It is not only in Mathematics that there is a distinction between the Pure and the Applied. Of recent in Researchgate discussions, somebody made mention of Structural geology as opposed to Applied geology. It is not wrong to rationalize such a distinction. The truth is that some areas are clearly more applicable than others. Abstract algebra through galois theory is becoming more and more applicable but mostly it is a body of theories yet to be applied to real-life problems. In future the theory-application gap is likely to be narrower not just in Mathematics but also in other sciences as the frontiers of knowledge become extended.

Most mathematicians are now going back and forth between "pure" and "applied" to look for new insights or problems to solve.

Qualifying some mathematics of applied or pure is just a very crude and imperfect way of describing these domains... Do not waste sleep over it, mathematicians do not!

I think that although such division into pure and applied mathematics is a bit fuzzy, it makes sense. Applied mathematics develops in orientation on practical problems. You named correctly fluid dynamics. It is the area where most of models have asymptotic origin based on small parameter (like viscosity, nonlinearity, etc). In this sense it is already physical approach; they try to solve a problem approximately, while pure mathematicians either try to find an exact solution or to prove that it is non-solvable.

Among the new fields of applied mathematics of the 20th century I would mention numerical methods, theory of information, theory of algorithms, game theory. Abstract algebra, theory of numbers, topology are abstract fields of math, although some parts of them are used in applied math too.

It is not only in Mathematics that there is a distinction between the Pure and the Applied. Of recent in Researchgate discussions, somebody made mention of Structural geology as opposed to Applied geology. It is not wrong to rationalize such a distinction. The truth is that some areas are clearly more applicable than others. Abstract algebra through galois theory is becoming more and more applicable but mostly it is a body of theories yet to be applied to real-life problems. In future the theory-application gap is likely to be narrower not just in Mathematics but also in other sciences as the frontiers of knowledge become extended.

Theoretical and Applied Mathematics is not it? But if the former is called pure in real sense or does it just the name or any deeper meaning is attached to it? I mean applied Mathematics is not pure?

Indeed the people in different classes or categories of mathematics do all mathematics, the logical study of structures of objects in reasoning. What differentiates them is the objects used in the structures they study. In applied mathematics,  objects are physical behaviors of realities/phenomena, while in pure mathematics, objects are abstract concepts, regardless of what they might represent in reality. For instance with no intent of purpose, or not caring what fits in the model or not, a pure mathematician  can study a quadratic equation : y = ax^2 +bx +c, simply its algebraic or analytic behaviors in x , with a, b,c  arbitrary real numbers, across the x-axis, while an applied mathematician replaces the variable x with a suitable variable say t  and y with s  and the coefficients with meaningful constants that have natural meanings to the phenomena under study. 

However, it is clear that a shortcomings in a mathematical theory to study and describe a perceived behavior of natural phenomena causes a development of a new mathematical theory, and the existence of a pure mathematical theory out there will be found by applied scientists to be a magic bullet to solve their problem. Pure mathematics and applied mathematics are superimposed galaxies of mathematics in a subliminal way where one nourishes  the other for necessity.

Regards, 

Dejenie A. Lakew

As long as it was confined to things like

v = u + ft, and

s = ut + 1/2 (ft2 ),

with f for acceleration, v for final velocity and u for initial velocity, as solutions of dv/dt = f and d2 s/dt2 = f, it was General Physics (Linear Motions).

Then came Taylor's Mean Value Theorem in Newton's differential calculus..

In that line, following Taylor's Theorem, Newton framed his finite differential calculus, starting with Newton's Forward Interpolation Formula. That became Numerical Analysis. 

Now, where exactly do we separate the findings as pure and applied?

I think this classification due to focusing of "Applied Mathematics" on solving applied problems and "Pure Mathematics" on theoretical studies. In my opinion, a researcher in applied mathematics field must has sufficient skills in pure mathematics to present good solutions to applied problems, and a researcher in pure mathematics field should has good sense to, at least, signs applications to his pure research. 

My personal opinion is that mathematics is not divisible by Pure and Applied. Faithful and value mathematical ideas are always a reflection of the reality that surrounds us and in this sense they sooner or later will be applied. Moreover, mathematics are "the language spoken by all sciences."

I agree with many of the opinions above. There is no strict borderline between pure and applied mathematics. I see myself more as applied, but some too applied people (from business, for example) can see me as a pure mathematician.

Hemanta, your example is exactly on intersection between pure and applied math. Perhaps all secondary school program in math is there. And the first year university course in math (calculus) for engineers or economists is also there. The division starts in university, at the faculty of math. There are some "esoteric" departments. I will name you some concepts that are not common for not-narrow specialists in some field of pure math. There are Hilbert and Banach spaces, Kantor set, Borel algebra, Lie group, tensor, Poisson bracket, manifold  and similar notions. When a person talks to you in this language and cannot do it simpler even knowing that you are from other branch - it is a sign of pure mathematician. If he is also applied, then he can translate this into a simpler language and provide examples understandable for  the 1st year university student of any profession.

Applied mathematician will start talking about the field of application. It may be hydrodynamics or electromagnetic waves, dynamics of fish stock in the ocean or fire in a forest, stochastic field in some media or a game between two firms. Then he will be able to tell "on fingers" about the simplest mathematical model there, what assumptions are used for its derivation, when it is valid and how to model the solution on computer.