The Number Theory

There are a lot of number theories you can find if you talk about the branch of pure mathematics. The theory will be concerned to deal with the properties of numbers in general and also the integers in particular. In this case, the Number theory can be divided into some fields and each field is named based on the methods used as well as the type of questions which are investigated.
If you have ever heard about the term “arithmetic” or even the “the higher arithmetic”, it can be classified into the elementary number theory. But, those terms were used in the past and no longer popular recently. Some of the examples are such as arithmetic geometry, arithmetic of elliptic curves and also the arithmetic functions.

1. Analytic Number Theory

The Analytic number theory will employ the machinery of both calculus and complex analysis to tackle the entire questions about integers. The two most popular examples are such as the prime number theorem (PNT) as well as the related Riemann hypothesis. Further, the Proofs of the transcendence of mathematical constants are classified in the analytic number theory as well, such as such as π or e. Anyway, the use of term “transcendental numbers” is currently removed from Integers study.

2. Algebraic Number Theory

If you talk about the algebraic number theory, you would have found that the concept of a number is greatly expanded to the algebraic numbers where you will find the idea about the polynomials with rational coefficients too. This theory consists of the elements analogous to the integers and thus the theory is called algebraic integers.

3. Combinatorial Number Theory

When someone talks about the combinatorial number theory, they will need to deal with the number theoretic problems which mostly involve combinatorial ideas for both formulations and solutions. This branch of theory will bring people to recognize the founder called Paul Erdős. Some of the typical topics are such as the partitions, zero-sum problems, covering systems, various restricted sumsets as well as the arithmetic progressions in a set of integers. Please remember that the algebraic or analytic methods become the very powerful method in this field.
The study about number can be traced from the history too. You can learn more about the Greek Number Theory, Islamic Number Theory and also Classical Indian Number Theory. There are also some other theories such as Early European Number Theory. The study is commonly talks about and deals with the symbols which are used to refer to the numbers. The study will also talk about the scientists which find some theories in some fields and how those theories are developed into the more extensive and comprehensive patterns.

The study about number is not only a study about the symbols and some patterns to come up with particular results. It is about the study of culture and how science is developed from time to time as well. The theories are actually almost the same in all parts of the world, but the difference is probably (only) about the symbol. Numbers are easily found in our daily life and it completes the two other most essential attributes in human civilizations: picture and words.