THE RESEARCH OF MATHEMATICS

INTRODUCTION
The nature of mathematics consist of formal mathematics/axiomatic mathematics/pure mathematics, applied mathematics, school mathematics/ concret mathematics/real mathematics.
1. Formal mathematics/axiomatic mathematics/pure mathematics.
Mathematics is a deductive system consists of definitions, axioms, and theorems in which there is no contradiction inside. It is very easy to establish mathematical system step by step like make a definition then use the axiom and theorem, proof the theorem. The substance of formal mathematics such as numbers theory, group theory, ring theory, field theory, Euclidian Geometry, Non Euclidian,Geometry,etc.
2.Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. The are a lot of applied mathematics in this world, in optic, mechanic, astronomy, engineering, etc. For example we use angle in compass if going to sea, mountain,etc.
3. School mathematics/ concret mathematics/ real mathematics.
School mathematics is really different with formal mathematics. In school mathematics we just focus in mathematics phenomenon. We must transform the mathematics phenomenon with its abstraction and idealization to the student’s mindset. So, the student just need to aware about the characteristic of the object. It is need awareness and intention from the student.
According to Ebbute Straker (1995), school mathematics is about :
- pattern / relationship
- problem solving
- investigation
- communication
To identify mathematics problem we need mathematics knowledge, mathematics system, and mathematics characteristic. The three aspects above can we get easily if we have a will, attitude, knowledge, skill, and experience.
PREFACE
In this time, I try to take a research of formal mathematics, especially about number theory. Formal mathematics builds on formal logic. It reduces mathematical relationships to questions of set membership. The only undefined primitive object in formal mathematics is the empty set that contains nothing at all.
BACKGROUND
There is a lot of mathematical knowledge. This knowledge is mainly stored in books and in the heads of mathematicians and other scientists. Mathematics should be more readily available to be used by others. In this respect, a positive thing about mathematical knowledge is that it has a rather formal, which makes it easier to formalize.
A number is a mathematical object used in counting and measuring. There a lot of kinds of number. If we study mathemathics, we can not without number, whereas there are many kinds of number. So, in this time, i try take a research about formal mathematics, especially about number theory.
PURPOSES
The aim of the research of mathematics is to examine and develop mathematics. If we want to be a mathematician we must doing the research of mathematics to develop our subject. To be able to productively develop mathematical theories with all theorems and proofs formal
METHOD
There are a lot of methods that we can use in the research of mathematics, such:
- by analyze the data,
- by collect data/ literature,
- deductive method /syntetic method
-by hermeunitika method, etc
DISCUSSION
Number theory is one of the oldest branches of pure mathematics, and one of the largest. Of course, it concerns questions about numbers, usually meaning whole numbers or rational numbers (fractions).
Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated, such us: Elementary number theory, analytic number theory, algebraic number theory, geometry of number, combinatorial number theory, computational number theory, Arithmetic dynamics, Modular forms, etc.
Elementary number theory involves divisibility among integers. ", the Euclidean algorithm (and thus the existence of greatest common divisors), elementary properties of primes (the unique factorization theorem, the infinitude of primes), congruences (and the structure of the sets Z/nZ as commutative rings), including Fermat's little theorem and Euler's theorem extending it. But the term "elementary" is usually used in this setting only to mean that no advanced tools from other areas are used not that the results themselves are simple.
In algebraic number theory, the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. These domains contain elements analogous to the integers, the so called algebraic integers. In this setting, the familiar features of the integers (e.g. unique factorization) need not hold.
The geometry of numbers incorporates some basic geometric concepts, such as lattices, into number-theoretic questions. It starts with Minkowski's theorem about lattice points in convex sets, and leads to basic proofs of the finiteness of the class number and Dirichlet's unit theorem, two fundamental theorems in algebraic number theory.
Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers.
Computational number theory studies algorithms relevant in number theory. Fast algorithms for prime testing and integer factorization have important applications in cryptography.
CONCLUSION
From the above we have field has application in mathematics. Number theory has been applied to: physics, chemistry, biology, computing, engineering, coding and cryptography, random number generation, acoustics, communications, graphic design and even music and business. Most of common people can not applicate the number theory in the dailyy life.








REFERENCES
Encarta
http://en.wikipedia.org/wiki/Number_theory(23-12-09)
http://www.math.niu.edu/~rusin/known-math/index/11-XX.html923-12-09)
http://www.mtnmath.com/whatth/node21.html(23-12-09)