uses prob/stats.

[@economicist2011]

8:25 Nope, you were right the first time. It's German so "Ree-mahn" is far more correct than what you corrected it to. In fact, I think English is the only language where the digraph "ie" is routinely prounounced the way it is in "lie" (as in lie down, not lie algebra).

[@bendavis2234]

That’s what I was thinking, I’ve actually never heard it pronounced “Riey-man” before.

[@leeming1317]

5:49 Some of the students in my physics class did this too, I was like wait.... What?! You can do that!? Hahaha

[@Wandering_Horse]

I didn't see the link for the Velleman book, but I did find free pdf of it from a unnamed university in the middle east ;) Also was able to find the book of proof by Hammack at Virginia Commonwealth university. Thanks again math sorcerer, you really are an inspiration on my math journey. I am signed up for PreCalc next semester!

[@tvviewer4500]

I love how this channel exists.

[@ValenceFlux]

I knew an ac theory instructor that once shouted at an apprentice 'if you can't stay sober for this career go do something else with your life. If you can't uphold the purpose of the code, go be a plumber. At least there your mistakes won't get someone killed.' I heard that instructor was also a helicopter pilot who knew how to shout for a living and that experience was hard to forget.

[@raexn]

Hah "it's math, everyone is confused."

[@TheMathSorcerer]

yes lol

[@pushkarnagpure2357]

A reference after or than vellemann/ hammack would be Derek goldrei...predicate and propositional calculus

[@mingmiao364]

Lol James Stuart leaves all important proofs in the appendix so going through the appendix actually helps you learn real math!

[@icybrain8943]

I'd say when in doubt, be as explicit as possible in your solutions - especially for your own benefit when you look back at them in the future. That would mean for key steps in your solution (even if what you're doing is purely computation as opposed to an explicit proof) to identify theorems being used and explaining why you get to use those theorems by showing how the mathematical objects you're manipulating fit in to the conditions defined in those theorems. Different professors will have different standards for things like allowable shorthand and implicit steps, but I think you should identify all the different key "types" of problems you encounter in a book or course and have a record of at least a couple prototype questions for each type where you write out a solution as explicitly as possible for your benefit.

[@dandrost646]

i love the website! bookmarked

[@highviewbarbell]

I gotta say, "Ree-mahn" is correct. In german EI is always like "eye" and IE is always like eeee A is like "ah", so something like "reemahn" from an american perspective would be correct. It would have to be spelled something like Reimän to be pronounced the way someone "corrected" you to

[@rivierasperduto7926]

You have a very natural sounding english accent so I have to ask what was your first language? I am guessing it was Spanish since you said you like to collect math books in Spanish.

[@topdog5252]

Could you review Hugh Neil’s Calculus book? It’s part of the Teach Yourself series.

[@frfrpr]

Reading Surely You Are Joking Mr. Feynman led me to sobriety.

[@ronaldjorgensen6839]

THANK YOU BOTH QUESTION AND SUMMARY OBSERVATIONS AS WELL AS BOOK RECOMENATION , I AM A NUT CASE ADDICT IN /OUT ALSO GLAD TO SEE YOU HOLD TO MATH AS YOUR CORE AFTER LIFES CONFUSIONS AND DISSORDER REMEMBER REAL WORLD CAN NEVER REACH MATH PRECISION ALSO

[@unsalad]

Hey, I am in 9th grade and I am curious about mathematics, I had already read Pre-Calculus in previous classes and I am doing Thomas' Calculus (The Indian Subcontinent Edition), although I am having some troubles understanding some of the concepts (like the epsilon-delta definition of the limit and the fundamental theorem of calculus part-2), what should I do in these situations? And, what other books would you reccommend me.

[@ardiris2715]

"I am having some troubles understanding some of the concepts (like the epsilon-delta definition of the limit and the fundamental theorem of calculus part-2)." You are lucky. Enter that sentence exactly as you wrote it into ChatGPT. Ask it to clarify further anything you still don't understand. Tell it you are 8 years old, if you have to. Also, ask your high school math teacher. You will absolutely make her/his day. (:

[@unsalad]

@@ardiris2715 Alright! Thanks

[@nagamanikomarla5376]

Does the university you get your bachelor’s degree (in math) from matter a lot if you want to become a mathematician? I really want to get into pure math research, but not many universities in my country offer BSc. Honours in Mathematics. I did try to get into the few reputed universities offering these courses, but millions of people write them, and only about 200 get in. Unfortunately, it does not look like I will be one of those 200. However, I have found a private university that does offer a BSc. Honours in Mathematics. It’s a very new university and I am concerned if it will hamper my chances of getting into a good master’s course later on. I met the dean of the mathematical department today, and he said that the university will not be able to teach math at my level, because other students may find it difficult. Any tips? As for my mathematical proficiency, I do consider myself to be very good at math (though I have much to learn). I have a deep interest for the field and am always happy to learn something new in math. The answer key for the one of entrance exams I mentioned just came out, and I missed the mark by a very small margin. I am wondering if this small margin is the difference between me making it in academia and me not making it.

[@videogame7778]

I really am curious, what is your first language. You mentioned how English wasn’t your first language.

[@surrealistidealist]

Accept the fact that there's going to be hardship, but it's normal, healthy, it won't last forever, and it's all worth it! Failure needs to be your friend. Struggle helps you grow. But make sure you've got a good set of healthy coping mechanisms, especially if you're still coming out of a vulnerable situation! Journaling helps me cope. It even helps me organize my thoughts before and after talking to a friend. Get plenty of exposure to nature. Exercise is one of my favorite things in the world, but nutrition is even more important. Beef liver and egg yellows are the best and the cheapest sources of choline, which the body uses to make acetylcholine for focusing ability. Most people are extremely deficient here. Get lots of potassium and magnesium. Talk to your doctor about how much sodium is right for you. As for math itself, get a few cheap, used high school and college algebra textbooks so that you can really master your trigonometry and calculus. Sheldon Axler's books on algebra, algebra and trig, and college algebra are great. They're designed to be read for comprehension and to prepare you for an advanced math education.

[@ass09tube]

I make it a point to begin reading from Preface. Then contents and so on. Also I check appendix and bibliography or references. They help me find sources of ideas, relevant papers, books to explore. Historical context n various other mathematicians plus their work. Any book that doesn't share references - i don't like such book. Its like a fiction, a work of imagination or criticism.

[@arturo9608]

Can we found you in Apple podcast?

[@ardiris2715]

Etiquette: As a former TA at GA Tech, my greatest bane was horrible handwriting. Students with poor skills should at least learn to write CAPITAL LETTERS legibly. The very first thing I do when I open a new textbook is go through it page by page and look at the pictures, about three minutes. Then I go through the book again and read everything that is a title or is written in bold letters, less then ten minutes. I would speedread the entire text in the third pass, about 15 minutes. All told, less then 30 minutes. My best friend at GA Tech was stripping her way through school as a math major. She had been a coke addict for 6 years with multiple hospitalizations. Her overall strategy was to be the biggest pain in the ass that office hours allowed. Profs, TAs, study groups, she extracted all she could from each. She graduated summa cum laude, and her first job was designing minefields for the US Navy. (:

[@jamieg2427]

so, she's a war criminal now? oof

[@jamieg2427]

good advice about previewing texts though. i usually do this on a per-chapter/section basis, but maybe i'll try it on a few textbooks in their entirety.

[@jamieg2427]

i don't know: many textbooks are 1000 pages. three minutes per such a text would be 1/5 of one second per page to look through the figures. how are you turning pages?? i just tried this on a small 300 page math text and it took me five minutes, and it was a pdf, so "turning pages" was faster than normal. 🤔 what

[@ardiris2715]

@@jamieg2427 Only if you are on the wrong side, sugar. (:

[@user-po2nq2kf5o]

Please guide me so that i can study mathematics atleast 2 hours daily

[@divyansh3_6_9]

How can a person can guide you to study Mathematics 2 hours daily?

[@TheMathSorcerer]

I will make a video on this!

[@user-po2nq2kf5o]

I'll be waiting 🙏

[@anniesizemore3344]

@@divyansh3_6_9 maybe he or she needs math book recommendations and recommendations about how to schedule study time.

[@bayzul7405]

​@anniesizemore3344 Entry level recommendations per level of math. A hardcore, in-depth book may dissuade those newly interested

[@nexusnexus9221]

LoL the proofs are in the appendix. Yes, I'm looking at appendices now.

[@TheMathSorcerer]

hehehehehe

[@dont-want-no-wrench]

it helps if you have six fingers

[@dominicdon6916]

Where do you get the thumbnail photos/paintings ? They don't look to be created by AI. And if they are, what AI are you using.

[@leeming1317]

6:45 LOL Math Sorcerer, That must be the final form for a math professor, You arms become so long so you don't need to drag the overhead whiteboard down/up.

[@charlesacaranci9039]

This guy is addicted to math in a good way !

[@highviewbarbell]

If you have an addictive personality anyway, and are finding it hard to quit something, just transfer the addiction to something like helping others, learning, or working

[@mochizzmo5266]

Real. There’s not much else to do but drugs here in Alberta😂

[@kristianpahe4639]

Wonder why your graduate classmate tried to make fun of you since you were right in pronouncing as REE-MANN instead of RAI-MANN. Was introduced to German a while back but never took it further. However using Forvo is fun in looking for pronunciation for a specific word.

[@joelhenderson3723]

English isn't your first language? You had me fooled!

[@TheMathSorcerer]

Yeah it's Spanish:)

[@marxkarx797]

"English wasn't my first language" *mind blown* Sir, your english is better than most of the english speaking people I've met.

[@dubdq3844]

I though he said it might be AI generated.

[@economicist2011]

@@dubdq3844 Math Sorcerer later (4:55) says in the video that English wasn't even *his* (i.e. the Math Sorcerer's) first language.

[@alanguages]

His first language is Spanish, as he was born in Cuba and moved to Florida afterwards.

[@dubdq3844]

@economicist2011 Yeah, sorry I am an idiot, my bad.

[@mrtienphysics666]

The amazing thing is almost all maths book has an appendix on set theory, and almost all physics books has an appendix on calculus. Yes, serious people study appendix.

[@davidgrenier]

Yeah, I've found most of my textbooks have an appendix with some of the core basic things you may have forgotten from earlier maths (or may never have learned). A lot of it is just memorizing formulas and such... great to add to a flashcard program if you use one.

[@mrtienphysics666]

The other approach which some authors use is: instead of an appendix, they have a introductory chapter on mathematical preliminaries.

[@matteogirelli1023]

Proofs by Jay Cummings is good and very cheap, the math sorcerer has reviewed it

[@vaimast2825]

With this video I feel like a true mathematician even if im not lol

[@Tateworld189]

🙏🙏 guru ji

[@Titanic_shirohige]

loving the ai thumbnails

[@TheMathSorcerer]

thank you!

[@aracelimercadofernandez9928]

Isn't English your first language? Which is it?

[@christressler3857]

Pretty sure it's pronounced like, e-ti-kit.

[@TheMathSorcerer]

Thank you:)

[@Haiderali-qd1jy]

Man you look like isaac Newton

[@sicko_the_ew]

Apropos the Pron: It's "Ree-man". "IE" - just like in "piece". A piece of Riemann. "Ry-man" would have the E before I. "Reiman". I can't think of an English word right now that has the EI pair. (If there is one, it's probably from German?). Zeitgeist? (Although the English version of that is painful to hear. :-) It's meant to sound like "Tsite-guy-st" but I think most people using it would say "Zzzite..." And they'd be rite, too. Because it's English, not German.)

[@skaus2184]

LaTeX

[@whatever_it_takes6691]

What are your thoughts on The Oofy Doofy Theory? Does the modern woman increasingly prefer the more submissive partner?

[@aidancollins7545]

From meth to math

[@english3082]

Please stop using those cheesy abhorrent AI generated images in your thumbnails.

[@iklides9709]

Why?

[@english3082]

​@@iklides9709Because they are over the top and contrived. Their aesthetics as it is today stand out a mile as cheesy and corny. They're becoming ubiquitous and convey the idea that the person making use of them has no personality, has poor sense of self-awareness, makes poor or questionable content, and just doesn't know better. It diminishes human dignity.

[@AR-rg2en]

@@english3082 I love it

[@AR-rg2en]

@@english3082 “Diminishes human dignity” now that’s ridiculous.

[@english3082]

@@AR-rg2en Although I think it does diminish human dignity in not one, but many senses, I understand that it may sound ridiculous. Maybe I shouldn't have written it here seemingly out of the blue.

[@Blend33]

Fortin sounds 👌 bo surprises. SDFX2!! 💯