# Mathematics/Applied sciences/Introduction/Section

Mathematics has a multitude of applications. E.g., it helps to find a suitable room for a given audience, by comparing the number of seats with the number of persons. Both sets are counted with the natural numbers and mathematics provides methods to compare the two numbers directly, so that one does not have to try several rooms with the people. Mathematics helps to balance monthly debits and credits into balance by computing from the monthly income how much one may spend per day.

Mathematics has applications in the technical area, in image processing, in medicine, in data transmission and data storage, in the design of search algorithms, in solving logistic problems, in data science, machine learning, artificial intelligence. Here, we are mainly interested in applications of mathematics in other sciences. Mathematics is used in many different sciences, in the natural sciences like physics, chemistry, biology, in computer sciences, it is used in the form of statistics in psychology, sociology and history, as geometric device in cartography, in economics. Mathematics provides a language to describe empirical incidents in a quantitative way, it helps *modelling* theories about observable phenomena. Using techniques like *interpolation* and *extrapolation*, it allows giving predictions like for the weather or the development of the population on earth.

In these different areas, a huge variety of mathematical methods is applied, which differs strongly in its complexity, in terms of the mathematical structure and in terms of the application. E.g., statistics is a complex mathematical discipline, but in many sciences it is used
(for good reason)
directly as a *black box* without deeper reflection, for example to evaluate the reliability of a psychological test. On the opposite side, in
(theoretical)
physics there is such a tight connection with mathematics so that the theories in physics can not even exist without the mathematics and that a large portion of mathematics originates from physics. Regarding differential equations, one might be only interested in their numerics
(concrete approximating techniques of computations),
or whether they are adequate for a certain problem, or for the qualitative behavior of a solution. So mathematics is used in a broad range regarding depth in the sciences.

However, for most mathematical applications, it seems fair to say that an appropriate and reasonable use of mathematics requires a profound mathematical education, which can not be acquired in the context of concrete problems alone. It is the target of this course to build a mathematical base in order to persist in several scientific contexts.