##### Chapter 2

## Algebra

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

#### Introduction

**Algebra**, which deals with number theory, geometry, and analysis, is one of the oldest branches in the history of mathematics. The definition of algebra sometimes notes that the study of symbols and laws in mathematics requires the manipulation of these symbols in mathematics. If you observe carefully, Algebra covers nearly everything from solving elementary equations to the analysis of abstractions.

In order to rewrite the equations, the algebra variables should be used to describe the unknown quantities that are coupled in such a way.

In our everyday lives, algebraic formulas are used to find the distance, the number of containers, and to work out the market rates when and when necessary. By making use of letters or other symbols describing entities, Algebra is very helpful in expressing a mathematical equation and relationship. It is possible to solve **problems on algebraic identities**.

In a contemporary elementary algebra tutorial, Elementary Algebra covers the typical topics covered. Along with logical operations like +, -, x, ÷, multiplication requires numbers. Yet numbers are also represented by symbols in the field of algebra, which are referred to as variables such as x, a, n, y. It also facilitates the traditional formulation of arithmetic laws such as a + b = b + a and is the first step that illustrates the systematic discovery of all the properties of a real number system.

As opposed to pre-algebra, this algebra has a high level of equations to solve. This section has **algebraic identities for two variables.** Advanced algebra allows you to go through other algebra elements, such as:

- Equations of differences
- Matrices
- System Solving of Differential Equations
- Function graphing and linear equations
- Conic Sections
- Equation with Polynomial
- Functions of inequalities Quadratic
- Polynomials of extremists and expressions
- Series and sequences
- Expressions of rationality
- Trigonometry
- Discrete and probability mathematics

Abstract algebra is one of the algebra divisions that discovers the truths of algebraic structures regardless of the particular existence of certain operations. Such operations have certain properties, in particular situations. We should also assume that these properties have certain ramifications.

Abstract algebra covers algebraic structures such as fields, sets, modules, rings, lattices, spaces for vectors, etc.

The abstract algebra principles are below—

**Sets-**Sets are characterized as the selection of objects that are identified by a particular set property. For eg, a set of all 2-2 matrices, the set of two-dimensional vectors present in the plane, and the various finite groups form.**Binary Events-**When the definition of addition is conceptualized, binary operations are given. Without a package, the definition of all binary operations will be pointless.**Identity Element-**To give the idea of an identity element for a particular operation, the numbers 0 and 1 are conceptualized. Here, 0 for the addition operation is called the identity element, while 1 for the multiplication operation is called the identity element.**Inverse Elements-**A negative number comes up with the definition of Inverse components. We write “-a” as the inverse of “a” for addition, and the inverse form is written as “a-1” for multiplication.**Associativity-**There is a property known as associativity when integer numbers are applied, in which the aggregation of added numbers does not affect the sum. Consider (3 + 2) + 4 = 3 + (2 + 4) as an example.

A subset of algebra that refers to both applied and pure mathematics is linear algebra. The linear mappings between the vector spaces are dealt with. It also deals with the analysis of aircraft and lines. That is the study of linear sets of equations with the properties of transformation. It is basically seen in all fields of mathematics. This refers to linear equations with their expression in vector spaces and via the matrices for linear functions. The following are the important topics discussed in linear algebra:

- Linear Equations
- Spaces of Vectors
- Relations
- Matrices and decomposition of matrices
- Relationships and Computations

One of the divisions of algebra that explores commutative rings and their ideals in commutative algebra. The theory of algebraic numbers, as well as algebraic geometry, relies on the algebra of commutations. It covers algebraic integers rings, polynomial rings, and so on. There are several other fields of mathematics, such as differential topology, invariant theory, order theory, and general topology, that depend on commutative algebra in various ways. In modern pure mathematics, it has occupied a remarkable position.

This chapter helped us in learning the basics of algebra. We learned the **proof of algebraic identities, **and how to use **algebraic identities for three variables **in solving questions.

**1. What are the basics of algebra? **

Algebra is a mathematics division designed to make it faster and easier to solve certain types of problems. Unlike arithmetic, which is entirely based on known number values, algebra is based on the concept of unknown values called variables.

**2. What are the branches of algebra? **

- Pre-algebra.
- Elementary algebra.
- Abstract algebra.
- Linear algebra.
- Universal algebra.

**3. Who invented algebra? **

Al-Khwarizmi is algebra’s father.

**4. What is the main purpose of algebra? **

By using letters of the alphabet or other symbols to represent entities as a form of shorthand, the purpose of Algebra is to make it easy to state a mathematical relation and its equation. Algebra then allows you to substitute values for unknown quantities in order to solve the equations.

**5. How is algebra used in real life?**

Algebra is used every day in our morning routine. When you wake up, by the end of the day, at least you have some objectives to achieve. The alarm is another good instance. People set up the alarm to wake up in the morning, but they do not understand that algebraic addition has just been executed.

Learn more about **algebra** and algebraic equations through simple, interactive, and explanatory visualizations, download the MSVgo app.