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

So far, you have learned basic arithmetic operations like multiplication, division, etc. You have also learned about two and three-dimensional shapes and calculated their area etc. This branch of mathematics is called geometry. Now, you will learn about the third branch of mathematics called **Algebra**.

You might have come across specific formulas or equations where the problem is solved using alphabets and symbols. These alphabets and symbols are called ‘variables’ in **algebra**. When you form an equation, variables are used to determine the unknown values. However, since **algebra **is a vast concept, it is divided into various branches to make it easier to understand and find the correct approach to solve the problem.

**1. Elementary Algebra:**

**Elementary algebra** combines the basic arithmetic operations like addition, subtraction, multiplication, and division with the variables of **algebra **like a, b, x, and y to form the equations.

For example: a=1, b=5, x=2, y= 4 then the equation can be written as

a+ b= x + y

**Elementary algebra** is used to evaluate the equations, the equalities, and inequalities of the properties, solving equations with one or more variables, etc.

**2. Advanced Algebra:**

**Advanced algebra** helps you understand the higher level of equations to solve the other parts of **algebra **like:

- Matrix
- Linear Equations
- Polynomial Equations
- Series and Sequences
- Rational Expression
- Trigonometry
- Quadratic formulas
- Graph functions
- Conic Sections
- Probability

**3. Abstract Algebra**:

**Abstract Algebra** is the **algebra **field that deals with the algebraic structure like vectors, lattices, rings, modules, groups, etc. These algebraic systems are not dependent on a specific nature of the algebraic operations, and hence it follows certain concepts to solve the problem.

The concepts of **abstract algebra** are as follows:

- Sets:

A collection of objects that are determined by a specific property is called a set. For example, a 3×3 matrix or a few two- dimensional vectors for a set. - Binary Operations:

Binary operations combine two elements to produce a third element. - Identity Element:

In a set, an identity element is the one that does not change elements when combined with it. For example, 0 and 1 are the identity elements for additions and multiplication as they do not change the value of the element. - Inverse Elements:

The inverse elements are used as a concept of negation in terms of addition and multiplication. For example, the inverse of ‘a’ will be ‘-a’ while the inverse of ‘a’ in multiplication form will be ‘a-1’. - Associative Algebra:

The basic premises of the associativity is that when numbers are added, their grouping does not affect the sum of the integers. For example, 1+(2+3) will yield the same result as (1+2)+3.

**4. Linear Algebra:**

**Linear Algebra** is a branch of **algebra **that deals with the spaces between the vectors and their mapping. **Linear algebra** lets you study lines and planes while also giving you an opportunity to understand the transformation properties of the linear equation. In **linear algebra**, you will study:

A. Vectors

B. Relations

C. Linear equations

D. Matrices

E. Computations and relations

**5. Commutative Algebra:**

**Have you ever wondered if there is any branch in mathematics that studies the rings? Well, commutative algebra is the branch that studies the rings, such as polynomial rings and their corresponding ideals. **

Two algebraic expressions can be multiplied to find the unknown variable, and the expressions can consist of integer variables and constants. For example, 2ab+5 is an expression, where a & b as variables and 4 & 9 as constants. There are certain rules that must be followed while multiplying the algebraic expression:

- The brackets in the expression must be simplified first.
- If there are no brackets, then the expression can be solved through the BODMAS rule.

When solving the algebraic expressions with brackets, it is necessary that the bracket is simplified first using the BODMAS rule. **BODMAS and simplification of brackets** are used for expressions that have multiple arithmetic operations.

By the rule, the expression needs to be solved in the order Bracket, Order, Division, Multiplication, Addition, and Subtraction. If there are multiple bracketed expressions in an expression, then the innermost bracketed expression is solved first and then moved towards the outermost bracket.

Exponents give the idea about how many times the base number will have to be multiplied. For example, in the expression 3^{2}, 3 must be multiplied 3 times to get the result. Here, 3 is called the base number, and 2 is called the exponent.

The algebraic expressions with exponents follow a particular rule, and the rules are as follows:

- RULE 1: If the numbers being multiplied have the same base, then the exponents must be added. For example, 3
^{2}x 3^{3}= 3^{(2+3)}. - RULE 2: If the numbers being divided have the same base, then subtract the exponents. For example, 3
^{3}x 3^{2}= 3^{(3-2)}.

RULE 3: When the exponent of the number is raised by another exponent, then multiply the exponents. For example, (3^{2})^{3}=3^{(2x3)}.

**Exponents in algebra** are used when a large number has to be represented in a simple way that can be easily understood. It shows how many times a number must be multiplied by the same number to achieve the result. For example, for the number 100, 10 must be multiplied twice, and it can be denoted as 10^{2}.

- What are the different types of algebraic equations?

There are five types of algebraic equations: Polynomial, exponential, trigonometric, relational, and logarithmic equations. - Are there any branches in algebra?

There are five different branches in algebra: pre-algebra, elementary, abstract, universal, and linear algebra.

Formulas are critical, and one must understand the concept behind them. MSVgo is a learning app that is built on the philosophy that understanding a concept is the core of learning and therefore explains the concepts with examples, animations, or explanatory visualisation.

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