List 8 basic branches of algebra and its applications

Algebra is a wide branch of mathematics, and that is finding the value of two or more variables accordingly. Every branch has its own focus and uses. Here’s a brief introduction to and application of the branches of algebra.

Branches of algebra

1- Elementary Algebra

          Elementary algebra is the most common and basic type, and it forms the basis for more complex branches of mathematics and algebra. And another name for elementary algebra is college algebra.
           Elementary algebra includes the general properties and the fundamental concept of algebra, which deals with the basic study of mathematical operations on numbers, solving equations, and algebraic expressions. Elementary algebra is taught at the entry level of education. In this type, we use constants and variables. Constants are fixed values that do not change, and variables are letters used to represent unknown, changing quantities.
For example, 8y + 2 = 9.
where 8, 2, and 9 are constants and y is a variable.

Application:

          The following list of fields is where elementary algebra is used:

• • • Problem Solving in Everyday Life
    • Business and Economics
    • Finance and Budgeting
    • Statistics
    • Environmental Science

2- Abstract / Modern Algebra

          Abstract algebra, or modern algebra, is the study of algebraic structure, basic properties, and relations that are present in these structures in a general and abstract manner. They are considered distinct areas because they focus on distinct ideas. Abstract algebra includes many abstract structures like fields, rings, vector spaces, groups, modules, and other algebraic structures.
          Abstract algebra refers to a group of advanced and common algebra structures and patterns that arise in various mathematical systems rather than traditional number systems because abstract algebra focuses on specific numbers and operations. Sets, binary operations, identity elements, inverse elements, and associativity are the common concepts of abstract algebra.

Application:

The applications of abstract algebra in mathematical systems, and other scientific fields and the study of this algebraic branch provide a great understanding of the fundamental framework of mathematical systems.

• • • Cryptography
    • computer science
    • Vector space
    • Algorithms and Computation
    • Game Theory

3- Linear Algebra

          The study of linear combinations is known as linear algebra, in which we deal with lines and planes, matrices, vector spaces, and some linear mappings between these spaces that are necessary to perform the linear transformations and analyze the properties of matrices.

An equation in linear algebra is generally expressed as

Ax = b

A represents a matrix, x represents unknowns, b represents a vector

Application:

The applications of linear algebra in graphs and networks, matrices in engineering, linear programming, fourier series, linear algebra for statistics and probability, and statistics It forms the basis for many advanced mathematics branches, and it is an important topic and tool in data analysis, computer graphics, and machine learning.

3- Commutative Algebra

          The study of algebra focuses on commutative rings, modules, and their ideals over such rings; on the other hand, commutative algebra also deals with polynomial rings and maximal ideals; and the study of algebraic structure involves commutative multiplication. It is known as ideal theory.

         The commutative property works only on addition or multiplication because an expression's value is unaffected if the order of value is changed.

• • Addition:

a + b + c = a + c + b = b + a + c = b + c + a = c + a + b = c + b + a

• • Multiplication:

a b c = a c b = b a c = b c a = c a b = c b a

         Commutative property property did not apply to subtraction and division because when we change the order of value then the value of expression is affected.

• • Subtraction

a - b  ≠  b - a

• • Division

a / b  ≠  b / a

Application:

The application of commutative algebra is in various branches of mathematics and beyond. Some important applications are here.

• • • Algebraic Geometry
    • Algebraic Number Theory
    • Coding Theory
    • Topological Methods in Algebra
    • Computer algebra systems
    • Differential Equations
    • Combinatorics
    • Cryptology
    • Ring theory

5- Noncommutative Algebra

          In the branches of algebra, a noncommutative algebra focuses on the structure where the multiplication operation is not necessary to be commutative. A noncommutative ring is the basic algebraic structure of noncommutative algebra.

In noncommutative expressions, the answer is affected if the order of the value is changed, so that value order can matter in this branch. In simple words, if we multiply ab and ba, then the answer is equal in commutative algebra, but this commutative property is not assumed in this branch.

Application:

The application of noncommutative algebra is in various branches of theoretical physics and mathematics. Some applications are written as follows:

• • • Operator Algebras and Functional Analysis
    • Quantum mechanics and quantum field theory
    • Signal Processing
    • Noncommutative Geometry
    • Representation Theory
    • Topological Insulators
    • Coding Theory

6- Boolean Algebra

          Boolean algebra, or logical algebra, is a mathematical structure and system that focuses on binary variables and logical operations. A binary variable is a variable that consists of two values: true or false. True is also represented by one (1), and false is also represented by zero (0).

There are three important logical operations of Boolean algebra written as follows:

• conjunction or AND operation (⊥),
• Disjunction or OR operation (⊦)
• Negation or Not operation (¬)

Application:

Boolean algebra is used in different fields; there are some specific fields written as follows:

• • • Robotics
    • Mathematics and Set Theory
    • Digital Logic Design
    • Network Design
    • Control Systems
    • Artificial Intelligence and Boolean Logic

7- Homological Algebra

          Homological, or homotopical, algebra is the study of homological and categorical techniques in a general algebraic setting. This branch of algebra is used for extracting information from complexities by focusing on concepts such as homology, exact sequences, functors, and chain complexes and putting it in terms of homological invariants of modules, topological spaces, rings, groups, and other 'physical' mathematical objects.

          Homological algebra provides a framework for understanding algebraic objects in different mathematical conditions.

Application:

The application of homological algebra is in different branches of mathematics, such as the following:

• • • Topological Data Analysis
    • Functional Analysis
    • Algebraic geometry
    • Category Theory
    • Singularity Theory
    • Representation Theory

8- Computer Algebra

          Computer algebra is the development and research of software and algorithms for performing symbolic manipulation of expressions and other mathematical objects. Symbolic computation and algebraic computation are other names for computer algebra. It is commonly used in scientific research, physics, computer science, and engineering disciplines. They are necessary for the automation of symbolic computations. The symbolic algebra system (SAS) and computer algebra system (CAS) are two systems that work with symbolic computation.

          In conclusion, every branch of algebra plays an important role in mathematics, and they collectively contribute to the rich and carried landscape of algebra and a deeper understanding of relationships and mathematical structure. The primary application of algebraic identities is to calculate polynomial factors in numerous scientific and technological fields.