What is Pure math: Definition and 91 Discussions

Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.
While pure mathematics has existed as an activity since at least Ancient Greece, the concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with a systematic use of axiomatic methods. This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics.
Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. A famous early example is Isaac Newton's demonstration that his law of universal gravitation implied that planets move in orbits that are conic sections, geometrical curves that had been studied in antiquity by Apollonius. Another example is the problem of factoring large integers, which is the basis of the RSA cryptosystem, widely used to secure internet communications.It follows that, presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference than a rigid subdivision of mathematics. In particular, it is not uncommon that some members of a department of applied mathematics describe themselves as pure mathematicians.

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  1. N

    B Is There an Alternative Approach to Learning Pure Math for Beginners?

    I've wanted to learn pure math for a while. I bought the book "A Concise Introduction to Pure Mathematics by Martin Liebeck." The first chapter or so misled me into thinking that finishing the rest of the book would be doable. Chapter 1 gets into definitions of sets and related vocabulary and...
  2. A

    Intro Math Plan to revise my learned math & start learning pure math

    Hello everyone, I'm planning to review learned mathematics from high school & start leaning pure math. You can reply to red colored text if you don't want to read full thread. My math level : Graduates as engineer 10 years ago. Got A in Calculus 1 & 2 (Those were single variable calculus), Got...
  3. Z

    Admissions Applying to a PhD in Pure Math with an undergraduate degree in physics

    Not from the US. I have a Bachelor of Science in Physics and a Master of Engineering Science. Mid-way through my undergraduate study, I realized that I am more interested in pure math than physics and wanted to apply for a PhD in pure math after my graduation. I had done research and written a...
  4. S

    Studying Good research areas in pure math

    My name is shabani makwaru from Tanzania i am master student in Pure mathematics at University of Dar es salaam Please can help me in advising which area is good in research in pure math
  5. F

    Should I avoid a pure math route?

    So I'm still doing some soul searching with finding a major. Currently I enjoy physics and math. However, while I'm good at the mathematical side of physics, conceptually I'm pretty frustratingly slow at it. Often I make the mistake of jumping right into the math without considering the actual...
  6. M

    Prime Factors Bounds in a Recurrence

    Let $$g(n)$$ be the numerators of the elements of the recursion $$i(n)=i(n-1)+\frac{1}{i(n-1)}$$ when they are expressed in simplest form, with $$i(0)=1$$. Let $$p$$ be the smallest prime factor of $$g(m)$$. Show that $$p>4m-4$$.Homework Equations Euler's Theorem? The Attempt at a Solution OEIS...
  7. Oats

    Going from pure math to applied math

    Hello, I am a second year PhD student in pure mathematics, and I'm going through a bit of a dilemma. Up to this point I felt that I wanted to work in Mathematical Logic. To be honest, I think working in any field of math would be fun, but logic had extra cool topics like computability theory...
  8. Ericamathstats

    Studying Applied math or pure math? + Stats

    Hi, I am wanting some advice on about studying applied math or pure math. I have finished my 2nd year at uni and will be starting my 3rd year soon and have been thinking about changing my major from pure to applied (I also major in statistics as well). After I graduate I would love to do...
  9. R

    Math Worth of doing pure math if not going into academia?

    Is it worth doing pure math if you're not going into academia? Would it be a bad idea to get your bachelors in pure math and stop there? Is pure math just something for people who want to go on to grad school? I'm just not sure if there are good job prospects for someone with a bachelors in...
  10. E

    Courses Taking Statistics vs Pure Math for Experimental Physics

    Currently I am studying a double major in Math and Physics, hopefully leading towards experimental physics, however come next semester I will have to pick my math "specialisations" from two of applied math (mostly chaos and optimisation etc.), pure math(mostly abstract algebra and analysis) and...
  11. H

    B Which Branch of Math Should I Study for a Foundation in Functional Analysis?

    looking at notes on functional analysis for self study. the authors I have stumbled across so far launch straight into the formal theory. I can't find any middle ground of intuitive motivations for the theory as yet. I am assuming the following is part of set theory? I can't follow the arguments...
  12. MidgetDwarf

    Programs Pure Math or Applied Mathematics

    I have a big choice to make which is not letting me sleep or study properly. For the last few years I have been attending community college. For financial reasons I left school during the 9th grade. Both of my parents were terminally ill. My father passed away, my mother still lives. I placed...
  13. TheMathNoob

    Programs Is it a good idea to major in computer science and pure math

    I am not an expert in the field, but intuitively I think that a resume with both math and computer science has a lot of impact. I also think that math can help me understand computer science better. Next quarter, I am planning to take classical geometry, so I am just wondering if this class...
  14. I

    Applied Math or Theoretical Physics?

    Hello everyone, as those of you who have helped with my other threads know I'm currently a high school student and an aspiring physicist. I'd like to start out my post by saying that I completely understand that I am very early to be thinking this far ahead, especially when I haven't experienced...
  15. M

    MHB Need help writing a pure math formula

    Hello, I am really struggling to write a math formula that I need for my website plugin. A=(Length) B=(Width) C=(Square Inch) A*B=C 0<=C<=16 then C*.84 17<=C<=34 then C*.70 35<=C<=65 then C*.56 66<=C<=99 then C*.42 100<=C<=150 then C*.35 151<=C<=500 then C*.31 I wrote this formula in excel...
  16. kade

    As a pure math phd student, how to get into the quantum research

    As a fresh pure math phd student, I aim to become a mathematical physicist, and currently I try to schedule out what I should do step by step in this four years on phd program. But in my university, there is no professor in this field could help me as my advisor. So I wish to get some advice...
  17. L

    Tailoring Pure Math Ph.D experience for jobs outside of academia after

    Hey guys! Let me throw in who I am before I introduce my question. I finished my year of graduate school last May, figured out an advisor and a field (something related with Brownian motion on manifolds), and I'm overall happy with it all. No plans of dropping out, I enjoy the company of my...
  18. Q

    Making the right decision Pure Math

    Hi guys I'm here to ask the age-old question, namely, I am thinking of transferring from my highly ranked university where I finished my first year in an Engineering Physics type program to the university in my hometown where I want to study Pure Math (it is too late to transfer faculties at my...
  19. T

    Importance of pure math books for aspiring physicists

    Hi everybody! I was wondering how much importance there is in studying pure math, for the budding theoretical physicist? mathematical physicist? Also I was wondering whether a mathematical physicist would work more on mathematical methods that physicists can use or do work like Einstein...
  20. C

    MHB Could someone recommend me pure math linear algebra book?

    this probably is a wrong section but i did not know where else to put it (feel free to move it to where it belongs). that said could someone recommend me a pure math linear algebra book? preferably with LOTS and LOTS of examples! i think i have decent grasp of the basic theory but i have been...
  21. chimath35

    After calculus first pure math course, having a hard time.

    I just started number theory, and I feel completely lost. I never did proofs in calc. I am used to watching the professor do examples, then with solutions manual at home see how the problems are done and use the manual if I get stuck. This book on NT has no homework problems or solutions. I just...
  22. E

    Applied or Pure Math for Future Teacher?

    Hi everyone! Thank you clicking on this thread. I am a community college student seeking to transfer by the end of this upcoming spring semester. I plan to major in math; my main goal is to become a high school teacher or community college professor. Only recently did I find out that...
  23. I

    Programs Is this considered a 'pure math' major?

    Is it considered a 'pure' math major if classes like calc 1-3, diff eq, linear algebra, analysis, and discrete maths are a part of the major? If so, would an engineering or physics major be considered 'applied' mathematics?
  24. D

    Computer Science with Pure Math or Applied Math?

    Greetings Physics formus, I am a high school learner currently in my final year of schooling. I have decided that pursuing a double major in math and computer science would be a good option for my tertiary studies, as, my passion lies in the Mathematical Sciences. However, I have not decided on...
  25. 1

    What is the value of pure math (besides just being awesome?)

    So, as I have stopped focusing on physics and shifted over to math I've found myself much more motivated, happier, and relaxed. Furthermore, I am much better at it. I have not taken any math classes that have been as difficult as the basic E&M physics classes (for me). I have made such a...
  26. Z

    Engineering Engineer Studying Pure Math: Benefits and Application

    I have recently completed my first year of Eng. Physics taking the standard math courses: Calculus, Linear Algebra 1 and 2, Multivariable Calculus and Numerical Analysis. Recently though I have been self studying Rudin's "Principles of Mathematical Analysis"/Abbott on my own and I enjoy this...
  27. B

    Math Which Major Should I Choose for a Balanced Academic Career: ChE, Physics or Pure Math?

    I will become a professor regardless of what major i choose except for maybe engineering. My major will be either ChE, Pure math or Physics. I am currently taking the prequisites for all of these majors so i can change at anytime. With pure math it may be hard to compete for a position against...
  28. J

    Is Vector Calculus useful for pure math?

    Hi everyone, I have the option to take a vector calculus class at my uni but I have received conflicting opinions from various professors about this class's use in pure math (my major emphasis). I was wondering what others thought about the issue. I appreciate any advice.
  29. D

    Applied Math undergrad to pure math grad school?

    The best instate college in my state for mathematics is Georgia Tech. However, it does not have a pure mathematics degree. Can I still go to grad school for pure mathematics as an applied math major? Will it prepare me for pure mathematics?
  30. B

    Is Math Useless? A Troubling Belief & Its Evidence

    I was not too sure what to title this but I just skimmed through a thread with the same title so I just used this. I am not sure whether to go into math or physics. I would prefer math but how will I ever know if I "know" something. This is a troubling thought that discourages me from math. On a...
  31. C

    Math Double Major in Physics and Pure Math vs Computer Science and Pure Math

    I am currently stuck between the two choices. I have to pick my majors soon, as I am going to third year. I originally wanted to go into theoretical physics, but now I am leaning more towards theoretical Computer Science. I have always loved Physics, but I am just more at home in Computer...
  32. S

    Schools Do you know Good schools for Pure math and/or physics theory? .

    Do you know Good undergrad schools for Pure math and/or physics theory? Please Help. Hunting for colleges, I'm a definite pure math or theoretical physics major. To give you an idea of my academic standing: MIT would be a stretch, U Penn is a mid reach, Brown is a low reach, Northeastern is a...
  33. B

    Programs Can a Pure Math undergrad get into an Applied Math PhD program?

    I want to get my undergraduate in Pure math (I've taken and will take some additional CS courses). My question is, would I still be a competitive candidate for a PhD program in computational biology? (mathematical modelling of biological/behavioral/social systems) or should I really get my...
  34. S

    How much Pure Math is in Physics, pure math / physics major?

    I really like pure math, like analysis, topology, and number theory, but I also like physics. I am wondering how many applications pure math has in higher level physics, because I don't want it to leave my life. And also, would a pure math /physics double major be intuitive or a waste of my...
  35. M

    The way math is used in physics vs. pure math

    Is it just me, or does it seem much different when doing math in a pure math class vs doing the same type of math in a physics class? The way things are done seem different to me..I can't pinpoint exactly what it is but it just doesn't seem like the same thing.
  36. L

    Engineer stumped by pure math, need to find some way to get started

    Hi guys, I am reading some paper on Mellin Transform and its application in signal processing and I am totally stumped by phrases such as lebesgue space, square integrable, isometry, automorphism, measurable and essential supreme and such. Of course, when I look them up one by one I sort of...
  37. D

    B.S. Applied Math to Pure Math Grad?

    Hello, I've been pursuing a second bachelor's degree part-time for a while now at a Group III school. I just found out, somewhat unexpectedly, that I might be able to transfer to Georgia Tech (a Group I school) with relative ease to complete my degree as a full-time student. GT is...
  38. G

    Math Pure math to financial math (quants) questions

    I'm an undergrad in college and I've been studying pure math for some time now. I do not really like the job outlook that this path is taking me. My school offers a Masters program in Math Finance (Stochastic Calc and all that). It seems that this more applied field has a greater job outlook...
  39. P

    Applied or Pure Math: Which is More Beneficial for a Chemistry Major?

    I'm just starting out right now as a math and chemistry major. Should I choose applied or pure math as my concentration? I've been told that applied math will help me with chem, and I agree. However, I also feel comfortable with the theoretical side of things, and it's a large part of what draws...
  40. J

    Just Want to learn Pure Math Graduate level

    Good evening, The biggest question for me right now is what to do now? I got a bad experience from high school and was never a hardworking student. I got admission to a top school for undergrad and as we all know college is somewhat different, I started to really learn as from Junior year. I...
  41. D

    Is Switching from EE to Pure Math the Right Move for Me?

    I'm finishing up my second year in EE, but realizing that it may not be the right field for me. When I applied to colleges, I didn't really put much thought into what I wanted to study; my parents suggested engineering and I stuck with it. I've always been very strongly mathematically...
  42. D

    Programs Probability from EE or Pure Math department? EE major

    This probably (60%?) seems like a strange question but I have the option of taking probability from either the EE or pure math department and want to know how each will deal with the subject. Here's the course descriptions: EE - Introduction to Probability Theory. Covers the fundamentals of...
  43. H

    Programs Physics major preoccupied with pure math

    I lack talent in pure mathematics. (Although applying mathematical methods has been natural for me.) I am a physics freshman. I can prove and I can self study mathematics given enough time. I guess it's still too early to decide, but, I suppose, I'll be doing theoretical/mathematical physics...
  44. K

    Programs Does a pure math major need to take ODEs?

    Right now I'm a sophomore at a state uni with hopes of getting into graduate school in pure mathematics. When I was a freshman, I surveyed the three major areas of math - analysis, algebra, and topology - and I decided that analysis was for me. Although I did very well in Algebra, I found it...
  45. R

    Programs Pure math PhD going into the industry. What are the chances?

    So I'm a junior in mathematics and I've reached a point to decide about graduate school. I love pure math and am considering getting a PhD in it. But I do not want to go into academia because I don't think I will enjoy teaching. So I want to go into the industry after a PhD. But one of my...
  46. P

    Math Intent on getting pure math PHD, start plan B now?

    Basically what I mean by my title is that I am intent on getting a PHD in pure math because I love the subject, and am still young enough to do it without throwing the rest of my life into disarray. But beyond that I need to have a plan. Plan A is clearly to go on to do several post-docs...
  47. R

    Pure math also used in physics

    Hello. I'm a physics major. But I'm really interested in maths. Lately I've been neglecting physics for Set Theory by Jech & Hrbacek, math is just awesome. Majoring in math is not possible (I already switched major twice). So yeah, I'm a bit indecisive but this time its more of an...
  48. Z

    Is Pure math of any use in Theoretical high energy physics?

    Is Pure math like functional analysis and real analysis of any use in Theoretical high energy physics research ? Do I need to learn them?
  49. O

    Waterloo Pure Math Co-op vs Regular?

    Hi I am a grade 12 student in Canada and goint to apply for University of Waterloo. I am not supposed to declare my major yet but I am quite certain that I will major pure math. Now waterloo asks us to choose between co-op and regular. 1)co-op I don't know what kind of jobs are...
  50. O

    Programs Pure math major + music minor = no merit?

    Hi Do I really need to take economics/statistics/finance course in my undergraduate years if I am considering the possibility of landing on a finance job? I really do not want to worry about getting job in my undergraduate years but would like to put some serious thoughts after my graduate...