Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.
A computer is a device that can be instructed to carry out sequences of arithmetic or logical operations automatically via computer programming. Modern computers have the ability to follow generalized sets of operations, called programs. These programs enable computers to perform an extremely wide range of tasks.
Computer science is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. It is the scientific and practical approach to computation and its applications and the systematic study of the feasibility, structure, expression, and mechanization of the methodical procedures (or algorithms) that underlie the acquisition, representation, processing, storage, communication of, and access to, information. An alternate, more succinct definition of computer science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems. See glossary of computer science.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change. It has no generally accepted definition.
Science (from Latin scientia, meaning "knowledge") is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Interviewer: Is studying computer science the best way to prepare to be a programmer? Bill Gates: No. the best way to prepare is to write programs, and to study great programs that other people have written. In my case, I went to the garbage cans at the Computer Science Center and I fished out listings of their operating system. You got to be willing to read other people's code, then write your own, then have other people review your code. You've got to want to be in this incredible feedback loop where you get the world-class people to tell you what you're doing wrong.
Bill Gates cited in: "Programmers at Work: Interviews With 19 Programmers Who Shaped the Computer Industry", Tempus, by Susan Lammers (Editor)
The purpose of computing is insight, not numbers.
Richard Hamming (1962) Numerical Methods for Scientists and Engineers. Preface
[Computers] are developing so rapidly that even computer scientists cannot keep up with them. It must be bewildering to most mathematicians and engineers... In spite of the diversity of the applications, the methods of attacking the difficult problems with computers show a great unity, and the name of Computer Sciences is being attached to the discipline as it emerges. It must be understood, however, that this is still a young field whose structure is still nebulous. The student will find a great many more problems than answers.
George Forsythe (1961) "Engineering students must learn both computing and mathematics". J. Eng. Educ. 52 (1961), p. 177. as cited in (Knuth, 1972) According to Donald Knuth in this quote Forsythe coined the term "computer science".