Are then, Physics, Economy... also "branch" of Applied Mathematics? They use mathematics in their research, also.

Computer Science, where programming is, also use mathematics. Applied mathematics use computer programming to proof something, to analyse something, to "build" new theories, etc.

If you use "invert" logic, then applied mathematics is part of CS, or the circle above the part of Computing. I'am joking of course.

This question is "story of chicken and egg"... The origin of Computing is Mathematics, but over time, Computing (CE, SE, CS, IS, IT) has become a science. Dot.

I guess some type of computer programming can be arguably seen as pure applied mathematics. However, I think that computer programming is vast and cannot be considered entirely as a scientific discipline. On the other hand, numerical analysis is the field of mathematics that treat the use of algorithms and computer-logic to solve problems. In fact numerical analysis is older than computers and thanks to it computers were made possible. This area can treat non-computer problems, as approaching the solution of analytically problems by discrete approximations. It definitely could be the real branch of mathematics dealing with all the programming stuff related with mathematical and logical computation.

First, to clearly the terms Computer Programming look at the https://www.researchgate.net/post/Is_Information_Technologies_part_of_Computer_Science_or_vice_verse

Computer Programming is part of Computer Science.

Goes to its philosophy and terminology computing and programming is mathematic applied and developed by. But in application and diversification it will be on computer science.

Historical and in essence it goes all the way to applied mathematic.

Are then, Physics, Economy... also "branch" of Applied Mathematics? They use mathematics in their research, also.

Computer Science, where programming is, also use mathematics. Applied mathematics use computer programming to proof something, to analyse something, to "build" new theories, etc.

If you use "invert" logic, then applied mathematics is part of CS, or the circle above the part of Computing. I'am joking of course.

This question is "story of chicken and egg"... The origin of Computing is Mathematics, but over time, Computing (CE, SE, CS, IS, IT) has become a science. Dot.

I forgot... The field of Computing includes computer engineering, software engineering, computer science, information systems, and information technology.

Sure Prof. Jerinic , i dont say its an applied math just philisophy,term and history.

It has its own root now that is computer as Prof. Jerinic says before.

But for informatics and informations system some university define its in business and management for univ in canada and social sciences in sweden and denmark.

There are number problems particularly in Fluid Dynamics, which were not solved because hand calculations were impossible. Applications of numercal methods and estimations were made upto certain number o steps only. Iterative procedures were worked out for a few number of times only. Lot of calculation are involved to solve problems in Operations Research. Advent of superfast Computers helped researchers to solve and analyse many problems in these fields. Development of Computers lead to Computer Science and is spreading in many directions such as hardware, software. Today there may not be any hardware without a software glued in it. Of course, the basis for computer science is the Mathematical Logic. However, I will not consider Computer Programing as a branch of applied mathematics. But both applied mathematics and computer programming are supplementing each other to help researchers to tackle the problems on hand. Some engineers also does accept that computer programming (Software Engineering) as an Engineering subject, since they feel design and development of software requires a paper and a pen with human brain (skill).

Mathematics has evolved since the beginning of civilization and throughout the centuries. The computer age is a technological extension to mathematics and accordingly computer programming, data structures, algorithms, artificial languages, et al. ought to be considered under applied mathematics.

Nowadays all science and engineering students must be competent in writing computer programs. Therefore it is necessary by all (agriculture, engineering, Bio-medical, ...). In some cases especially engineering the students must take "Numerical Analysis" which maybe regarded as part of Applied Mathematics is very helpful. So it is a basic course by almost all students and should not regarded as a branch of applied mathematics. In engineering fields I think lecturers who have engineering degrees are more successful teaching computer programming to the students of their own department than lecturers coming with pure computer science or mathematical degrees background..

It depends on the way a particular problen is being solved, If there are formal results proved mathematically with rigor then it is Applied Mathematics. If here is just " an ad hoc" solution problem for a practical objective then it is an Engineering issue. Since Enginnering and Applied Mathematics have common frontiers ( for instance, Control Theory, Lyapunov stability, Nonlinear stability etc.) , it can be very difficult to classify in some cases or , may be, it can be said that a problem belongs to the two science fields. This is the reason because many journals are classified in both disciplines in the scientific databases. If a computer programming problem is just a programming one based on using models and results from outside, then I would not say that this is " Applied Mathematics".

Computer programming itself isn't Computer Science (it is an art, not a science; but definitely applied many ideas), so I'd doubt itself being a branch of applied mathematics in itself. Now on the other hand, if you are talking about how programming languages are formed (can be seen in the realms of Pure or even Applied Mathematics with formal systems in Theoretical Computer Science), the development of algorithms (Theoretical Computer Science), and proving the "reach" of a programming language; these in the end are all mathematical in nature. Theoretical Computer Science underpins Computer Science, and of course can be seen as a flavour of Pure Mathematics and Applied Mathematics (especially Discrete Mathematics and Foundations).

So following this, I'd say definitely no; computer programming isn't a branch of applied mathematics as it isn't even Computer Science (it's an activity that is applied CS).

Dear @Anup, Nowadays, it is not easy to draw borders between the fields; it depends from which point of view you are dealing with computer programming. Myself as an example, I am working in the fields of computer graphics and computer aided geometric design, and this field fits in both computer science as well as mathematics. I prefer to be a mathematician.

Undergraduate and graduate students sometimes wonder what applied mathematics is. Students often form an impression of applied math based on conversations with other students or conversations with and courses taught by particular faculty members. Unfortunately this impression can be wrong in the sense that a subset of applied mathematics may seem to represent all of applied math.

Applied (or perhaps Applicable) Mathematics consists of mathematical techniques and results, including those from "pure" math areas such as (abstract) algebra or algebraic topology, which are used to assist in the investigation of problems or questions originating outside of mathematics.

Mathematical and computational techniques from ordinary and partial differential equations are often (justifiably) used as examples of applied mathematics. Mathematicians occasionally disagree on the answers to the questions in the title and some may view applied math (outside of probability & statistics) as being largely limited to ordinary and partial differential equations, continuum mechanics, the calculus of variations and numerical computations. In my opinion, this view is wrong. For example, it ignores the contributions of algebra to physics and to the recently completed Human Genome Project.

Various definitions of applied math including its relationship with computer programming can be found below.

1. Applied Mathematics concerns the application of mathematics in a wide range of disciplines in various areas such as science, technology, business and commerce. Applied mathematicians are engaged in the creation, study and application of advanced mathematical methods relevant to specific problems. Once this referred mainly to the application of mathematics to such disciplines as mechanics and fluid dynamics, but currently, applied mathematics has assumed a much broader meaning and embraces such diverse fields as communication theory, theory of optimisation, theory of games and numerical analysis. Indeed, today there is a remarkable range and variety of applications of mathematics in industry and government, involving important real-world problems such as materials processing, design, medical diagnosis, development of financial products, network management and weather prediction.

2. Human beings have innate natural tendencies to count, to quantify, and to apply logic in their attempts at understanding the world. Mathematics is the human endeavor which has come to provide definition and scope to these activities, in terms which employ the utmost precision of thought. A few ancient civilizations developed mathematical systems to some extent, for example some of the Babylonians, but the first great step in the establishment of mathematics was made by the Greeks between 600 and 300 BCE. The contribution of Euclid was to state theorems in geometry and to construct their proofs from a small number of basic statements, called axioms, taken as the starting point of the subject. This showed that new knowledge could be obtained by pure reasoning about the basic axioms. Though geometrical ideas and structures have since been greatly broadened and deepened Euclidean geometry remains an important and useful part of modern mathematics and science.

Attention to geometrical ideas predates Euclid and arose from a desire to quantify physical space. Arithmetic, the subject of operations with various kinds of numbers, arose from the need to count and quantify any objects. The roots of probably all major divisions of mathematics go back to concern about practical matters or knowledge of the natural world. In a few situations serious investigation of the natural world has lead directly to the creation of whole areas of mathematics that not only produced methods for formulating and solving important physical problems, but lead, by further development, to new advanced mathematical subjects. The most striking example of this is the invention of calculus in the 17th century to solve the problem of motion, particularly the motion of the planets under their mutual gravitational attractions. The development and extension of calculus, along with the construction of a mathematical foundation for it, lead to the subject called analysis, a major part of mathematics.

But mathematics itself is not physical theory. The growth of mathematical roots, as mathematics, involves abstraction away from concern with particular objects towards an emphasis on relations among abstract objects, axiomatization, and establishment of the basic characteristics and facts of a subject by rigorous proof. A piece of mathematics developed this way may be considered to stand alone as a coherent logical structure independent of any connections to the physical world that may be possible. Its inherent content often suggests to mathematicians ways for further axiomatization and logical development which lead to new interesting structures, often to ones for which no relation to the physical world seems possible. However, experience has shown that areas of mathematics developed from purely mathematical motivation do find, with surprising frequency, significant application in the real world, sometimes many years after their development. Thus there are two intrinsic connections of mathematics with the real world, the direct one, illustrated by the invention of calculus, and what might at first be called the serendipitous. But the latter is so prevalent that it forces the recognition that any area of mathematics may prove useful. In addition to geometry and analysis the other major subject areas of mathematics are algebra (the study of equations and associated abstract structures), discrete mathematics (the study of sets of discrete quantities), topology (the study of continuous transformations of objects), number theory (the study of the properties of numbers), and set theory and logic (the basis for how to make precise arguments starting from the foundations of mathematics). Each of these subjects contains areas of fruitful application to real world problems.

This is the setting in which the meaning of the term "applied mathematics" is to be understood. Modern applied mathematics is two things. It is the attempt to use mathematics to quantify and solve problems which arise in investigation of the physical world and human enterprise. It is also the study and further development of those areas of mathematics that have proven the most useful in solving real world problems or seem to offer promise for present problems. Thus a distinction between applied and pure mathematics is one of the interest and purpose of the practitioner, it is not a fixed dichotomy or division of the subject areas of mathematics. The two different interests, one in applying mathematics and the other in mathematics for its own sake, have always progressed together and probably could not progress without each other. Attempts to solve real world problems have created important and lasting aspects of mathematics, and the purest of mathematical constructs have found application.

Historically, applied mathematics mostly concerned theoretical physics and physical problems. The fundamental laws of physics are formulated as mathematical equations governing the behavior of physical quantities. These provide our deepest understanding of the physical world and our most accurate predictions of physical phenomena. While problems arising from physics and ranging from the study of classical fluids to quantum systems remain a significant part, applied mathematics has grown to include a wide variety of other areas such as bio-mathematics, cryptography, scientific computation, mathematical modeling, economics, financial mathematics, operations research, and engineering problems.

3. Mathematics is used to help understand the real world and to help to change parts of it for Man's benefit. It is used in such diverse fields as engineering, physics, biology, economics, environmental studies, chemistry, political studies, medicine etc, etc, etc. The first step in this process is often the construction of a mathematical model, i.e. a description of the problem in mathematical terms. This model is then studied by using analytical or numerical methods to obtain exact or approximate solutions. Finally, the conclusions are interpreted in the language of the original problem in terms more familiar to a client or user. Often the model is changed to be more realistic or to include more features of the problem. Thus, the modeling process may involve false starts, modifications, and simplifications.

Mathematics enters primarily in the second stage: the solution of mathematically well-formulated problems and the development and analysis of the underlying theory. This stage may include analytical or numerical methods. The approach can range from specific algorithms and formal methods to abstract, general theories. It is often not clear which mathematical skills will be useful in the study of a new problem; thus, applied mathematicians need to be broadly trained so they will have a wide variety of mathematical tools available.

The mathematical scientist must not only be a competent mathematician, but must be knowledgeable in the area to which mathematics is being applied.

The art of formulating models requires that the modeler make choices about which factors to include and which to exclude. The goal is to produce a model that is realistic enough that it reflects the essential aspects of the phenomena being modeled, but simple enough that it can be treated mathematically.

Often the model is constructed to answer a specific question. Sometimes the modeler must either simplify the model so it can be analysed, or devise new mathematical methods that will permit an analysis of the model. Often a combination of analytical and numerical methods are used. The modeling process may involve a sequence of models of increasing complexity. Problems sometimes lead to new mathematical methods, and existing mathematical methods often lead to new insights into the problems. The successful applied mathematical scientist must be comfortable and confident in both mathematics and the field of application.

Applied mathematical and computational sciences is a name used to encompass the many analytical and numerical methods used to solve certain types of scientific problems. This name more accurately reflects the nature of modern "applied mathematics" since it also includes the area of scientific computing, which includes many components of computational processes, such as numerical analysis, algorithms for machines with vector and parallel architectures, visualization, simulation, and computer-aided design.

4. Applied mathematics is a branch of mathematics that concerns itself with the application of mathematical knowledge to other domains. Such applications include numerical analysis, mathematics of engineering, linear programming, optimization and operations research, continuous modelling, mathematical biology and bioinformatics, information theory, game theory, probability and statistics, financial mathematics, actuarial science, cryptography and hence combinatorics and even finite geometry to some extent, graph theory as applied to network analysis, and a great deal of what is called computer science.

The question of what is applied mathematics does not answer to logical classification so much as to the sociology of professionals who use mathematics. The mathematical methods are usually applied to the specific problem field by means of a mathematical model of the system.

5. Certainly, in the past, pure and applied mathematicians have argued at length over which endeavor is nobler. I think that there is not so much conflict along those lines today. It certainly is the case that mathematics is much broader in scope than the portion of it which is traditionally of highest interests to physicists. Physics is only one area of applications. There are lots of significant applications of mathematics to fields such as economics, business, biology, computer science, the social sciences, etc. There seems to be both an old and a new meaning to the term "applied mathematics".

6. Mathematics is an enigmatic field of research. On the one hand, it is a field of research in its own right in which fundamental mathematical studies are undertaken without particular applications in mind. On the other hand, mathematics is the essential language used in the development of abstract models of reality that underpin R and D in countless other fields, including applications in industry. Applied mathematics covers both the development of new mathematical techniques that can be used for practical applications and the application of such techniques. Industrial mathematics is the extension of applied mathematics to industrial applications ;

Down-voting a question/answer is a cowardly pathetic act if the down-voter is not able to share his point of view.

Dear Ziad. I cannot agree more with your comments.

Thanks Jorge!

Dear Ziad. If hitting Orange button was on some basis it could be justified. But going through the answers for this particular question it looks someone is doing it systematically or on purpose or s/he like pushing Orange button. My suggestion to you would be to ignore up or down votes altogether and concentrate on the answers.

Thanks Dear Mahmoud for your comment.

Personally, I care less if any one down voted my questions/answers - This is part of the democracy to be exercised in this platform and there is acceptance that there are other opinions that may be opposing to one's believes.

But, when someone remains anonymous and systematically exercises an unjustified down-voting for other colleagues, I think the issue becomes one of morality and professional ethics that we should all abide with.

Nevertheless, I agree with you that we should concentrate on the question and answers.

I am not quite following how someone being insulted by "up" and "down" voting a comment has any relevance to this question. People are entitled to their opinion and though I think it is a good idea for people to give their stance (like others have), they reserve the right to retain their opinion without disclosing it. It isn't cowardly. Is it cowardly when somebody goes onto a YouTube video and "thumbs down" a video? No, it just means they dislike it. They may or may not have a sound reason for doing so. No different here. If that offends you, I'd suggest ignoring it. This site is a social media site for researchers to communicate their ideas or a resource to ask questions to other researchers. If one cannot accept the idea that people may like or dislike an opinion, one may not find the Internet to be a welcoming place as even if something is correct, there will still be people who will dislike it; no matter how well convinced one can be. It isn't something to get offended at. This isn't a matter of morality, or professional ethics.

Hope this helps!

The issue of voting is given more attention than it deserves. I think that it is time to move on with the original question.

In my opinion, surely it is. Many of the topic of applied mathematics are related to large matrix which is best solved by using computer programming.

In EWD 498: "How do we tell truths that might hurt?" Professor Edsger W. Dijkstra, wrote "Programming is one of the most difficult branches of applied mathematics; the poorer mathematicians had better remain pure mathematicians".