Algebra - Definition, Basics, Branches, Facts, Examples

Edited By Team Careers360 | Updated on Jul 02, 2025 05:16 PM IST

Algebra can be defined as one of the oldest branches in the history of mathematics that deals with number theory, geometry, and analysis. Its origin starts from Egypt and Babylon where people used it first time in the history. It also means the application of mathematical symbols and various rules. In algebra, we assume the value of some unknown numerical constant to be a variable in an equation and then we solve the equation till we find the unknown quantity. This article is about the concept of algebra. Now, let us look in detail about algebra meaning, algebra formulas and its branches.

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What is Algebra?
Branches of Algebra
Algebra Formula Chart
Algebra Examples

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

Algebra Class 6	Algebraic Identities Class 8
Algebraic Expressions	Algebraic Identities Class 9
a cube plus b cube

Frequently Asked Questions (FAQs)

1. What is vector algebra?

Vector Algebra is the branch of Algebra that basically involves the analysis and study of vectors which are defined as quantities that have both magnitude and direction in physics.

2. Who is known as father of algebra?

Muhammad ibn Musa al Khwarizmi is known as father of algebra.

3. What is algebra calculator?

Algebra Calculator is a calculator that gives step by step solution to any algebra problem.

4. Mention the types of algebraic equations.

They include Monomial or polynomial equations, Exponential equations,Trigonometric equations, etc.

5. What do you mean by relational algebra?

Relational algebra in dbms means the use of algebra to contain the data and maintain database in computers with languages such as SQL. It is also known as RDBMS.

6. What is algebra and why is it important in mathematics?

Algebra is a branch of mathematics that uses letters and symbols to represent numbers and quantities in formulas and equations. It's important because it provides a way to solve complex problems, model real-world situations, and forms the foundation for advanced mathematics. Algebra helps develop logical thinking and problem-solving skills that are useful in many fields beyond math.

7. How does algebra differ from arithmetic?

While arithmetic deals with specific numbers and basic operations (addition, subtraction, multiplication, division), algebra introduces variables and focuses on the relationships between quantities. Algebra allows us to solve for unknown values and express general rules about numbers, making it more abstract and versatile than arithmetic.

8. How does algebra relate to other areas of mathematics?

Algebra is foundational to many areas of mathematics:

9. What is the significance of the zero product property in algebra?

The zero product property states that if the product of factors is zero, then at least one of the factors must be zero. Algebraically, if ab = 0, then either a = 0 or b = 0 (or both). This property is crucial for:

10. What is the difference between a rational and an irrational number in algebra?

A rational number is any number that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. Rational numbers have decimal representations that either terminate or repeat. Irrational numbers cannot be expressed as simple fractions and have non-repeating, non-terminating decimal representations. Examples of irrational numbers include √2, π, and e. Understanding this distinction is important in algebra for working with different types of numbers and solving equations.

11. What are the basic components of algebraic expressions?

The basic components of algebraic expressions are:

12. What is a variable in algebra, and why do we use letters instead of numbers?

A variable is a symbol, usually a letter, that represents an unknown or changing quantity. We use letters instead of numbers because they allow us to express general relationships and solve problems without knowing specific values. This flexibility makes algebra powerful for solving a wide range of problems and expressing mathematical concepts.

13. How do you simplify algebraic expressions?

To simplify algebraic expressions:

14. What is the difference between an expression and an equation in algebra?

An expression is a combination of numbers, variables, and operations without an equals sign. It represents a single value but doesn't make a statement. An equation, on the other hand, contains an equals sign and states that two expressions are equivalent. Equations can be solved, while expressions can only be simplified or evaluated.

15. What is a linear equation, and how is it different from a non-linear equation?

A linear equation is an equation where each term is either a constant or the product of a constant and a single variable (like x) to the first power. When graphed, it forms a straight line. Non-linear equations involve variables with exponents other than 1 or products of variables, resulting in curved graphs. Linear equations are simpler to solve and have consistent rates of change, while non-linear equations can be more complex and have varying rates of change.

16. What are the main branches of algebra?

The main branches of algebra include:

17. How does algebra apply to real-world situations?

Algebra applies to many real-world situations:

18. What are imaginary numbers, and why are they important in algebra?

Imaginary numbers are numbers that, when squared, give a negative result. They're based on i, the square root of -1. Imaginary numbers are important because they:

19. How do you solve a system of linear equations?

There are several methods to solve systems of linear equations:

20. What is a function in algebra, and how is it different from an equation?

A function is a relation between a set of inputs and a set of outputs, where each input corresponds to exactly one output. It's often expressed as f(x) = [some expression in x]. An equation, on the other hand, is a statement that two expressions are equal. Functions describe a specific relationship or behavior, while equations state a condition to be solved. All functions can be written as equations, but not all equations represent functions.

21. What is the purpose of factoring in algebra?

Factoring in algebra serves several purposes:

22. What is the quadratic formula, and when is it used?

The quadratic formula is x = [-b ± √(b² - 4ac)] / 2a, where a, b, and c are coefficients in the quadratic equation ax² + bx + c = 0. It's used to solve quadratic equations when they can't be easily factored. The formula gives the x-intercepts (roots) of the parabola represented by the quadratic equation, which can be real, imaginary, or equal depending on the discriminant (b² - 4ac).

23. How do you graph a linear equation, and what information can you derive from the graph?

To graph a linear equation:

24. How do you determine if a relation is a function?

To determine if a relation is a function:

25. What is the difference between domain and range in algebra?

In algebra, domain and range are key concepts for functions:

26. How do you solve absolute value equations and inequalities?

To solve absolute value equations (|x| = a):

27. How do you solve rational equations, and what precautions should you take?

To solve rational equations:

28. How do exponents work in algebra, and what are the rules for working with them?

Exponents indicate how many times a base number is multiplied by itself. Key rules include:

29. What is the significance of the discriminant in quadratic equations?

The discriminant in a quadratic equation ax² + bx + c = 0 is given by b² - 4ac. It's significant because:

30. How do you factor polynomials, and why is it important?

Factoring polynomials involves breaking them down into simpler expressions that, when multiplied together, produce the original polynomial. Methods include:

31. What is the relationship between algebra and geometry?

Algebra and geometry are closely related:

32. What are complex numbers, and how do they extend the real number system?

Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1). They extend the real number system by:

33. How do you interpret the slope and y-intercept of a linear function algebraically and graphically?

In the linear function y = mx + b:

34. What is the difference between solving equations and inequalities in algebra?

While equations and inequalities both express relationships between expressions, they differ in:

35. How do logarithms relate to exponents, and why are they useful in algebra?

Logarithms are the inverse operations of exponents:

36. What is the significance of the Fundamental Theorem of Algebra?

The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This theorem is significant because:

37. How do you determine the domain and range of a rational function?

To determine the domain and range of a rational function: