What is the most difficult mathematics?

[ice109]

meh none of it is difficult to learn. it's a whole other matter to contribute.

 

[Jame]

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.

 

[chota]

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."

is the other thing integrals :P

 

[hampton770]

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

 

[Cantab Morgan]

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."

 

[hampton770]

Actually I was a Pre-Med / Psychology Major / Sociology Min.

I did a double blind study to compare Math majors MATH GPA and PSYCHOLOGY GPA. I was surprised to find that people who typically made "As" in any either major were equally likely to have a HIGH GPAs average in Math! Surprise...Surprise. In fact, I think the Psychology Majors has the Math Majors beat in Calculus by a couple tenths of a point!

H

 

[CRGreathouse]

I did a double blind study to compare Math majors MATH GPA and PSYCHOLOGY GPA. I was surprised to find that people who typically made "As" in any either major were equally likely to have a HIGH GPAs average in Math! Surprise...Surprise. In fact, I think the Psychology Majors has the Math Majors beat in Calculus by a couple tenths of a point!

Doesn't match my experience at all, unless you're talking about Calc 1.

 

[dvs]

Don't math majors and psych majors take different calculus classes anyway?

 

[zetafunction]

the hardest or most difficult mathematics is the one you CAN NOT learn.

perhaps higuer Algebra, or Functional Analysis , .. almost any branch of mathematics is difficult

 

[qntty]

the hardest or most difficult mathematics is the one you CAN NOT learn.

What do you mean "cannot learn"? Pretty much anything can be learned with enough work.

I did a double blind study to compare Math majors MATH GPA and PSYCHOLOGY GPA. I was surprised to find that people who typically made "As" in any either major were equally likely to have a HIGH GPAs average in Math! Surprise...Surprise. In fact, I think the Psychology Majors has the Math Majors beat in Calculus by a couple tenths of a point!

Did you use Calculus classes or the proof-based classes that a math major takes?

 

[hampton770]

Did you use Calculus classes or the proof-based classes that a math major takes?

Used...each students average Calculus GPA. The sample size was relatively small, but the study brought the interesting question to light.

 

[AUMathTutor]

For me, it has mostly to do with interest in the subject.

 

[quasar987]

The hardest class I took was commutative algebra. I don't know if the difficulty is inherent to the subject or if it's just how the professor chose to present the material. What is other people's experience with commutative algebra (as in localization, notherian, artinian, dedeking rings, nulstelensatz, GU, GD, etc.)??

 

[swraman]

i liked calc 1 2 and 3.

i hated linear algebra/differential equations. at my school they combine linear algebra and differential equations into one semester...from the basics of matricies to solving higher order and PDE's all in one semester. I really think they should split it up, but I am not hte judge of that...

 

[adoado]

Integration...knowing which method, substitution, rule, etc. to use...

I guess its just practice before you get good at it, but this has got to be one of my weak points...

 

[slider142]

Number theory tends to be difficult in that the simplest of statements can take huge structures to get near a proof, and many simple statements are still open conjectures. Ie., Fermat's last theorem is part of number theory, but the machinery used to prove it draws from many high powered areas of mathematics.

 

[Gauged]

Remembering the fundamentals.

 

[jweygna1]

Like a lot of others said, it all depends on your personality. I struggled with abstract algebra and discrete a bit as an undergrad, but after a few weeks i got used to it. For me anything to do with calculus is very easy (ODE, PDE included), but it all depends on how your mind works.

 

[protonchain]

I had to take 3rd semester math, linear algebra and differential equations, and "math for physics and astronomy students" @ Berkeley.

3rd semester math was more like a review of second semester to be honest. You learned some new techniques for integration, and nothing really got to become major until the latter part of the semester with stoke's theorem and whatnot. I found that hard at first but after I sat down with it I got a grasp of it.

For Lin. Alg. I took a course in a community college over the summer and then again at Berkeley. I could have gotten away with taking just differential equations but because I was an idiot and didn't check, I took the whole shebang. Got a B in that class. The linear algebra wasn't hard at all, and diff EQ was probably the one topic that was new for me. Since we didn't spend ages and ages on it, not too many difficult topics were covered so, no problem there.

Lastly there was the math for physics and astronomy students. This class was a joke. It was just a review of everything that was covered previously with some very very minor applications in physics. I thought certain things like Fourier series would be difficult but I sat down and got it down.

It's all about hunkering down with the subject and understanding it. Nothing is really difficult once you are able to do that. Geniuses can get away with not studying because well... they're geniuses. Regular shmos like me have to work hard and understand itty bitty concepts, but once you do, it's nothin.

Watch out for partial diff eqns though. Those are a doozie.

 

[Jame]

For me number theory, it's very apparent that I don't have that intuitive feeling for numbers as some people do. Especially when it comes to modular arithmetic, I've been in the situation where some people just figure it out by common sense while I have to really work the algebra to see why a certain number has that divisor or not. I'm also bad at inequalities and I have a sense that these concepts have something in common, such as the number line.

Vector analysis on the other hand I grasp easier than others. It's interesting how there's no single mathematical talent but rather many different kinds.

And as someone already mentioned - remembering the basics. It's crucial and can save you so much time and despair not having to through elementary books feeling like you should start over from the beginning with the whole mathematics thing.

 

[wajed]

Please help me make these two lists
1)A list of the following desciplines from the hardest to the easiest
2)A list of the following desciplines from the most fundamental to the least

A)Discrete mathematics
B)Logic
C)Numbers analysis
D)Complex analysis

As I noticed many of you said its a matter of personality and so on..
But please dosn`t hesistate to show me your preferences..


Another question: I actually think of studying the four of these, do I still have to study something that must be fit in between in the list?

 

[Jame]

You can't answer such a question. All of the branches you listed are major parts of mathematics which means that they can be as hard or as easy as you like. The subjects are therefore very interconnected meaning that the answer to your second question is: it depends on the actual curriculum for the corse.

 

[flatmaster]

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

If you continue in physics, you'll use algebra quite a bit. The algebra is usually where you say "math happens" then report a result.

As far as the hardest part of math, it depends on the person. I'm very good at spatial reasoning, so Calc 3 was a breaze for me. I have trouble with more abstract thought, so higher math is a blur to me.

I suppose it comes down to if you think of math in terms of the actual physical world that motivates it, or as an abstract thing that stands alone.

 

[Fredrik]

I'm trying to learn functional analysis on my own right now, and it's by far the most difficult subject I have studied. (For those who don't know, it's basically linear algebra with infinite-dimensional vector spaces, but the methods used are more like the ones from an advanced calculus course than the ones from a linear algebra course). Seems like every page takes at least 2 hours to understand (sometimes a lot more) and the book has about 240 pages.

 

[wearethemeta]

Do you think it has (a lot) to do with age?
I'm 14 and I started integration a couple days ago and its fine when I do it; on the spot, but I have trouble remembering it and writing down formal definitions (etc...) later, like the next day. For example, I think the hardest bit of integration I've done yet is finding the area bounded by two curves, one negative and one positive, and while it was fine on the go and I didn't have too much trouble, if I was to look at it now without looking at my notes, I am sure it would take me much longer, or that I would not even find the answer.
Thank you

 

[arildno]

Do you think it has (a lot) to do with age?
I'm 14 and I started integration a couple days ago and its fine when I do it; on the spot, but I have trouble remembering it and writing down formal definitions (etc...) later, like the next day. For example, I think the hardest bit of integration I've done yet is finding the area bounded by two curves, one negative and one positive, and while it was fine on the go and I didn't have too much trouble, if I was to look at it now without looking at my notes, I am sure it would take me much longer, or that I would not even find the answer.
Thank you

Some maths might, indeed, be easier to learn for a bright teenager than other maths.

For example:
Maths that is strongly assosiated with visualization is generally easier to get a hold on than very formal proof structures, for example.

And, the capacity for abstract logical thought is still developing during your teens, until the age 18-20 or so. (And then, everything goes downhill again..)


But, I think you are well underway in developing that capacity, just being 14 and having a good grasp of integration already.

 

[kpinky]

My son scores good marks in all subjects except maths. He is afraid of maths. He is not good of analyzing problems. Last night he showed me a problem and told mom i am scared of this bigg problem, such a big problem "In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato, and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line (see Fig). A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?" Can anyone tell me how simple can we explain solution to this problem

 

[arildno]

Well, you could set it up like this, in order to preserve the "visual element" in the calculation:

Total distance:
2*5+2*(5+3)+2*(5+3+3)+2*(5+3+3+3)+2*(5+3+3+3+3)..and so on
Or, that is:
10+2*8+2*11+2*14+2*17 and so on.

Here, each term represents the total distance traversed in a particular potato-fetching run.

Another way of representing this requires a bit of thinking:

The "first five metres" are run by all 2*10 runs, so you get 10*2*5
The "next three metres" are run by 2*9 runs, so you get: 9*2*3
The next three metres: 8*2*3
The next three metres: 7*2*3

and so on..

Thus, when adding it all up, you get 100+6*(9+8+7+6+5+4+3+2+1)=100+6*45=380 metres in total