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So, I have been looking into college courses to see what I want my major to be, and I noticed that M.I.T offers two specified types of Mathematics. Pure and Applied. What is the major differences in Pure vs. Applied and what are examples of each?

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  • $\begingroup$ In general, applied math relates to the world in a very direct sense, while pure math is more theoretical. $\endgroup$
    – user242594
    May 28, 2015 at 13:35
  • $\begingroup$ So, would applied would be used more often? $\endgroup$
    – Hackaholic
    May 28, 2015 at 13:37

7 Answers 7

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The lines between these two divisions, much as in all divisions within mathematics, is often blurred. However, the traditional division between the two is that Applied Mathematics has a very clear connection to physical real-world problems. At its heart are PDE's, but also included are things like numerical methods and (once upon a time) what are now called computer science and statistics.

Pure Mathematics is mathematics for its own sake, pursuing questions based on the internal attractiveness of the questions. At its heart is number theory.

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  • $\begingroup$ Alright. Thank you, a lot. I think I understand now. $\endgroup$
    – Hackaholic
    May 28, 2015 at 13:42
  • $\begingroup$ Did you note you were writing "PDE's" in an answer for someone who writes he's looking into collage course. $\endgroup$
    – Nikolaj-K
    Jul 8, 2015 at 13:43
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    $\begingroup$ @NikolajK PDE's are studied by first, second and third year undergraduates nowadays. At my university anyway. $\endgroup$
    – user85798
    Nov 1, 2015 at 1:22
  • $\begingroup$ Certainly at MIT they are. $\endgroup$
    – vadim123
    Nov 1, 2015 at 1:30
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    $\begingroup$ I'm not sure that any subject can be said to be at the heart of Pure Mathematics, but if there was, it would probably be Logic & Set Theory. Even Group Theory is more central to Pure Math. Number Theory's distinction is that it is probably the only currently active major subject area that people outside of mathematics may have heard of and might have some understanding of. $\endgroup$ Aug 13, 2019 at 14:54
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The best way to tell the difference between "pure" and "applied" mathematics is to compare the kinds of problems faced in each. Consider the following list of questions:

  1. How can we model, mathematically, the movements of a human heart?
  2. How quickly might a particular virus spread through a population of people?
  3. What's the best way to throw a skipping stone? That is, how can we maximize the number of bounces on the surface of the water before the stone sinks?
  4. Why do whirlpools form? Is there an optimal strategy to escape from one?
  5. How can we accurately predict what the weather will be like in 3 days time? Is it possible to accurately predict what the weather will be like in 3 months time?
  6. Why do soap bubbles form spheres? In cases where they do not form spheres (YouTube link), what kinds of shapes might they form?
  7. Consider $k$ runners on a circular track of unit length. At time $t = 0,$ all runners are at the same position and start to run; the runners' speeds are pairwise distinct. A runner is said to be lonely at time $t$ if the distance along the track from them to the nearest other runner is at least $1/k$ at time $t.$ For which values of $k$ is every runner lonely at some time?
  8. For which $m\times n$ chessboards does there exist a knight's tour? What about if the knight is allowed to go off one side of the chessboard and come back on the other side (cf. Asteroids)?
  9. What is the area of a circle of radius $r,$ and what's its perimeter? Can we find a relationship between the two that also relates the volume of a sphere of radius $r$ to its surface area?
  10. Can we find integers $a$ and $b$ with $b\neq0$ such that $\pi=a/b?$ What about if we replace $\pi$ by $\sqrt{2}?$
  11. Given a number $n,$ is there some easy way to compute the number of prime numbers less than $n?$ Is there some easy way to estimate the greatest prime less than $n?$
  12. Pick a positive integer. If it's even, then divide it by $2;$ if it's odd, then multiply it by $3$ and then add $1.$ For which starting values do we eventually reach $1$ as a result of repeating this process?

All of these questions have some kind of mathematical answer. I would say the first half are "applied" problems and the second half are "pure" problems. Broadly speaking, a problem is "applied" if it is phrased in terms of physical phenomena. That being said, question 6 leads to some very interesting pure mathematics, and question 7 has been clearly stated in terms of physical, everyday things. Hopefully, this is enough indication that there is not always such a clear distinction between "pure" and "applied."

It should be noted that I have certainly failed to give an indication of the breadth of either area: applied mathematics is much more than those six questions above, and pure mathematics is much more than the other four. When the time comes, I'm sure you'll be able to find more examples of the kinds of problems solved in mathematics by looking through your university library. I don't see any reason why you should choose in advance; I advise you to keep your options open, if you can.


EDIT: I have just realized that I am more than a year late to the party. Nevertheless, my answer might be helpful to someone, so I'll leave it here.

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    $\begingroup$ Modeling is only one part of applied math -- applied math also has a great emphasis on developing efficient algorithms to solve problems such as linear systems of equations, eigenvalue problems, ODEs and PDEs, and optimization problems. $\endgroup$
    – littleO
    Aug 16, 2016 at 18:09
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    $\begingroup$ Yes, of course! I agree completely. However, it's somewhat difficult to state such problems in a way which is understandable to someone who is "looking into college courses." Moreover, I find it difficult to make such areas sound interesting without getting involved in the necessary background first. Similarly, pure mathematics is more than "problems with numbers" or the like; much of pure mathematics is to do with "understanding the mathematical universe," or, in less poetic terms, "theory-building." $\endgroup$
    – Will R
    Aug 16, 2016 at 18:22
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It can be boiled down to (my choice of) Algebra, Number Theory, Topology and Gemoetry versus Statistics, Mechanics and Applications ot Physics.

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As previous answers mentioned, there is a lot of overlap between Pure Mathematics and Applied Mathematics because they might use similar techniques or concepts. What makes them different is the goals set on in Applied Mathematics versus Pure Mathematics, I am just going to refer them to AM and PM respectively. For AM the goal is typically to advance mathematics for the sake of some practical purpose, now this might still be quite theoretical, for instance people in mathematical physics might be creating new math for the sake of advancing our understanding the physical world. While in PM is advanced for the sake of advancing mathematics without a concern for its practical applications, even if practical applications do exist or not.

What is important to also emphasize is that one field is not necessarily more difficult than the other one. For instance there are mathematicians who study the mathematical ideas behind General Relativity, this involves very technical concepts that require studying a lot of PM but serves AM as it allows us to have better physical understanding of our universe. Whether PM is harder or not than AM I believe is not a very useful conversation though.

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I would have to say that pure mathematics involves pure numbers (and other objects that don't have units of measurement) while applied mathematics involves quantities (numeral values and units of measurement such as volts or dollars).

For example, if you are studying physics or statistics without using any units of measurement, then these would be forms of pure mathematics (mathematical physics and mathematical statistics). But if you are studying them using units of measurement, then they are applied mathematics (applied physics and applied statistics).

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AFAIK "applied" means applied to problems outside mathematics, while pure deals with problems just relevant for mathematics (at that time).

Also there might be considerations of solving problems for specific instances or solving them efficiently instead of just knowing that they are solvable.

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Pure Mathematics means going deep into math only for the sake of stuying and advancing mathematics and only mathematics. Pure Math is studied not for the sake of any academic purpose, however applied math is studied in order to know how to advance other fields using math.

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