Not a math major, but an engineer. Far fewer engineers take analysis than you might expect. It was almost certainly my favorite course. Everything about it is inherently rewarding in a way that most classes where you need to chase rote memorization to stay afloat just aren't. I think it was the only course I ever took that barely asked me to learn any "content" in the scheme of things but rather process as content was the name of the game.
Best answer: It depends on how hard your professor makes the class. If the assignments and exams are hard, then the class will be hard. My first group theory class was most certainly harder than my first analysis class.
@@kenfrank2730 i semi-agree. A good instructor can definitely helo but math is still 90% your own input. A good instructor alone is nowhere near enough for me personally, but a bad instructor is also not the end of the world.
Functional analysis, algebraic topology, differential geometry, algebraic number theory , analytic number theory, complex analysis, category theory, measure theory etc.
In Calculus there is usually a mismatch between the argument and the technique. Arguments that justify differentiation and integration are analytic as they are based on infinitary limiting processes, but the techniques to find derivative or to integrate are algebraic. Students are shielded from the messiness of limiting processes. As soon as they learn the algebraic manipulation they don't want to know the limiting processes (with some exceptions like epsilon-delta definition of limits where the argument and the technique align as they are both analytical). But when you get to the Analysis proper, there is no escaping the fact that you have to look under the hood and deal with estimates rather than exact formulas, with inequalities instead of equalities.
For a different approach to abstract algebra, try Introductory Modern Algebra: A Historical Approach by Saul Stahl. Instead of starting with abstract concepts, it starts with what people were investigating / trying to achieve. What problems were they grappling with and how did that lead to what we now call abstract algebra. For some, it might make a lot more sense than just being given definitions of algebraic structures out of the blue.
@@julianwilson9919 oh my Euler ... thank you very much again. That day I got sad because I couldn't find the book anywhere. I just saw your reply, tried again and found it!!! There are Excerpts about mathematicians!!! Yaaay! I don't know where I will start, but definitely not at page 1. I am a little wild. Hahahahahaha. I guess I will start at 12. Number Theory ... page 273 Thank you!
I took Calculus through Vector Calculus (because of Maxwell's Equations); it was certainly challenging, but I "got it". The "easy part" is that Calculus is very geometrical; one can draw pictures to visualize a situation. I took Abstract Algebra because I needed to learn Group Theory (for crystal symmetries); I never "got it". During many visits to my Professor's Office Hours, stuff didn't stick. He'd say, "That's a good example for a *representation* of Group G. Now generalize it." I hindsight, I didn't understand "The Proof Game" and the level of detail that I needed for my application can be found in many "Mathematical Methods for Physics" texts.
I found algebra much harder tbh. Grasping the ideas in algebra really took quite a lot of mental effort. Being able to apply the ideas to problems was hard to do creatively simply because understanding the ideas was difficult. Analysis on the other hand isn’t all that hard to understand. The problems can be difficult but you never feel lost reading through material
I feel like since you're often much more mathematically mature when you learn abstract algebra, you often learn it much more thoroughly. Calculus is often explained through intuitive arguments which makes rigorous proofs much harder
I found Abstract Algebra to be the harder of the two. I took Abstract Algebra as a junior in undergrad and took Advanced Calculus (Real Variables) as a senior. Granted, Abstract Algebra was my very first theory math class I ever took. I remember struggling a bit in Abstract Algebra I (Sets, Groups, and 1st part of Rings) and got my first and only non 'A' grade in a math class in undergrad. In Abstract Algebra II (rest of Rings, Fields, and other special topics,) I did just fine and got an 'A'. When I got to grad school, I avoided taking Abstract Algebra related courses except for one. I did take a course in Group Representation Theory in grad school, which I found to be quite interesting. In Grad School, I mostly took courses that were Calculus/Analysis related (Complex Analysis, Theory of Ordinary Differential Equations, Numerical Analysis, etc.) I also did take a full year of Mathematical Statistics in addition to other topics that interested me (Applied Partial Differential Equations, Mathematical Programming (Linear and Non-linear,) Graph Theory, Combinatorics, Applied Probability, Regression Analysis, Applied Linear Algebra, and maybe a few others I am forgetting.) Edit: Forgot to mention, when I took Abstract Algebra, we used the classic book by I.N. Herstein "Topics In Algebra." That is a very condensed book that covers a lot. Very hard book for first timers taking a theory course. I still have that book to this day, along with all my other math books from undergrad and grad school. This coming fall, will be the 40th anniversary of me starting undergrad. Time sure flies. Hard to believe its been 40 years already since I graduated from High School and began my college journey.
@@kenfrank2730 While I was in Grad School I also had to teach a class each term, usually Calculus or Introductory Calculus (more casual for non engineering/math science majors.) After I got my Masters in Math, I took a year off and then went back to get my Masters in Computer Science, while also teaching introductory programming classes. After I received my Masters in Computer Science, I started working as a Software Engineer.
Le-see ... 2024 - 1981 = 43. WHAT?!?! That can't be right! I kept nearly all of my Physics and Math library from when I was a young, sexy Physics Major. Well, young, at least. Still use them.
It always comes down to the teacher and their ability to convey the course material to the student. The class raw average, shows the quality of the teacher teaching, as well as the student learning I was always a proponent of a 2 course, six week program, instead of a 5 or 6 different courses alternating during the day. That allows more focused study and homework, and feedback to the teacher.
The best thing about reading proofs is that you can understand the mind of the person who wrote the proof. How did he come to this conclusion? etc. There are conclusions that are very easy to understand and other conclusions that may take days to confirm. A persons mind is reflected in his proof style. Have a nice day : )
Analysis is easier than abstract algebra, but most universities it is clear that they rush abstract algebra so much that they only touch the surface, and thus making it easier on surface level. In my opinion and experience ofc
Personally, I like that book for abstract algebra. However, in my opinion, for introductory level, the book by Charles C. Pinter could be the best, especially for self-learning. Thanks for the video❤
For me algebra is definitely easier to start learning. I actually know more analysis because I knew it was harder for me. I started with just some basics and jumped into measure theory so undergraduate analysis would seem easier. Weird approach but actually kinda worked out
I failed my first course in abstract algebra, not because I didn't study but because it was hard for me to disprove and prove statements when it came to the tests. I like applied mathematics to be honest like ODE/PDE's, numerical analysis, combinatorics. Abstract algebra due to my learning disability was a lot tougher to grasp concepts and use definitions and theorems. I would try to prove theorems from the book we used and I would only get 2 lines of the proof correct.
IMHO abstract algebra as a whole is way harder than calculus. Because the amount of abstraction involved. It's not about the formula, but the actully imagining the algebric structures in the mind is the real challenge.
Depends the person, book and sometimes even the moment that the person is studying. For example, im undergraduate student in physics. And im doing scientific iniciation(i dont know if there exists it or not ...or if its other name) in math(specifically analysis 1) and im currently using rudin's book(because after reading some part of some books, rudin's and tao's book are the type of books that i like for this part) and yeah... really tough book, im at same time learning how to write proofs with a proof writting book. I really dont know if im an analysis person or algebra person
If the abstract algebra professor decides to go into group actions, semidirect product and the free group, you'll be dealing with a pretty difficult class. But most professors generally don't go there in a first course in abstract algebra, so people get the impression that abstract algebra is easy compared to analysis.
There was no course on proofs in my Math department so that made Real Analysis, and all my other advanced Math courses, harder than they should have been. Real Analysis was harder, if for no other reason than that was the first course I took that was proof based.
My Daughter is a Senior at her University. Dual majoring in Biology and Mathematics. She is taking abstract algebra. I've watched several videos trying to get a grasp of it. I have no clue!
For me: I don't really care about polynomials. The only thing I like about polynomials is that they are easy to differentiate. So I took the one Algebra class I needed and I really tried to give abstract algebra a shot, it just wasn't for me so I ended up dropping the class.
What's the point of doing math if someone looks the answers up? But those Humongous book of math answers book if you want to see the answers without working it out. I'd still suggest going through the motions of figuring the problem out even if you can see the solution first. I hardly look at the answer before doing the problem. If its something I really can't see a path through, I will look the answer up. Most of the time if I can see the answer to a math problem I can figure out on my own how they go the answer. I view doing math like the most fun game of all. Figuring it out on my own is the fun.
Teaching myself abstract algebra right now. Vector calculus was way more difficult for me, but that's because I lack a good foundation in geometry/trigonometry(polar coordinates specifically). If it weren't for the poor foundation in those topics, vector calculus would be much more equal to abstract algebra. The only difficulty in abstract algebra so far is proof writing, and that's mainly because I'm trying to learn the language used within the field of math to describe the concepts, which is an easy fix. I also come from a psychology/philosophy background, so abstract algebra is very much like metaphysics, and easier to navigate for me because of my experience(the "abstractness" is very comfortable to me).
@@stevematson4808 There is nothing that is called "abstract" calculus. To be honest, you can interpret that in a few ways. "Abstract" analysis can be seen as studying topology. But you can in fact do calculus on more abstract surfaces, instead of just lines and planes.
Abstract Algebra and Advanced Calculus are two easy subjects. It's not that the subjects are difficult, it's something else. To clarify the problem, I'll show you an example: "Suppose 300 electrons had to move from one energy level to another in several groups. But the quantum transition was made by the number of groups two less, so each group included 5 more electrons. What is the number of electron groups? This is a difficult problem still unsolved." The listeners-highly qualified specialists-stated that they would not undertake to solve such a hard problem. Then they were given the following problem: "Several buses were ordered to send 300 schoolchildren to a camp, but since two buses did not arrive by the appointed time, 5 more schoolchildren than expected were put on each bus. How many buses were ordered?" The problem was solved instantly! The two problems are identical. It's not about the class, it's about something else.
Not a math major, but an engineer. Far fewer engineers take analysis than you might expect. It was almost certainly my favorite course. Everything about it is inherently rewarding in a way that most classes where you need to chase rote memorization to stay afloat just aren't. I think it was the only course I ever took that barely asked me to learn any "content" in the scheme of things but rather process as content was the name of the game.
Most engineering students will struggle with taking either of the two classes.
Engineers are math majors at heart
Best answer: It depends on how hard your professor makes the class. If the assignments and exams are hard, then the class will be hard. My first group theory class was most certainly harder than my first analysis class.
yeah that makes a HUGE difference!!
Fa sho
Especially when the professor says you don't need a book for the class lol
@@kenfrank2730 i semi-agree. A good instructor can definitely helo but math is still 90% your own input. A good instructor alone is nowhere near enough for me personally, but a bad instructor is also not the end of the world.
Functional analysis, algebraic topology, differential geometry, algebraic number theory , analytic number theory, complex analysis, category theory, measure theory etc.
In Calculus there is usually a mismatch between the argument and the technique. Arguments that justify differentiation and integration are analytic as they are based on infinitary limiting processes, but the techniques to find derivative or to integrate are algebraic. Students are shielded from the messiness of limiting processes. As soon as they learn the algebraic manipulation they don't want to know the limiting processes (with some exceptions like epsilon-delta definition of limits where the argument and the technique align as they are both analytical). But when you get to the Analysis proper, there is no escaping the fact that you have to look under the hood and deal with estimates rather than exact formulas, with inequalities instead of equalities.
For a different approach to abstract algebra, try Introductory Modern Algebra: A Historical Approach by Saul Stahl. Instead of starting with abstract concepts, it starts with what people were investigating / trying to achieve. What problems were they grappling with and how did that lead to what we now call abstract algebra. For some, it might make a lot more sense than just being given definitions of algebraic structures out of the blue.
Oooh really??? That seems interesting ... Will check your recommendation right now.
Got a 1st edition of the book for $10! Cool!
@@samueldeandrade8535 If you like it (or not), feel free to post an update here.
@@ladyoftheflowers9781 That's cool! What do you think of it?
@@julianwilson9919 oh my Euler ... thank you very much again. That day I got sad because I couldn't find the book anywhere. I just saw your reply, tried again and found it!!! There are Excerpts about mathematicians!!! Yaaay! I don't know where I will start, but definitely not at page 1. I am a little wild. Hahahahahaha.
I guess I will start at
12. Number Theory ... page 273
Thank you!
You really are a sorcerer. You always post when the question pops up in my head. So many videos you've done this... ya damn WIZARD
I took Calculus through Vector Calculus (because of Maxwell's Equations); it was certainly challenging, but I "got it". The "easy part" is that Calculus is very geometrical; one can draw pictures to visualize a situation.
I took Abstract Algebra because I needed to learn Group Theory (for crystal symmetries); I never "got it". During many visits to my Professor's Office Hours, stuff didn't stick. He'd say, "That's a good example for a *representation* of Group G. Now generalize it." I hindsight, I didn't understand "The Proof Game" and the level of detail that I needed for my application can be found in many "Mathematical Methods for Physics" texts.
I found algebra much harder tbh. Grasping the ideas in algebra really took quite a lot of mental effort. Being able to apply the ideas to problems was hard to do creatively simply because understanding the ideas was difficult. Analysis on the other hand isn’t all that hard to understand. The problems can be difficult but you never feel lost reading through material
I feel like since you're often much more mathematically mature when you learn abstract algebra, you often learn it much more thoroughly. Calculus is often explained through intuitive arguments which makes rigorous proofs much harder
I found Abstract Algebra to be the harder of the two. I took Abstract Algebra as a junior in undergrad and took Advanced Calculus (Real Variables) as a senior. Granted, Abstract Algebra was my very first theory math class I ever took. I remember struggling a bit in Abstract Algebra I (Sets, Groups, and 1st part of Rings) and got my first and only non 'A' grade in a math class in undergrad. In Abstract Algebra II (rest of Rings, Fields, and other special topics,) I did just fine and got an 'A'. When I got to grad school, I avoided taking Abstract Algebra related courses except for one. I did take a course in Group Representation Theory in grad school, which I found to be quite interesting. In Grad School, I mostly took courses that were Calculus/Analysis related (Complex Analysis, Theory of Ordinary Differential Equations, Numerical Analysis, etc.) I also did take a full year of Mathematical Statistics in addition to other topics that interested me (Applied Partial Differential Equations, Mathematical Programming (Linear and Non-linear,) Graph Theory, Combinatorics, Applied Probability, Regression Analysis, Applied Linear Algebra, and maybe a few others I am forgetting.)
Edit:
Forgot to mention, when I took Abstract Algebra, we used the classic book by I.N. Herstein "Topics In Algebra." That is a very condensed book that covers a lot. Very hard book for first timers taking a theory course. I still have that book to this day, along with all my other math books from undergrad and grad school. This coming fall, will be the 40th anniversary of me starting undergrad. Time sure flies. Hard to believe its been 40 years already since I graduated from High School and began my college journey.
@@kenfrank2730 Exactly what I wanted to ask!
@@kenfrank2730 While I was in Grad School I also had to teach a class each term, usually Calculus or Introductory Calculus (more casual for non engineering/math science majors.) After I got my Masters in Math, I took a year off and then went back to get my Masters in Computer Science, while also teaching introductory programming classes. After I received my Masters in Computer Science, I started working as a Software Engineer.
Le-see ... 2024 - 1981 = 43. WHAT?!?! That can't be right!
I kept nearly all of my Physics and Math library from when I was a young, sexy Physics Major. Well, young, at least. Still use them.
It always comes down to the teacher and their ability to convey the course material to the student.
The class raw average, shows the quality of the teacher teaching, as well as the student learning
I was always a proponent of a 2 course, six week program, instead of a 5 or 6 different courses alternating during the day. That allows more focused study and homework, and feedback to the teacher.
The best thing about reading proofs is that you can understand the mind of the person who wrote the proof. How did he come to this conclusion? etc. There are conclusions that are very easy to understand and other conclusions that may take days to confirm. A persons mind is reflected in his proof style. Have a nice day : )
Love your English and Spanish channel! Gracias hechicero!
Analysis is easier than abstract algebra, but most universities it is clear that they rush abstract algebra so much that they only touch the surface, and thus making it easier on surface level. In my opinion and experience ofc
Yeah both CAN be very challenging but usually analysis is a harder class.
Personally, I like that book for abstract algebra. However, in my opinion, for introductory level, the book by Charles C. Pinter could be the best, especially for self-learning.
Thanks for the video❤
For me algebra is definitely easier to start learning. I actually know more analysis because I knew it was harder for me. I started with just some basics and jumped into measure theory so undergraduate analysis would seem easier. Weird approach but actually kinda worked out
I failed my first course in abstract algebra, not because I didn't study but because it was hard for me to disprove and prove statements when it came to the tests. I like applied mathematics to be honest like ODE/PDE's, numerical analysis, combinatorics. Abstract algebra due to my learning disability was a lot tougher to grasp concepts and use definitions and theorems. I would try to prove theorems from the book we used and I would only get 2 lines of the proof correct.
IMHO abstract algebra as a whole is way harder than calculus.
Because the amount of abstraction involved.
It's not about the formula, but the actully imagining the algebric structures in the mind is the real challenge.
Personally I find analysis harder than algebra. Algebra is more vocab and random factoids whereas analysis I find the proofs more challenging.
Depends the person, book and sometimes even the moment that the person is studying. For example, im undergraduate student in physics. And im doing scientific iniciation(i dont know if there exists it or not ...or if its other name) in math(specifically analysis 1) and im currently using rudin's book(because after reading some part of some books, rudin's and tao's book are the type of books that i like for this part) and yeah... really tough book, im at same time learning how to write proofs with a proof writting book. I really dont know if im an analysis person or algebra person
If the abstract algebra professor decides to go into group actions, semidirect product and the free group, you'll be dealing with a pretty difficult class. But most professors generally don't go there in a first course in abstract algebra, so people get the impression that abstract algebra is easy compared to analysis.
Abstract Algebra is harder in my view but no one should feel intimidated by how other people perceive a subject
While I was an electrical engineering student I took advanced calculus from Apostol's book, Mathematical Analysis. It was an attention getter.
Which is the harder exercise? Lat pulldowns or machine rows?
Thanks for the video.
You are welcome!
@@TheMathSorcerer what are the Prerequisites of Differential Geometry?
There was no course on proofs in my Math department so that made Real Analysis, and all my other advanced Math courses, harder than they should have been. Real Analysis was harder, if for no other reason than that was the first course I took that was proof based.
Do you have a video with the "greatest hits" of well written math books?
Nice camera upgrade man, watching your content for quite some time now
My Daughter is a Senior at her University. Dual majoring in Biology and Mathematics. She is taking abstract algebra. I've watched several videos trying to get a grasp of it. I have no clue!
boy, do you keep me hitting the books!
Abstract algebra is more interesting for me so it is easier
I got a C in my first Analysis class and an A in my first Abstract Algebra class.
Inspiring video many thanks
For me: I don't really care about polynomials. The only thing I like about polynomials is that they are easy to differentiate.
So I took the one Algebra class I needed and I really tried to give abstract algebra a shot, it just wasn't for me so I ended up dropping the class.
Is it better to study from a real analysis textbook or advanced calculus textbook? I know it’s the same subject but the textbooks seem different
I definitely think abstract algebra is much harder. Still, I prefer it to analysis at any level.
What's the point of doing math if someone looks the answers up? But those Humongous book of math answers book if you want to see the answers without working it out. I'd still suggest going through the motions of figuring the problem out even if you can see the solution first. I hardly look at the answer before doing the problem. If its something I really can't see a path through, I will look the answer up. Most of the time if I can see the answer to a math problem I can figure out on my own how they go the answer. I view doing math like the most fun game of all. Figuring it out on my own is the fun.
having the answers is so you can check your work
@@highviewbarbell I thought he said some people looked up answers before the finished the problem. sorry if I was wrong
I find analysis much harder than algebra
Teaching myself abstract algebra right now. Vector calculus was way more difficult for me, but that's because I lack a good foundation in geometry/trigonometry(polar coordinates specifically). If it weren't for the poor foundation in those topics, vector calculus would be much more equal to abstract algebra. The only difficulty in abstract algebra so far is proof writing, and that's mainly because I'm trying to learn the language used within the field of math to describe the concepts, which is an easy fix. I also come from a psychology/philosophy background, so abstract algebra is very much like metaphysics, and easier to navigate for me because of my experience(the "abstractness" is very comfortable to me).
Too lighten things up did you see Terrence howard creating a proof for Neil degrasse Tyson with physics and mathematics.😊
What about abstract calculus?
What do you mean by abstract calculus? Manifold theory?
@@ClumpypooCP I don't know, but it has to be out there.
@@stevematson4808 There is nothing that is called "abstract" calculus. To be honest, you can interpret that in a few ways. "Abstract" analysis can be seen as studying topology. But you can in fact do calculus on more abstract surfaces, instead of just lines and planes.
@@ClumpypooCPHe is making what is commonly referred to as a "joke"
@@madallas_mons hahah nice try bud but no, he is not making a joke.
Jokes on your the Harder math class is the one you Failed😂
Make a video about your thiccest books plss
maybe he changed his major because he thought it was too easy and he was bored LOL
Abstract Algebra and Advanced Calculus are two easy subjects. It's not that the subjects are difficult, it's something else. To clarify the problem, I'll show you an example:
"Suppose 300 electrons had to move from one energy level to another in several groups. But the quantum transition was made by the number of groups two less, so each group included 5 more electrons. What is the number of electron groups? This is a difficult problem still unsolved."
The listeners-highly qualified specialists-stated that they would not undertake to solve such a hard problem. Then they were given the following problem:
"Several buses were ordered to send 300 schoolchildren to a camp, but since two buses did not arrive by the appointed time, 5 more schoolchildren than expected were put on each bus. How many buses were ordered?"
The problem was solved instantly! The two problems are identical.
It's not about the class, it's about something else.