What is the most difficult mathematics?

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In summary, the conversation discusses the most difficult subject in mathematics according to experienced mathematicians. The concepts of Advanced Calculus, algebra, and mindless manipulation of symbols are mentioned as potential candidates. However, it is also noted that what is considered difficult may vary depending on an individual's background and experience in different areas of mathematics. It is also suggested that the difficulty of a subject may decrease once it is understood and mastered.
  • #1
silverdiesel
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I am about a year into an undergraduate degree in Physics and I am wondering what might lay ahead. What do the more experienced mathematicians think is the most difficult subject?

a couple things happended recently that made me ponder the subject.

- My calc2 professor was lecturing on applicaitons of the deginate integral, specifically in regard to the work function. In doing so, he admitted to the class that this was his least favorite lecture becuase he was not as comfortable with the Physics. I thought it was an odd coment becuase he seems like the most intelligent professor I have ever had. W=Fd is so simple, how could that seem difficult to a guy that knows all the ins and out of calculus?

- I was reading an interview of a Mathematic PHD. He was asked what he thought was the most difficult mathematics. "Definetly Advanced Calculus" was his reply. I had always just assumed the math gets more and more difficult as you progress. I may be showing how green I am, but what is 'Advanced Calculus'? Is that Calc 3? Or, are there more higher level calculus classes? If calc3 is as hard as it gets, that does not seem too difficult.

Honestly, I really enjoy calculus -when you really get to apply it, as in optimization and applications of the definate integral. They just make a lot of sence. I enjoy them because it is like writing an essay, except in the most efficiant of language. I like applying the concepts of calculus, rate of change and the limit of summations. You can spend all this time in math playing logic games, but what is the point unless you can use it to tell you something about the world. I'll work all day trying to figure out a problem if the answer will actually tell me something interesting like how much work is required to move an object.

It is the nitty gritty algebra at the end that always causes me trouble. Algebra is what I would consider the most difficult. My physics professor is always setting up problems for us, and then saying "the rest is just algebra, and if you can't do that, you should not be in this class" Which is true, no doubt, but I don't like the implication that it is "just algebra". Algebra can be a major in pain in the arse.

Anyway, just curious, what someone with more experience might think.
 
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  • #2
The most difficult mathematics is that which you do not know.

A surprising amount of mathematics is actually easy once you've learned it. Of course, once you learn the easy stuff, then you have to start tacking the deep stuff, and that gets harder. :smile:

One teacher I had was introducing a new concept, and we did an example in class. (and this was a class for good mathematicians -- not your average students) There was a lot of blank stares, and not everybody seemed to follow all the way through.

The very next thing he asked was for us to differentiate the function x² with respect to x. Of course, everybody could do that very easily.

His response? "The reason you can do differentiation, but not the other thing, is that you've differentiated things hundreds of times, but you haven't done this other thing very much yet."
 
  • #3
silverdiesel said:
My physics professor is always setting up problems for us, and then saying "the rest is just algebra, and if you can't do that, you should not be in this class" Which is true, no doubt, but I don't like the implication that it is "just algebra". Algebra can be a major in pain in the arse.

Which is exactly why he doesn't want to do it. I would like to clarify that when you say algebra, you are not for instance denigrating group theory etc, but you are referring to the mindless manipulation of symbols such as as simplifying an equation. Now, for my money, a better name for it is 'bookkeeping'. It requires no intellect just the ability to follow a simple set of rules (which actually, is like a lot of maths apart from the simple part).

If for instance I were to take such a class and write on the board ...=42/64, I would leave it as that and would be mightily annoyed if any student pointed out that that is the same as 21/32 since that shows that they're focusing on the wrong thing.

Mathematics is such a huge subject with so many opinions you're not going to get a simple answer. Perhaps a more reasonable question would be: what is the hardest part of mathematics that I'm likely to need to master?
 
  • #4
Hurkyl said:
The most difficult mathematics is that which you do not know.

A surprising amount of mathematics is actually easy once you've learned it. Of course, once you learn the easy stuff, then you have to start tacking the deep stuff, and that gets harder. :smile:

yeah that sounds right. it doesn't matter what part of math you study, there will always be pages in a textbook that take a solid day or two to really understand. i guess it could be slightly easier for someone to study a subject & then study a subject that is relatively close to it. like some sort of algebraist might not have as much trouble working on some other kind of algebra because of their background. it would probably be harder for an analyst to start working on graph theory because they don't have a lot to do with each other.
 
  • #5
matt grime said:
and would be mightily annoyed if any student pointed out that that is the same as 21/32 since that shows that they're focusing on the wrong thing.
Yet there always seems to be such a person in the lecture theatre... :cry:
fourier jr said:
it would probably be harder for an analyst to start working on graph theory because they don't have a lot to do with each other.
Also bear in mind that some different areas of maths require different ways of thinking, so to someone whose good at one area, it might take an inordinately long time to get as good in another area, if even possible at all (though that depends on what you might consider 'good').
 
  • #6
It also depends on your "personality". Some analysts loathe discrete mathematics and would gladly not learn anything about it, and vice versa.
 
  • #7
Here's a related question: what is the mathematics that depends on the most other mathematics?
 
  • #8
I've had a batch of math classes, and so far, the most difficult IMO is differential equations. Lots of plug-and-chunk, and that's the problem for me: most of the time I don't know where to plug things.
 
  • #9
arildno said:
It also depends on your "personality". Some analysts loathe discrete mathematics and would gladly not learn anything about it, and vice versa.

Exactly.

It all depends on what you like.

If you truly hate it, well certainly it's going to become difficult after awhile. You'll never give it some thought because you hate it.
 
  • #10
difficulty is relative to the individual. for me analysis is the most difficult and the easiest is geometry topology, and in between is algebra.

but complex analysis is to me easier than real analysis. and so on...:wink:
 
  • #11
mathwonk said:
difficulty is relative to the individual. for me analysis is the most difficult and the easiest is geometry topology, and in between is algebra.

but complex analysis is to me easier than real analysis. and so on...:wink:

I can't get anywhere with Complex Analysis right now. Maybe it's too early to tell.

I've been asking for a good book for awhile now. Something that is not too rigorous though.

Sure, I might do well in the course, but that means nothing to me if I don't know what's going on.

We don't have a textbook in our class, and we seem to be jumping all over the place. A nice thorough textbook would be what I'm looking for. I want a good one. I've seen free ones and cheap ones, but there are reasons for them being free and cheap. They aren't very good.
 
  • #12
mathwonk said:
but complex analysis is to me easier than real analysis. and so on...:wink:

I'll agree with you on that count because I'm taking a complex analysis course right now and I'm really starting to enjoy it, and I've also tried to teach myself bits of real analysis but have been having some problems but in complex analysis I'm starting to make connections with other branches of mathematics and everything seems to be coming together for me.
 
  • #13
JasonRox said:
A nice thorough textbook would be what I'm looking for. I want a good one. I've seen free ones and cheap ones, but there are reasons for them being free and cheap. They aren't very good.

I take it you didn't like the one in french I referred to you. I agree, it stinks. The proofs are not easy to follow and his definitions are scatered randomly throughout the text. But what do you mean by
we seem to be jumping all over the place
? Give an exemple.
 
  • #14
quasar987 said:
I take it you didn't like the one in french I referred to you. I agree, it stinks. The proofs are not easy to follow and his definitions are scatered randomly throughout the text. But what do you mean by
? Give an exemple.

Yeah, even though I understand French, it still becomes hard to follow. I'm not the best in French, but I do know lots.

We jump around in the sense that we don't know where we are going or heading.

We didn't have a course outline either, so that doesn't help either.
 
  • #15
Math/formal logic=by the far hardest math course I have ever taken. You really have to think way far outside of the box to follow what is going on in math logic. Proving godel's theorems and learning recursion theory was the most challenging thing I have ever learned in my entire life. Next to logic, learning about Hilbert Spaces was also very hard, but not as bad as logic.

Honestly, I really enjoy calculus

we'll see if you say this after you suffer through advanced calc. :biggrin:
 
  • #16
Speaking from what little I have done, I found algebra hard, specifically just logrithms. Took me weeks to realize how change of base worked...well not really but you get the idea. I just finished roots of complex numbers using DeMoivre's Theorem (begining trig).

Speaking from what I have heard, everyone says Cal II is a (certain inappropriate word that starts with a capital "B"). All I hear is 'Cal III is some much easier than Cal II, what a "B" it was'.
 
  • #17
Cal III is easier mainly because by the time you get there, you are used to integration and differentiation. Cal III doesn't teach anything conceptually new, unlike cal 1 and 2.
 
  • #18
The most difficult maths is the one you haven't learn and you are not going to learn...so, learn more practise more, and all will be clear...
 
  • #19
Treadstone 71 said:
Cal III is easier mainly because by the time you get there, you are used to integration and differentiation. Cal III doesn't teach anything conceptually new, unlike cal 1 and 2.
Depends on who is teaching it... my Calc III class was one of the most difficult classes I have ever taken. In addition to the easy stuff it covered curl & divergence, Stokes theorem, Green's theorem, the divergence theorem, with an emphasis on proving things. I got an A but just barely, and it wasn't for lack of effort.
 
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  • #20
Calc II is the most difficult of the 3 to some because practically every day something new is introduced. Memory is very important in this class. Calc III also has some new concepts but a lot of it is based upon what you slaved over in calc I & II. I tutored many in calc III and found that the only people who suffered through the class were those who could not live without their Ti-89, graph 3d-functions or barely made it through calcII.
 
  • #21
0rthodontist said:
In addition to the easy stuff it covered curl & divergence, Stokes theorem, Green's theorem, the divergence theorem, with an emphasis on proving things.

But those are the easy things in calculus, surely? I'm guessing you weren't doing this on an arbitrary manifold, but in R^3
 
  • #22
But those are the easy things in calculus, surely? I'm guessing you weren't doing this on an arbitrary manifold, but in R^3
Yes... it was still hard. One thing that didn't help is that the first half of the course was spent studying discrete math and linear algebra type material that I already knew, so that the rest of the course was additionally compressed. It was also the first time the professor had taught the course. The homework assignments typically took me eight or ten hours.

Incidentally I still don't _really_ understand divergence and curl... the theorems prove what they mean and I can look at special cases like restricting a function to a plane where it makes sense but just from looking at div cross F I still don't see any intuitive clue as to why that should measure curl, or why div dot F should measure divergence.

daveyp225 said:
Calc II is the most difficult of the 3 to some because practically every day something new is introduced. Memory is very important in this class. Calc III also has some new concepts but a lot of it is based upon what you slaved over in calc I & II. I tutored many in calc III and found that the only people who suffered through the class were those who could not live without their Ti-89, graph 3d-functions or barely made it through calcII.
Well, I did an independent study for Calc II in high school and got exemption through the Calc AB & BC exams (took both), but I also had a yearlong course in mathematical statistics last year which was a refresher. I also found that difficult but mainly because it required a fair amount of Calc III material before I had taken Calc III. The problem in Calc III was not my Calc II skills. Much of the time Calc II didn't even seem all that relevant, besides basic integration and concepts of area and volume.

I did get an A, anyway.
 
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  • #23
0rthodontist said:
The problem in Calc III was not my Calc II skills. Much of the time Calc II didn't even seem all that relevant, besides basic integration and concepts of area and volume.

Basic integration as learned through Calc I? I guess it also depends on who is teaching it and what text you are using. Many of the 3-dimentional application problems (double/triple integrals) in my text require things such as trig substitution, change of variables, integration by parts, etc. All things I was taught in Calc II. Unless you meant this as basic integration?

In other ways besides integration techniques, Calc II can be considered a perparation for Calc III in that you do many application problems involving surface areas, volumes and the like.
 
  • #24
Well, I'm not totally clear on the distinction between Calc I and Calc II. But the focus in this course was not on techniques of integration very much. 95% of the integrals were just polynomials or sums of e^x or sin x, cos x type stuff. The focus in this course was on understanding and proof. "Focus on understanding" makes it sound like a concept course for biology majors or something... don't think that.

In fact I think that if I had the second part of that Calc III course to take over again, it wouldn't be a waste of time. I am taking Calc IV now and it is a month into the semester but we have only introduced one topic that wasn't done in Calc III (the frenet frame). Hopefully the pace will increase.

Surface areas and volumes as done in single variable calculus didn't seem all that relevant in calc III.
 
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  • #25
My calc III class started by stating the l-p norms and using them in proofs. We then moved on to proving everything we had ever done in calculus, and finally did the multivar / vector calc in the last 3 weeks.

It definitely wasnt easy.
 
  • #26
0rthodontist said:
Well, I'm not totally clear on the distinction between Calc I and Calc II.

possibly because, as I keep having to remind everyone it seems, there is no such thing as a universal bloody syllabus. For instance, in my experience Calc III (third course in calculus at university) ought to mean measure theory at the very least. Teichmuller spaces, or the Rietz representation theorem for Hilbert Spaces would also be acceptable, as would kernels and that theorem on when a set of functions is a basis (contain the constant function and separate points... can't recall the name, bloody obvious though).
 
  • #27
0rthodontist said:
Surface areas and volumes as done in single variable calculus didn't seem all that relevant in calc III.

Many of the formulas in Calc III can use single-variable integrations to derive multiple variable integrations.

For a simple example,

[tex]\int\limits_{x1}^{x2}\int\limits_{g(x)}^{f(x)}dA[/tex] is the same thing as [tex]\int\limits_{x1}^{x2}[f(x)-g(x)]dx[/tex]
 
  • #28
matt grime said:
(contain the constant function and separate points... can't recall the name, bloody obvious though).

Stone-Weierstrass I believe. I thought everyone learned about it in Math 433, you didn't? Seriously, I wish I could explain why people think there are universal courses and naming schemes.

Multivariable calculus, including greens theorem and "classical" stokes theorem, was covered in my 3rd term of my calculus stream. I didn't find it hard to work with but can't say I understood what was going on (with stokes) until I saw the more general version in a graduate differential topology course (shamefully where I first met differential forms as well). I'd say the most difficult mathematics as a student is the stuff you only know part of the story for.
 
  • #29
I recently finished a Mathematics BSc at a UK university. I think really there are a few things that affect how difficult you find the subject.

(i) Your depth of experience in that topic area previous to that module
(ii) The quality of the lecturing and if the way the lecturer communicates the material suits you.
(iii) Your natural aptitude for that area of Mathematics

I was not that good at Stats initially but I put that down to my lack of statistical training at that point in time.

I found Classical Mechanics really difficult as I am pretty mediocre at that area of maths. I found calculus pretty straight forward at all levels. However I think the area that I have heard a few mathematicians complain about is Real Analysis. Our Real Analysis was split into two modules and I recorded my lowest two marks for the whole degree in these modules.
 
  • #30
It depends on you. I'm not a mathematician (yet, knock on wood), but the hardest thing I've encountered is differential equations. And the easiest thing is set theory! Most people find it backwards.
 
  • #31
meh none of it is difficult to learn. it's a whole other matter to contribute.
 
  • #32
The more applied it is the more difficulties I have. I'm a "concepts person" and find it way easier to prove the general form of the implicit function theorem than actually realizing it on some particular system of equations. Let alone linear algebra, where the concepts usually are very intuitive but the details are so immense with all summations and indexes that seem to beg for getting mixed up.
 
  • #33
Hurkyl said:
The most difficult mathematics is that which you do not know.

A surprising amount of mathematics is actually easy once you've learned it. Of course, once you learn the easy stuff, then you have to start tacking the deep stuff, and that gets harder. :smile:

One teacher I had was introducing a new concept, and we did an example in class. (and this was a class for good mathematicians -- not your average students) There was a lot of blank stares, and not everybody seemed to follow all the way through.

The very next thing he asked was for us to differentiate the function x² with respect to x. Of course, everybody could do that very easily.

His response? "The reason you can do differentiation, but not the other thing, is that you've differentiated things hundreds of times, but you haven't done this other thing very much yet."

is the other thing integrals :P
 
  • #34
I have not taken college courses in about a decade or two. Reading the post makes me wonder what people are thinking around here.

It seems obvious that the age old question of nature v. nurture would come into play.
Depending on who YOU are what YOUR background is, you may find one class easier or more difficult than others. Also, it should be obvious that the teacher and class has a huge impact. I can tell you that my Cal III and IV professor was more difficult at that level than Advance Algebra taught by others. He boasted a class average of "F".

I miss school. You guys need to hurry up and solve some of the worlds problems.. bust ***!

H
 
  • #35
My girlfriend in college was a math major, and my roommate was majoring in the humanities. When my roommate made a comment over lunch one day that the liberal arts were far more intellectually demanding than math, my girlfriend piped up. "Exactly how much mathematics have you taken," she asked.

"Well, you know. Calculus," my roommate said.

"Ah," my girlfriend replied. "In other words, none to speak of."
 
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<h2>1. What is the most difficult branch of mathematics?</h2><p>The most difficult branch of mathematics is subjective and can vary depending on individual strengths and weaknesses. Some may find abstract algebra or topology to be the most challenging, while others may struggle with differential equations or calculus.</p><h2>2. What makes a mathematical concept difficult?</h2><p>A mathematical concept can be considered difficult if it requires a high level of abstract thinking, involves complex calculations, or has counterintuitive properties. It can also be challenging if it builds upon previously learned concepts and requires a deep understanding of multiple mathematical principles.</p><h2>3. Is there a limit to how difficult mathematics can get?</h2><p>There is no limit to how difficult mathematics can get. As our understanding of the universe expands, new mathematical concepts and theories are constantly being developed to explain and describe it. This means that there will always be new and challenging mathematical concepts to explore and understand.</p><h2>4. What are some strategies for tackling difficult mathematical problems?</h2><p>Some strategies for tackling difficult mathematical problems include breaking the problem down into smaller, more manageable parts, using visual aids or diagrams, seeking help from peers or instructors, and practicing regularly. It is also important to have a strong foundation in basic mathematical principles.</p><h2>5. Can anyone learn and understand difficult mathematics?</h2><p>Yes, anyone can learn and understand difficult mathematics with dedication, practice, and a willingness to learn. While some individuals may have a natural aptitude for math, it is ultimately a skill that can be developed and improved upon through hard work and perseverance.</p>

1. What is the most difficult branch of mathematics?

The most difficult branch of mathematics is subjective and can vary depending on individual strengths and weaknesses. Some may find abstract algebra or topology to be the most challenging, while others may struggle with differential equations or calculus.

2. What makes a mathematical concept difficult?

A mathematical concept can be considered difficult if it requires a high level of abstract thinking, involves complex calculations, or has counterintuitive properties. It can also be challenging if it builds upon previously learned concepts and requires a deep understanding of multiple mathematical principles.

3. Is there a limit to how difficult mathematics can get?

There is no limit to how difficult mathematics can get. As our understanding of the universe expands, new mathematical concepts and theories are constantly being developed to explain and describe it. This means that there will always be new and challenging mathematical concepts to explore and understand.

4. What are some strategies for tackling difficult mathematical problems?

Some strategies for tackling difficult mathematical problems include breaking the problem down into smaller, more manageable parts, using visual aids or diagrams, seeking help from peers or instructors, and practicing regularly. It is also important to have a strong foundation in basic mathematical principles.

5. Can anyone learn and understand difficult mathematics?

Yes, anyone can learn and understand difficult mathematics with dedication, practice, and a willingness to learn. While some individuals may have a natural aptitude for math, it is ultimately a skill that can be developed and improved upon through hard work and perseverance.

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