Importance of Number Theory:-
- Number theory is an important branch of mathematics. It is widely used in research field.
- Number theory is very important in mathematics as well as in daily life including security, memory management, authentication, coding theory, etc.
- Number theory is base of the algebra. In modern arithmetic also number theory is useful.
- In cryptographic number theory is used.
- In cryptographic number theory is used for security purpose. Using number theory we can hide information in cryptographic. Prime numbers, divisors and congruencies are used for this purpose. The congruencies are used for RSA public key cryptography and in Caesar ciphering key cryptography. Number theory and cryptographic together used in computer applications. .
- Number theory is quite useful in studying continued fractions, Diophantine equations,
- Fibonacci numbers and writing algorithms for computing the value of irrational number pi.
- Some basic concepts of number theory such as Pythagorean triplet, Fermat numbers, Divisibility and lattice points are quite useful in algebra and geometry.
- Number theory has large applications in understanding geometry.
- Number theory guide us to study the information theory which in turn useful for AI development and many other areas of study.
- Number theory gives us some interesting facts about the development of human thought and philosophy.
- Number theory is quite important in studying the number phi, which is also called the Golden Ratio.
- Phi is quite important in studying the Fibonacci numbers, study of nature, energy patterns seen in space. It is the most important number for researchers to understand how energy
- Synthesis occurs in the universe from pre-atomic wave forms.
- We all know about pi, the ratio of the circumference of a circle to its diameter is important part of number theory.
- Number theory is used extensively in network security.
- Number Theory is also used in code designs for Telecommunication theory.
- Public and Private Key encryption is based on number theory.
- Every theorem in elementary number theory used in natural way by computers to do High-speed numerical calculations. Number theory also used in numerical analysis.
- We all know that whole numbers are important part of our daily life mathematics.
Difficulties faced by a student while solving Number Theory problems
Base of number theory is not clear at early stage of learning mathematics. Most of the students are not able to understand the base of the number theory at an early stage. They do not understand the depth of the number theory. It requires extra coaching which is not affordable for many students
There are many concepts in number theory which are difficult to understand by an average student. In such cases they want to go for extra classes to understand the concepts clearly but the extra coaching is so expensive that generally many students are not able to afford it.
It is time consuming
Number theory requires lot of practice by a student to understand its concept clearly. Means number theory is one of the time consuming branches of mathematics. Many Concepts require lot of hard work and time from student side.
Students are not aware of its applicability
Many students are not aware of the applicability of number theory. They do not find it Useful in real life. They study number theory so that they can pass the exam.
Generally students do not score as per their expectation
Students generally find it difficult to understand number theory because of various theorems and new concepts. Therefore, generally students do not score as per their expectation.
Incomplete solutions
Generally, author does not give detailed solution of a problem by assuming that reader can solve it further.
Problems require extra skill
Some problems require use of calculator but generally in exam calculator is not allowed.
Some problems demands high level of thinking which is sometimes not possible for an average student. Geometric proofs require in depth knowledge of almost every branch of mathematics. Thus, students get demoralize.
Problems related to prime numbers
Two things which always bother many students about the prime numbers is that there are Infinitely many prime numbers and they cannot be generated by a simple polynomial function. Therefore, understanding the prime numbers at elementary school level is not sufficient.
Lack of craze and strategy knowledge
Many students are reluctant to go into the depth of the subject. Though students have good knowledge but they are inconsistent in analyzing and computing the problem.
Sometimes they make errors in reading the problem or taking wrong numbers. Hence, students struggle where basic computation and correct answers is demanded by the examiner.
Few important topics cover under Number Theory
- Arithmetic functions
- Perfect numbers and Fermat numbers
- Congruence
- Quadratic residue
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