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Because I'm OK at Applied Math but failed Pure Math two times, I asked my instructors and a brilliant friend studying the undergraduate Math Tripos at the University of Cambridge for advice. They all said that there's no secret or shortcut, that you must know your material, practice a lot, and keep trying. But I've been doing all this and still failed two times. So I Googled to see if even Cambridge math undergraduates fail PM. What should I learn from the quotes? Is Applied Math easier? Why or why not? I bolded the text.

Stealth77 ( Emmanuel [College] ).
I'm also in Maths IA (or rather just finished). A 2:2 isn't that bad. I have friends with 3rds and know of people not even on the class list. Remember, you can do mostly applied courses next year.

smilepea. a lot of people [,] who didn't get on very well with first year because it was too pure [,] really like second year as you can focus almost totally on applied.

Xavikadavi post 20 . [ This is the person who's suffering like me and responded below. ]
What you say about the pure/applied side makes a lot of sense - for final revision I focused on my strongest topic for each paper which in every case was the applied one- I've found these so much more managable over the year (hence why I thought maybe I should be on the natsci course doing physics . I found pure impossible- esp. groups and analysis.

smilepea post 22. I have a friend who only did applied this year (in IB) or as far as I can remember any way (she may have done one pure course) and got at 2.1. Doing only applied meant that she was much happier and did much better. And the applied courses are much, much better in 2nd year, I know the first year applied courses aren't brilliant and certainly not what I consider applied!!

Zhen Lin, post 23
I've found that applied questions consistently take longer to do. This isn't a disadvantage if you find pure questions impossible, but if you can do both, then, from a mark-maximising perspective, it is a disadvantage to focus on applied.

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    $\begingroup$ I am sure most people will agree it is highly speculative which is "easier", but I am also sure a lot of those same people would say they found the applied math more straightforward. Also, here is an obligatory xkcd comic that is not specific toward math, but "pure" and "applied" subjects in general. $\endgroup$ – PawnInGameOfLife Jan 18 '15 at 14:34
  • $\begingroup$ @CEM thank you. but why's Applied 'more straightforward'? I ask this at math.stackexchange.com/q/1109237/85079 $\endgroup$ – user4655 Jan 18 '15 at 16:26
  • $\begingroup$ I find it more straightforward personally because it's more algorithmic I suppose (or usually is in my experience anyway). It's not as abstract perhaps? Although people do find clever ways to solve computational problems that can be done by an algorithmic process, I tend to think of pure math as having less to do with computation (that's just my opinion). Without having an algorithm or a set of steps to follow (which is how I'd view a lot of applied problems), things can become less straightforward. $\endgroup$ – PawnInGameOfLife Jan 18 '15 at 17:18
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    $\begingroup$ There is a sense in which pure math is "easier" in that it requires less experience with the world. Whereas applied math often need extensive interaction with the application area to accurately assess the modeling options. So I can imagine a precious pure math talent; but it is less natural to imagine a precious applied math talent. $\endgroup$ – Joseph O'Rourke Jan 19 '15 at 2:20
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    $\begingroup$ It probably depends on who applies the Pure Maths and for which purpose. $\endgroup$ – Taladris Aug 2 '15 at 6:24
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When I was a student, I always found pure maths very difficult to study compared to applied mathematics, and as a result I gravitated mostly towards the applied courses during my degree.

Possible reasons why people might find applied mathematics more straightforward than pure mathematics:

  • The style of thinking required to succeed at pure mathematics is unfamiliar to many and can make students uncomfortable. This is particularly because it relies on logical definitions, and these definitions are very different to those that we think of in real life. Our definitions of things are pre-concieved images of what those things are, rather than a definition with several properties that must be satisfied. (Worth giving [1] a read to gain an understanding of why pure mathematics can be uncomfortable for students!)
  • Applied mathematics courses often involve solving differential equations, and doing calculations/simulations, with considerably less emphasis on why the methods work or the conditions that have to be met in order for them work. This is much more in line with the mathematics that pre-university students are taught (in the UK at least), so the sense of familiarity can make applied maths seem much more straightforward to digest than rigorous pure mathematics.
  • Many areas of pure mathematics are very difficult (if not impossible) to come up with a visualisation of. Many mathematics students need to rely on "concept images" in order to gain an intuition of what a definition means rather than blithely learning the definition ("concept definition"), and if they are unable to unite these, then this can make learning pure mathematics very difficult. However, there are some areas of pure mathematics that can be quite nicely visualised, which is why I really enjoyed point-set topology and aspects of analysis, but struggled much more with topics such as group theory, linear algebra or Galois theory.
  • Pure mathematical proofs as seen in lectures are often refined and concise. However, they do not do a good job of explaining or justifying the steps and it is often left to the reader to figure that out. This can put a lot of students off, as the way proofs are presented in lectures and books can seem arbitrary; the proofs do not outline the thought process undertaken by the person who wrote them, or explain why a specific tactic of proof has been used. It's often just "Proof: here it is."

However, one should be careful in just categorising topics as "pure" and "applied", as the two categories can really help each other and ties between them are essential in order to delve deeper into them. For example, partial differential equations very often form the basis of applied mathematics and physics courses, even though there are many areas of pure mathematical analysis that are required in order to be able to prove their properties or make further developments. Without a pure mathematical background, the best thing you can do is run simulations or solve PDEs in very specific cases.

Something I've felt in recent years is that I've shied away from learning pure mathematics for the reasons above, but it's a whole new beast that I still want to conquer and understand. Even after a maths degree (and a failed PhD attempt) this is something I want to overcome in the fullness of time.

[1]: Alcock, L. and Simpson, A., 2009. Ideas from mathematics education: an introduction for mathematicians. MSOR network.

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    $\begingroup$ This answer refers more to stereotypical ways that "the subjects" are taught, rather than the actual content. The distinctions and contrasts made are about the way the instructors choose to distinguish the things... which, yes, matters greatly, ... but is not at all innate. And the distinction will evaporate after the course is over. $\endgroup$ – paul garrett Aug 1 '15 at 19:06
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    $\begingroup$ I think my answers about the ways the courses are taught are valid, and with consideration to the pedagogical issues involved and why this means students may find pure maths harder than applied maths. However, as someone previously pointed out, this is highly subjective and probably more down to personal preference. I'm not just talking about the content, but the greater issues which influence students' experience of learning about the areas. $\endgroup$ – user5447 Aug 1 '15 at 19:10
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    $\begingroup$ But it's not about "the areas", but about cultural traditions in textbooks and instructors' preferences. Yes, that matters, but to play along with the implicit assumption that it's the content just perpetuates all sorts of confusion and misinformation. The issues are about particular university systems, particular instructors, particular books, not about mathematics of any sort. (E.g., in real life there is no simple dividing line between mathematics that "is applied", and math that isn't directly applied... It's just a school thing, not a scientific or intellectual distinction.) $\endgroup$ – paul garrett Aug 1 '15 at 19:43
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I do not think it would even make sense to answer your question literally. First applied and pure maths are intertwined and often difficult to tell apart. For example when we teach the spectral theorem, we often consider it as pure maths, but then there are lots of clearly applied maths where it is fundamental to understand this result intimately.

Second, I would say that given any two field, and any difficult concept A in one of them, it is possible to find a concept B in the second that is arguably more difficult than A. So some parts of so-called applied math could be easier than some part of so-called pure maths, but the converse is true too.

Nevertheless, there is an interesting and common question behind yours: What may explain your feeling?

A possible explanation is that the expectations of teachers are different in pure and applied maths (e.g. because more pure math majors will try to do research, while applied math majors have more diverse and arguably less selective professional objectives such as engineering, or because the teachers are different).

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    $\begingroup$ I agree that the question behind the OP's ("What may explain the feeling that applied math is easier than pure math?") is more interesting/tractable. $\endgroup$ – Benjamin Dickman Jan 18 '15 at 21:48
  • $\begingroup$ @BenjaminDickman thanks. please don't hesitate to edit and revise my question. $\endgroup$ – user4655 Jan 19 '15 at 3:45
  • $\begingroup$ I strongly second the appraisal that the perceived difference is a perception of artifacts, of instructors, of (possibly local) cultural traditions, artifacts of textbooks, and such. Further, faux-ironically, I'd have to pose as partly disagreeing with your last paragraph, because, in some situations, it's the self-styled "applied" students who "do research", while the "pure" students feel they're not yet up to speed. So, yes, I think the question is impossible to parse except as a question about artifacts that can result in such a perception. [cont'd] $\endgroup$ – paul garrett Aug 3 '15 at 19:03
  • $\begingroup$ ... [cont'd] In other contexts, the analogous question often is "which is harder, analysis or algebra" [sic] from students who've had just one course of each, one instructor hostile, one congenial, etc., but forming opinions about the mathematics based on the environment, not on the content. Still, yes, operationally it may not matter much... until one leaves that immature context. $\endgroup$ – paul garrett Aug 3 '15 at 19:04
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I have taught basic Group Theory (pure maths) to primary school kids of varied ability. It is a beautiful self-contained subject that you can play with and easily prove some interesting theorems (ie. properties of very small groups which even kids can enumerate). It is much easier than multiplying 2-digit numbers, let alone 1st year calculus.

The difficulty that you have in an area of maths depends on several things: the expectations of the lecturer, the style of presentation, and the type of leaner that you are. Given the right expectations and style of presentation, pure maths is not inherently hard.

I found pure maths easy because there is so little to learn. Get the basic concepts and a few fundamental theorems and everything else flows. The downside is that if you don't "get" the key concepts, the whole thing is near impossible. Pure maths just might not have clicked for you yet.

When it came to calculus and all the tricks to allow you to calculate the exact integral of all the special cases - I hated it! Lists of trig identities!! I thought I may as well be studying organic chemistry with everything I was supposed to memorise. You might be fine with all of that.

It also comes down to motivation. For a Platonist that enjoys exploring the world of pure forms, pure maths is as good as it gets, applied maths is dreary by comparison. Other people get their motivation from the way mathematics models the real world, and applied maths is awe inspiring from that perspective.

Mainstream culture in the English speaking world doesn't really care for either endeavour, but is especially antagonistic to a Platonist perspective. Even among those of us who like maths, the "pure maths is hard" sentiment might be an acceptable refinement of "maths is hard", but in my perspective it is just as unjustified.

It is better to stick with the perspective that some of us are good at some things, and not at others, but sometimes we get good at the things we previously had trouble with.

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    $\begingroup$ I don't agree that there is little to learn in pure math. I remember that I drowned in all the notions in my algebra course (centralizer, normal subgroup, cyclic and whatnot). In calculus you have just a very few objects (sequence, series, function) and have quite complicated questions about them... Interesting to see how different few on the subjects exist... $\endgroup$ – Dirk Feb 5 '15 at 5:56
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    $\begingroup$ @Dirk It is possible that my lecturers' expectations for pure maths were lower than yours were, so my lecturers didn't put as much in the syllabus I did. It is very hard to untangle the different factors that make a subject feel easy or difficult. It is all relative, so in the end it comes down to our relative abilities and motivations, and sometimes just the luck of how well we develop some key mental models. $\endgroup$ – Richard Feb 5 '15 at 7:23
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My two cents are: There is not a real dividing line between "pure" and "applied" mathematics. I think the real distinction is between mathematicians (or "want-to-be-mathematicians") who are inspired by an application and those who are not.

Of course, this answer does not help you with the question you posed. But well, there are people who are bad at algebra (considered "pure") and stochastics (considered "applied") at the same time and also good in optimization (considered "applied") and functions of a complex variable (considered "pure")–one example is myself.

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One guide is that all areas are developed until they are near the limits of what is possible for a bright, motivated human. Be it math, playing the violin, or planting roses.

More than looking for "which one is harder/easier", you need to find out what you like most. Somebody said that to get to be a great piano performer, you need to put in 10,000 hours of practice [the number is made up, but large anyway]. You see biographies of virtuoso performers (or star athletes, or whatever), and see they train for untold hours a day, every day. Same for any other area, I presume. If you aren't sufficiently interested in the subject to put in such hours, you won't excel. At the very least, you'll grow miserable in the process.

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