## Applied Mathematics Program

They include Babylon and the Greek empires. The first step in the establishment of mathematics was taken by the Greeks. Euclid’s contributions were to state theorems in geometry and construct their proofs from a small number of basic statements called axioms. Axioms in the Applied Mathematics Program are taken as the starting point of the subject. This showed that knowledge could be obtained by pure reasoning about the basic axioms.

# The Genesis of subjects in the Applied Mathematics Program

Euclidean geometry remains an important and useful part of modern mathematics and science even though geometrical ideas and structures have since been greatly widened and deepened. Attention to geometrical ideas predates Euclid and arose from a desire to quantify physical space in the Applied Mathematics Program. Arithmetic in the Applied Mathematics Program is the subject of operations with various kinds of numbers. It arose from the need to quantify and count objects. The roots of arguably all major divisions of mathematics in the Applied Mathematics Program concern practical matters or knowledge of the nature.

In some cases, serious investigation of nature has lead directly to the creation of whole subjects of mathematics that not only produced methods for formulating and solving important physical problems, but lead to new advanced mathematical subjects by further improvement. The invention of calculus in the 17^{th} century to solve motion problems, especially the motion of planets under their mutual gravitational attractions is the most striking example of the creation of new advanced mathematical subjects by further improvement in the Applied Mathematics Program. The development and expansion of calculus, along with the construction of a mathematical base (foundation) for it, lead to the subject called analysis. Analysis in the Applied Mathematics Program is a major part of mathematics.

Mathematics though is not physical theory. The growth of the mathematical base, involves abstraction away from concern with particular objects (towards an emphasis on relations among abstract objects) and the use of axioms in the establishment of the basic definitions and facts of a subject by serious proof. A piece of math developed this way may be seen to stand alone as an independent logical structure in the Applied Mathematics Program i.e. with no connections to the physical world. However, experience has shown that areas of math developed from purely mathematical interests do find, with surprising monotony, significant application in reality, sometimes many years after their development.

## Connections of Mathematics in Studies in Applied Mathematics Program

There are two intrinsic connections of mathematics with the real world in the Applied Mathematics Program: –

- The direct one. This is illustrated by the development of calculus.
- Serendipitous.

The direct connection is so prevalent that it forces the conclusion that any area of mathematics may be useful. In addition to geometry and analysis: -

- Algebra (study of equations and abstract structures),
- Discrete math (study of sets of discrete quantities),
- Topology (study of continuous changes (transformations) of objects), number theory (study of the properties of numbers), and
- Set theory and logic (the basis for how to make precise arguments starting from the base of mathematics) are the other major subject areas of mathematics in Studies in Applied Mathematics.

The above define the term ‘applied mathematics’ in the Applied Mathematics Program.

### Conclusion on the Applied Mathematics Program

Modern applied mathematics in the Applied Mathematics Program has two definitions: -

- It is an attempt to use math to quantify and solve problems which arise when studying the physical world and human enterprise.
- It is also the study and further development of those areas in math that have proven the most helpful in solving problems of the real world.

Being able to distinguish between applied and pure mathematics should be one of the interest and purpose of any practitioner in the *Applied Mathematics Program*.