# What is the most difficult mathematics?

by silverdiesel
Tags: difficult, mathematics
 Emeritus Sci Advisor PF Gold P: 16,091 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."
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 Quote by silverdiesel 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.
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?

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What is the most difficult mathematics?

 Quote by Hurkyl 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.
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.
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 Quote by matt grime 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...
 Quote by fourier jr 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').
 Sci Advisor HW Helper PF Gold P: 12,016 It also depends on your "personality". Some analysts loathe discrete mathematics and would gladly not learn anything about it, and vice versa.
 Sci Advisor P: 1,253 Here's a related question: what is the mathematics that depends on the most other mathematics?
 P: 304 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.
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 Quote by arildno 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.
 Sci Advisor HW Helper P: 9,470 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...
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 Quote by 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...
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.
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 Quote by mathwonk but complex analysis is to me easier than real analysis. and so on...
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.
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 Quote by JasonRox 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 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
 we seem to be jumping all over the place
? Give an exemple.
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 Quote by 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.
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.
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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.
 P: 166 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'.
 P: 304 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.
 P: 15 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...

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