I often see PhD students and faculty members describe their research interests on their web pages as something like:

I am broadly interested in applied and computational mathematics.

What is the difference between applied math and computational math?

Do applied mathematicians not necessarily work on computations and computational mathematicians not necessarily work on applied math?

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    I guess one way to (potentially) see the difference would be to ask yourself: was all math theoretical until computers came along? – user541686 May 6 '18 at 20:20

"Applied" math is in opposition to "pure" mathematics: that is, applied mathematics is used for a specific ("real-world") problem, such as fluid mechanics, statistical physics, or even the biosciences. This need not be computational: in many cases, it involves the study of differential equations and other analytical tools (expansions and other techniques).

In contrast, "computational" mathematics implies mathematics that cannot be solved by "pen and paper" techniques but instead require the use of a computer—either desktop or cluster—to solve.

  • I will just add that computational math need not be applied math. – Jonathan Landrum May 7 '18 at 23:53

My father worked on algorithms for calculating millions of digits of pi. That's computational, but not applied, since it's hard to find applications that require even twenty digits.

I used probability theory to find a strategy for making use of imperfect information in order to sample a rare population as cheaply as possible, while meeting certain accuracy requirements. This is applied but not computational, since I didn't use a computer to help figure out this strategy.

In practice, the line is often blurred; methods developed for "pure" mathematics often turn out to have practical applications (one of my father's other hobbies is factoring integers into prime numbers...) and computers are often handy for implementing methods developed by non-computational means.

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    Factoring prime numbers or semiprimes? – Peter Taylor May 6 '18 at 17:08
  • I guess the sentence was meant as "factoring integers into prime numbers". – Paŭlo Ebermann May 6 '18 at 19:53
  • @PeterTaylor whoops, fixed! – Geoffrey Brent May 6 '18 at 21:47

It's possible to work on the development theoretical analysis of numerical methods for differential equations (and other mathematical problems) without any immediate application in mind or in response to specific problems posed by other applied mathematicians, scientists, or engineers. Researchers who do this typically call themselves "numerical analysts" rather than applied mathematicians. You'll also see "computational science" used to describe research that might focus on applications to particular scientific or engineering disciplines and "high-performance computing" to describe research on how to effectively use supercomputers to solve problems in scientific computing.

It's also possible to develop and analyze mathematical models without using computational methods. Researchers may use analytical methods and study the qualitative behavior of models in ways that don't require computation, or they might work with specialists in numerical computing when necessary. There are many mathematicians who engage in this activity and call themselves "applied mathematicians" without adding "computational."

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