# What is the most difficult mathematics?

1. Feb 25, 2006

### silverdiesel

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 cant do that, you should not be in this class" Which is true, no doubt, but I dont 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.

Last edited: Feb 25, 2006
2. Feb 25, 2006

### Hurkyl

Staff Emeritus
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.

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. Feb 26, 2006

### matt grime

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. Feb 26, 2006

### fourier jr

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. Feb 26, 2006

### AlphaNumeric

Yet there always seems to be such a person in the lecture theatre...
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. Feb 26, 2006

### arildno

It also depends on your "personality". Some analysts loathe discrete mathematics and would gladly not learn anything about it, and vice versa.

7. Feb 27, 2006

### 0rthodontist

Here's a related question: what is the mathematics that depends on the most other mathematics?

8. Feb 27, 2006

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. Feb 27, 2006

### JasonRox

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. Feb 27, 2006

### mathwonk

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...

11. Feb 27, 2006

### JasonRox

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. Feb 28, 2006

### d_leet

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. Feb 28, 2006

### quasar987

I take it you didn't like the one in french I refered 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.

14. Feb 28, 2006

### JasonRox

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. Feb 28, 2006

### gravenewworld

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.

we'll see if you say this after you suffer through advanced calc.

16. Feb 28, 2006

### Plastic Photon

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. Mar 1, 2006

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. Mar 1, 2006

### Mafer

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. Mar 1, 2006

### 0rthodontist

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.

Last edited: Mar 1, 2006
20. Mar 1, 2006

### daveyp225

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. Mar 1, 2006

### matt grime

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. Mar 1, 2006

### 0rthodontist

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.

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.

Last edited: Mar 1, 2006
23. Mar 3, 2006

### daveyp225

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. Mar 3, 2006

### 0rthodontist

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.

Last edited: Mar 3, 2006
25. Mar 4, 2006

### tmc

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.