Algebra - Definition, Basics, Branches, Facts, Examples

What is Algebra?

Algebra includes almost everything ranging from solving elementary equations to the study of abstractions. Algebra formulas are included in most of the maths chapters which students at various stages of their life come across during their academic journey. The concept of algebra has wide ranging applications in our daily life. It is really helpful in finding the unknown quantities and we come across such situations almost daily in our lives.

Algebra Formula

Algebra helps solve complex mathematical equations or expressions within no time and also enables us to derive unknown quantities, for example bank interest, percentages, etc.

The algebra formulas are most commonly used in our daily lives. Algebra has always been helping us with forming a mathematical equation and relationship by using letters or symbols that represent various entities. We also know that we can calculate or find out the unknown quantities in the equation with the help of algebra. Few algebra formulas are as follows:

- $(a+b)^2=a^2+2 a b+b^2$
- $(a-b)^2=a^2-2 a b+b^2$
- $(a+b)(a-b)=a^2-b^2$
- $(a+b+c)^2=a^2+b^2+c^2+2 a b+2 b c+2 c a$
- $(a+b)^3=a^3+3 a^2 b+3 a b^2+b^3$
- $(a-b)^3=a^3-3 a^2 b+3 a b^2-b^3$

Branches of Algebra

We know that the concept of algebra is based on equations that contain some unknown terms. It is these terms in algebra that are known as variables(something that is not constant , vary or change accordingly).It has various branches like abstract, linear, advanced, commutative, etc. We will discuss them in detail and learn more about them in the following sections.

Elementary Algebra

It covers the very first stage or the elementary stage of algebra, involving basic operations such as +,×,÷,- which are also known as elementary operations. The examples like p+q, g÷p are few algebraic expressions. This acts as the foundation for other branches since we develop our basic knowledge at this stage. It comprises of concepts like linear equation in one variable, inequalities, etc.

Advanced Algebra

It is a step furthur in our study of algebra. It involves slightly higher level of concepts and calculations which are little complex in nature. Few examples include sequence and series, conic sections, matrices, trigonometry, etc. These are the topics that we come across in advanced study of algebra in classes such as 10th, 11th, 12th and furthur. So if we have deeper interest in mathematics we can go for the study of this branch as a career option in research.

Abstract Algebra

It is a branch of algebra in which we discover the truths regarding certain statements or system. Which means we learn to distinguish between true and false statements and also analyse our conclusions accordingly. We must know that it does not depend on that system and hence we say that it is independent. They also have certain properties. For example, groups, rings, vector space which is a topic of study in class 12th, etc.

Linear Algebra

Linear Algebra is that branch of algebra that deals with planes, lines and hence the linear mapping between vector spaces. It involves linear equations along with their respective transformation properties.So, we can say that it can involve topics like linear equations in one variable or two variable as well. For example, vector space, relations, etc to list a few. It is really well discovered, analysed and researched.

Commutative Algebra

This branch of algebra deals with commutative rings. The geometric and number theory of algebra is based on branch of commutative algebra. Few examples include topology, polynomial rings, their ideals, modules, Z and p ardic integers, etc. It is totally independed of all the other branches and is a part of various higher level competitive exams. It is used both in number and geometric theory.

Vector Algebra

Vector algebra class 12 involves algebraic operations across vectors, involving the magnitude and direction of vectors. Vector algebra helps for various applications in physics, and engineering to perform addition and multiplication operations with physical quantities, represented as vectors in 3-D space. Length of line between the two points P and Q is called the magnitude of vector and the direction or displacement of point P to point Q is called the direction of vector PQ.

Relational Algebra in DBMS

When we look at the applications of algebra in computers and applied fields, in database theory we use algebra structure for the process of modelling and it helps us to define certain queries which are famously known as query languages or SQL. This is known as relational algebra. This helps us to store and preserve huge quantities of data in an easy and efficient way. It is often meant for computer enthusiasts and IT domain people who wish to pursue career in computers and related domains.

Algebra Formula Chart

Algebra Formula Chart is given below:

Now, let us look into some algebra questions and answers.

Algebra Examples

Example 1: Solve the equation $2 x-6=6 x-8$.
Solution:

Given,

$
2 x-6=6 x-8
$

Adding 6 on both sides,

$
\begin{aligned}
& 2 x-6+6=6 x-8+6 \\
& 2 x=6 x-2 \\
& -4 x=-2 \\
& x= \frac{1}{2}
\end{aligned}
$

Example 2: Using the $(\mathrm{a}+\mathrm{b})^2$ formula in algebra, find the value of $(102)^2$.
Solution:
Given: $(102)^2=(100+2)^2$
Using algebra formula $(a+b)^2=a^2+2 a b+b^2$, we have,

$
\begin{aligned}
& (100+2)^2=(100)^2+2(2)(100)+(2)^2 \\
& (102)^2=10404
\end{aligned}
$

Example 3: Find the value of x in the following equation using the Algebra concepts. $3 x+8=$ 20

Solution:

$
\begin{aligned}
& 3 x+8=20 \\
& 3 x=20-8 \\
& 3 x=12 \\
& x=4
\end{aligned}
$

Therefore, the value of $x=4$

Example 4: Ten less than a number equals to two. What is the number?
Solution:
Using the concepts of Algebra, we will assume the number to be a variable. Let the number be y. As per the question, we can write $y-10=2$. On solving this, we get $y=12$. Therefore, the required number is 12 .

Example 5: Find the value of x in the following equation using the Algebra concepts. $45 \mathrm{x}+4=$$
25
$

Solution:

$
\begin{aligned}
& 45 x+4=25 \\
& 45 x=25-4 \\
& 45 x=21 \\
& x= \frac{21}{45}
\end{aligned}
$

Therefore, the value of $x= \frac{21}{45} $

For more such questions, refer algebra worksheet.

List of Topics Related to Algebra