When talking about mathematics, we must first define what mathematics is, what its purpose is, and what its tasks are.

The usual definition is the most general: mathematics is the science of quantities precisely measured.

To measure any quantity means to compare it to a quantity similar to it, taken as a unit, and to express the resulting ratio as a number. Hence the more particular definition: mathematics is the science of numbers in general.

Applied mathematics is the application of the mathematical method in various fields, such as physics, engineering, medicine, biology, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term applied mathematics also describes a professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.

In the past, practical applications motivated the development of mathematical theories, which then became the subject of pure mathematics, where abstract concepts are studied in and of themselves. Thus, the activities of applied mathematics are closely related to research in pure mathematics.

Today, the term “applied mathematics” is used in a broader sense. It includes the classical areas mentioned above as well as other areas that are becoming increasingly important in applications. Even areas such as number theory, which are part of pure mathematics, are now important in applications (e.g., cryptography ), although they are not usually considered part of the field of applied mathematics per se. Sometimes the term “applicable mathematics” is used to distinguish between traditional applied mathematics, which developed along with physics, and the many areas of mathematics that are now applicable to real-world problems.

There is no consensus on what the various branches of applied mathematics are. This categorization is complicated by the fact that mathematics and science change over time and by the way universities organize departments, courses, and degrees.

Many mathematicians make a distinction between “applied mathematics,” which deals with mathematical methods, and “applications of mathematics” in science and technology. a biologist using a population model and applying known mathematics would not mean applied mathematics, but using it; but mathematical biologists have posed challenges that have stimulated the growth of pure mathematics. Mathematicians such as Poincaré and Arnold deny the existence of “applied mathematics” and claim that only “applications of mathematics” exist. Similarly, non-mathematicians confuse applied mathematics and applications of mathematics. The use and development of mathematics to solve industrial problems is also called “industrial” mathematics.

The success of modern numerical mathematical methods and software has led to the emergence of computational mathematics, computational science and computer science. They use high-performance computing to model phenomena and solve problems in science and technology. They are often considered interdisciplinary.