The division of “Mathematics” into two “kinds” is fictitious and misleading. A particular mathematical technique may turn out to be useful in "practical" situations and, conversely, the application of mathematics to the solution of a practical problem may lead to new mathematical discoveries. In either case, the adjectives “pure” and “applied” are not descriptors of the technique itself, but of its context.

Pure means it is inclined on logic completeness, applied means it is inclined to analysis and solve the particular problems.

Most of pure mathematicians recognize that their work is done for their own fun, whereas most of applied mathematicians think that their work is useful :)

Pure mathematics is the manufacturer of applied mathematics tools.

Pure mathematics is the raw material while applied mathematics is the produced good,so without pure mathematics I don't think we can have applied mathematics and other aspects of science.

In some occasions, the motive of a mathematician had been to contribute to pure mathematics but her/his contribution was later proved to be tailor-made for another field, as it happens, for instance, with the mathematics of Hilbert space.

As Hilbert himself admits [1] “I developed my theory of infinitely many variables from purely mathematical interests and even called it ‘spectral analysis’, without any presentiment that it would later find an application to the actual spectrum of physics!”

[1] Constance Reid, Hilbert-Courant, (Springer-Verlag New York Inc., 1986).

The division of “Mathematics” into two “kinds” is fictitious and misleading. A particular mathematical technique may turn out to be useful in "practical" situations and, conversely, the application of mathematics to the solution of a practical problem may lead to new mathematical discoveries. In either case, the adjectives “pure” and “applied” are not descriptors of the technique itself, but of its context.

The distinction usually boils down to the underlying motivation of its practitioners. Applied mathematics is mainly guided by problems with "real world" applications, in other words, its main priority is "utility" over "esthetic beauty". This is not to suggest that applied mathematics is necessarily bland and lifeless; it also has a firm theoretical foundation and many of its results may be regarded as intrinsically beautiful. However, beauty is rarely the focus of the applied mathematician.

In contrast, the pure mathematician focuses on beauty and rarely cares if there are any applications of the work, the only exception being to further the theory of pure mathematics.

As others have pointed out, it is not infrequent that results from pure mathematics play a significant role in applied mathematics. However, this does not occur overnight. Take, for example, the imaginary number i, which was discovered (invented?) by pure mathematicians long before anyone suspected it would have applications in electrical engineering and other fields. Its discovery was based on an attempt to extend the real numbers to an algebraically complete field, in other words, it was fabricated for the sake of beauty and beauty alone. It would be over a century before applications would arise, however the allure was unmistakable and instantaneous for the pure mathematicians of the time.

Applied mathematics is mathematics as a language that is used in the development of information programming systems. Pure mathematics is the fundamental science of spaces of various dimensions.

In approximately 1900, any graduate Ph.D. student in mathematics knows everything in mathematics. Nowadays the rapid development of all branches in science made the professionals will never be knowledgeable of the whole domain.

So, specialization is the solution to classify scientists and researchers.

Notably, we are witnesses of the emerge of many branches in science every (short) period.

All branches in science started to be classified and categorized into tiny domains. This is a natural action to bulk the extensive information in each domain.

Consequently, mathematicians divide mathematics into applied and pure to show that some topics are often used in our life and so-called applied, and some topics are not used explicitly in a clear popular application. And hence, called a pure Math.

This is an artificial difference, and it is an attempt to do some classifications in the same department.

Precisely the same, to say, this doctor is a specialist in kidney, and the other doctor is a specialist in the brain. Both doctors should know about the whole body organs and in particular the kidney and brain.

But each has more experience and more knowledgeable in his field. The classification of each one will be more beneficial to humanity and yields more accurate treatment for their patients.

A mathematician who is an expert in topology, know what is numerical analysis and conversely.

But to do research and to teach high-level courses, specialization is a must to produce excellent results.

In conclusion, pure mathematics is somehow the same as the applied mathematics and the applied mathematics is pure mathematics in its basics.

Thanks to all.

It is always Mathematics, which is divided into branches for easier reading (especially for those who do not math of Mathematics).

apparenlty this is a division. It is, maybe more properly, a Yin Yang polarization but this is not a point of view of wetern thought.

Western thought tends to divide,separe, categorize, up to quantum mechanics vs general relativity. I think there is little hope of achieving a synthesis that is not trivially descriptive at least through western thought alone

Pure mathematics is a fine art, exploring imaginary generalizations and relationships. Applied mathematics is a hard science, concerned with real-world examples.

Both of them are very important and researchers on two branches studied it, almost cases, by passion. But I guest that who want to do applied mathematics need a great manipulation of pure math.

I think that for even the same question , if you want to go up ,it becomes pure ,if you want to go down ,it becomes applied. The way you organize your information determines whether it is pure or applied.

I agree. Science is a downer.

dear Zhao Shi Hi

not only the way of information but the physico-mental structure of the scolar too

Today's pure math might be tomorrow's applied math if an application is found. The only difference is the motivation that created them, with pure interest the motivation for one case, and practical applications the motivation for the other. I have the same respect for both.

We might consider a third category of math (beyond pure and applied) that might be called "sloppy math". During my college years as a physics major I often encountered textbook explanations of the properties of some function with the explanations based on the idea that some series representing the function has nice convergence properties (e.g., term-wise differentiable). The nice convergence properties were just taken for granted without proof but used to reach some conclusions. The success (in the sense of getting correct answers to the textbook homework problems) of these kinds of arguments created the illusion that this method of analysis is respectable. In reality, this sloppy math that is often found in science and engineering textbooks works for the textbook homework problems but is not reliable other problems, even real-life practical problems.

Instead of making a distinction between pure and applied math, which are both respectable, I would rather see sloppy math being discouraged from our educational system.

I don't think it's right to exclude math from science. Also, we not only have pure math, but also pure physics, pure chemistry, even theoreticle engineering( you may think it as pure engineering). The aim determines determines whether it's pure or applied.

An interesting question and not really as simple as the previous answers might imply. Of course, they are correct in the sense that in everyday usage of math the distinction between pure and applied math lies mostly in the aim/goal of the efforts, the context, the likelihood that the results can be used "outside" of traditional math areas.

However, there is an important aspect that, in my opinion, adds an extra dimension to this. From Goedel's incompletness theorem we know that there are provably undecidable problems/theorems/statements, and we know today that these are uncountably many (https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems, https://en.wikipedia.org/wiki/List_of_undecidable_problems). Now, every time we encounter such a statement, we have only one way to go ahead: we split the whole math into two branches: one in which the statement holds and one in which holds its logical negation. These two new Math branches stem from the same common stem but are mutually incompatible. They can give rise to two incompatible Math's (yes, Math is not unique) which can grow to infinity. Moreover, there are uncountably many such branching points.

Now, to make this more concrete: we may have one Math in which there exist sets with a cardinality sharply inetrmediate between that of integer numbers and that of continuum, and another Math in which such sets do not exist. There is a rigorous proff of this; and the proof says that the two 'options' are both possible though, being incompatible, they can not coexist. The fact that despite many efforts we never succeded to make a real-world model of such a set (the problem that nearly drove Goerge Cantor nuts, they say) indicates that, most probably, the physical Reality is incompatible with the Math in which such sets exist.

Yet there are Mathematicians who study that branch of Math, and who will eventually write whole libraries about it. None of that work will certainly ever be applicable to anything "practical" even though it will be both rigorous and correct. Therefore, it will remain forever in the realm of pure and totally abstract Math.

So mathematicians will offer scientists an even wider range of theoretical tools for trying to explain the structure of perceived reality (things they are both remarkably good at).

Pure mathematics is the mathematics that has not yet found a scientific field to be applied in while applied mathematics is that mathematics which serves other sciences.

As a result, we say, the two are mathematicians.

Let us not overlook theological applications -- all those disputes over Trinitarian doctrines (Christian and otherwise) that have bloodied so much of human history.

I do agree, there is always despise from one to another , one says that applied is too low , while another thinks that pure is too unrealistic and fantasy.

I still agree that is no absolute distinction between them. To start a project, one must consider a project that may be useful. On the other hand , when we encounter something that may be useless , not to give an unpartial treatment to this but just neglect it is not a way to do researh. Pure may become applied and applied may become pure.

I don't think it's right to exclude math from science