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## Algebra and Number Theory

Algebra and number theory are two very important elements that make up the areas of mathematics. This in conjunction with geometry and analysis. These four areas are very important for the development of mathematics.

## Algebra

One of the oldest areas that mathematics covers is this. In turn, it is in charge of studying the composition of elements of uncertain groupings according to certain norms. At first, these elements were taken as numerical quantities and it is then that it is known that algebra was developed as an extension of arithmetic.

## Number theory

Number theory also belongs to a specific area of ​​mathematics, which is responsible for analyzing the properties of digits. Specifically, he is in charge of studying the problems that arise from integers. Algebra and number theory are closely related to arithmetic and it is the number theorists who are in charge of studying this branch.

## Fields of number theory

Depending on the techniques used and the solutions to problems, number theory can be divided into different fields.

### Elementary number theory

This branch is responsible for the analysis of the integer digits, without having to do one of the other methods that comes from the other areas of mathematics. Techniques such as simplicity, Euclid’s algorithm, exploration of perfect digits, and congruences are awarded to this field.

### Analytical number theory

In algebra and number theory, he uses the calculation and study of holomorphic functions as instruments, in order to face questions related to whole digits. The most widely used methods in this theory are the prime number theorem and Riemann’s theory. Waring’s problem and the prime number conjecture are also used.

In algebra and additive number theory it constitutes the problems of representation of the digits in a more difficult way. The methods used in this field have a lot to do with those that have been mentioned in the other techniques, such as the Goldbach conjecture. However, these are complemented by other methods such as the Hardy Littlewood circle.

### Algebraic number theory

This field is the one most closely related to algebra, since algebraic numbers are studied in depth. These are what belong to the roots of varieties with rational coefficients.

### Geometric number theory

With algebra and number theory, this field is known as the geometry of numbers, since as its name indicates, it is made up of geometric figures. As a method it uses the Minkowski proposition, which explains a lot about the frequent points in prominent ensemble and the studies that have to do with round planes.

### Computational number theory

This field deals with the analysis of the relevant algorithms of algebra and number theory. These quick figures for studying the primary digits and integer factoring are widely used in cryptography.

### Combinatorial number theory

It is based on seeking solutions to the problems of number theory, including combinatorial doctrines and their statements. The topic of this theory is based on the systems covered, the zero-sum problems, the numerous concrete sets, among others. In this method, algebraic and analytical techniques are usually combined in a good way.

Algebra and number theory are two important branches of mathematics, which has helped the progress and solutions of different problems, thanks to the use of specific equations.

Studying and learning about anything is essential for life. Obtaining knowledge and being able to multiply it is the way humanity has passed all its wisdom from generation to generation.

Mathematics is part of all this knowledge disseminated and studied throughout history, from ancient times to the present. Everything that exists both in the world and in the universe is related to mathematicians, so they constitute fundamental knowledge.

Categories

## Prime numbers

The prime numbers are those that only have 2 divisors, which are itself and the unit. Its definition is very simple, however, knowing exactly what these digits are becomes a bit complicated because there is no mathematical procedure that determines these numbers in a concrete way.

## How do you know what the prime numbers are?

The easiest way to find out if a digit is prime is by division. That is, test if it has its own divisor. The way to do this is by dividing a number, for example, by 2, 3, 4.

If the result is exact when performing this procedure, the chosen digit is composite. If, on the other hand, when performing the division, the result is inaccurate, then it can be said that it is a prime number.

One of the things to consider is that this method is usually most effective when it comes to short numbers. If instead, you find digits of more than 25 digits, you would have the tedious task of doing a lot of divisions to find out if you find a prime number, or not.

For this, it is best to find an alternative that can be applied without giving so much trouble.

A good procedure to use in these cases is Fermat’s theorem, which consists in asserting that the divisibility of the numbers is feasible.

### Sieve of Eratosthenes

It is based on an algorithm created by the ancient Greek mathematician, Eratosthenes, around 284 BC. This manages to find all the prime numbers less than the natural digit from which it was started.

This illustrious scholar drew up an ingenious table, in which they located the numbers from one to one hundred, from which he eliminated those that are multiples of the other.

In this way, only the corresponding prime numbers remain in the table. To begin, what you must do is dial the number one, as this is not considered a prime digit.

Subsequently, the multiples will be separated, starting with the number 2. If in the process you get a whole number that has not been eliminated, it means that it belongs to the prime numbers.

From the latter, we start again with the procedure of eliminating the multiples and so on repeatedly. This method ends when the box of the next whole digit, considered as a prime, is greater than the natural number from which it was started.

## How many prime numbers are there?

The precise answer to this question is that the number of prime numbers is infinite. Exhibitions such as that of Leonhard Euler, demonstrate by means of a proposition that the sum of the series of contrasts of the prime digits is bifurcated. Because of this, it can be concluded that a set of prime numbers will never be finite.

## Fundamental theorem of arithmetic

This is based on the fact that every natural digit has a unique spelling, as a result of the prime factors, minus the order. It also implies that the same element manages to arise several times and like the others also, it states that the number one is null. Also consider that prime numbers are the foundations that constitute natural digits.

The relevance of this statement to exclude the number 1 as a prime, is that necessary adjustments would have to be made so that the results were as required.

And it is from this same theorem that other sets are developed that are used in mathematics. Like, for example, the least common multiple, the greatest common divisor and the coprime numbers.

The importance of prime numbers is based on the fact that, without them, no type of algorithms or complex calculations could be carried out, which are usually used to establish the bases of any type of number.

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## Computational mathematics ¿How it works?

Two sciences that have been combined, to give way to a world where solving problems in the scientific and technological field, becomes easier. Mathematics and computer science have a lot to do with different aspects of life, even in everyday life. Combining the two, computational mathematics was born.

## Definition of computational mathematics

When hearing this science referred to under the name of computational mathematics, many may think that it is something extremely complex and although it has its difficulty, you will find that it is easy to understand if you pay attention.

Computational mathematics can be defined as a conglomerate of processor programs, which is capable of creating algorithms, methods and good hypotheses. These will serve for the solution of different problems that originate in science and technology. When we talk about science, everything is included, that is, finance, social sciences, health, physics, chemistry, among others.

The purpose of this specific area is to provide easy and direct solutions to difficulties. It is then that basic mathematics is discussed, because it includes numerical analysis. However, it can be mentioned that they seek to examine new algorithms, facilitating improved programs that develop them.

## Fields of computational mathematics

Being a branch of applied mathematics, it can be made up of different fields, to each of which it makes an important contribution that can add much value in terms of research and advances. These are:

Scientific engineering: in this case, computational mathematics is applied to solve difficulties through a program, through a computer instead of using applied mathematics techniques.
Numerical techniques: these are used in scientific computing, in addition, applied in areas such as algebra and equations.
Monte Carlo procedure: in this method, computational mathematics is applied together with other characters of scientific engineering.
Mathematical proofs: this means that you use numerical study and numerary methods.
Algebra systems: in addition to this type of system, it also applies to the processing of symbolic data and computing.
Computational linguistics: this field refers to the use of mathematical and computer methods in what is known as natural expressions.

## Importance of computational mathematics

Computational mathematics becomes truly exceptionally important thanks to innovative technologies. Daily life is increasingly computerized and more mechanized tasks are performed. At this point is where this method becomes relevant, as it becomes a basis in the scientific progress that is being incorporated into daily tasks. Some of these are:

### Change in scientific method

It is known that the main thing about scientific technique is research and information. With computational mathematics, computational simulation must be applied. It means that, since it is a scientific research technique that requires computers, it will be able to analyze physical, biological or social programs, always based on the rules of research and information.

It means that this method of computational simulation, despite being relatively new, can pay off in scientific and specialized benefits.

### Benefits in the industry and the economy

Computational mathematics is now used in industry sketches. As, for example, in the virtual construction of airplanes, satellites, automobiles and ships. This means that these types of industries save a great deal of money before moving to the physical construction of these specimens. In addition, it applies in the construction of large buildings and in the research of new drugs.

From these points it can then be said that computational mathematics, being specialized in the creation of new algorithms, represents an incredible development. Its only limitations become the minds of individuals and at the level of physical development that computers can present, limitations that sooner or later can be solved.

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## Numbering systems

It is known that for many years, ancient civilizations used different methods of counting. These were the settlements of the numbering systems that are known today, since many of these counted in units, tens, hundreds, only that the writing of the numbers was different from the one used today.

## Definition

It can be defined as an aggregate of symbols and generation rules, which supports the creation of valid digits in the system. The numbering systems are made up of various elements, which are governed by a series of rules that allow the establishment of operations and relationships between the different elements.

When these two definitions are combined, the conclusion is then reached that the numbering systems are a set of compendiums, whether they are symbols or digits, as well as operations, as well as relationships that those already assigned numbers can carry out.

## Classification of numbering systems

These numbering systems can be classified into 3 different groups, which are:

Non-positional numbering systems: these are based on the numbers that have the value of the symbol used. That is, they do not take into account the column they invade in the number. An example of this group can be the Egyptian numbering system.

This system is the most archaic known, since they basically used the fingers of the hands.

They used knotted ropes to represent the quantity of numbers and they used a lot the ability to coordinate between the elements.

Semi-positional systems: a system born in ancient Rome and that is flexibly considered positional. This meant that it was much more complex to develop the algorithms for addition, subtraction, multiplication, and division. An example of this is Roman numerals, which from the beginning to the end of the digit have the same value.

Positional numbering: also called as the weighted numbering system, it is when the value of a number has to do with the symbols used. In other words, what matters is the position that element occupies in the digit. In this system the amount of elements that are allowed are called as base.

Within this type of system others can be known, one of them is the binary system, as well as the decimal, even the octal and hexadecimal systems. The positional decimal system is the one used in most of the countries of the world, except in languages ​​such as computer science, where systems such as hexadecimal and binary are used.

Binary system: the digits that are used only with the numbers zero and one are represented here. Normally it is only used in computers, since they work internally with 2 voltage levels. This is basically its on that they identify it with the number 1 and with the off that is represented by the number 0.

Hexadecimal system: in positional numbering systems, this is base 16, which means that it is based on 16 symbols. Its use today also leans towards computing. Since these sciences use bytes and bytes as a basic memory device.

## Fundamental numbering theorem

It institutes the ordinary way of constructing digits in positional numbering systems, as long as the basic denominations of the valid number, the base, the various symbols, the numbers of the whole part, and the digits that make up the number are established and clear. decimal part. All of these applied to a specific formula.

The importance of numbering systems is based on the fact that all natural digits can be stated in a unique way, as a linear combination.

Number systems are a fundamental part of science, while science is the foundation of society and the world as we know it therefore it is important to know and study number systems.