Algebra in Math: Definition, Branches, Basics and Examples

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is a unifying thread of almost all of mathematics and includes everything from solving elementary equations to studying abstractions such as groups, rings, and fields.

It helps represent problems or situations in the form of mathematical expressions. It is different from Arithmetic as Arithmetic deals with specific numbers and simple operations such as addition and subtraction. Algebra, on the other hand, introduces more complex operations and includes the use of variables, equations, and functions.

What is Algebra

Algebra mainly focuses on variables, which have unknown values. These variables may change. Variables are symbols like x, y, z, p, or q.

Mathematical operations like addition, subtraction, multiplication, and division are combined with variables like x, y, and z to form a meaningful equation. It follows a set of laws when performing mathematical operations. When analyzing data sets with two or more variables, the rules are used.

Algebra Definition

Algebra is a branch of mathematics that deals with symbols and the rules to solve equations and it focuses on operations with variables, constants, and mathematical expressions.

Algebra Branches

The various branches of Algebra based on the use and complexity of the expressions are as such:

1. Pre Algebra
2. Elementary Algebra
3. Abstract Algebra
4. Universal Algebra
5. Linear Algebra
6. Commutative Algebra 
7. Boolean Algebra

Pre Algebra

Pre Algebra includes the fundamental concepts of arithmetic and algebra, such as the order of operations, basic operations with numbers, and simplifying expressions.

Algebra assists in turning day-to-day problems into mathematical expressions that use algebraic techniques and algebraic expressions. Pre-algebra specifically involves creating an algebraic expression for the provided problem statement.

Elementary Algebra or Algebra 1

Goal of elementary Algebra is to find a solution by resolving Algebraic expressions. Simple variables like x and y are expressed as equations in elementary Algebra.

• Equations are divided into polynomials, quadratic equations, or linear equations depending on the degree of the variable.

• Formulas for linear equations are ax + b = c, ax+ by + c = 0, and ax + by + cz + d = 0.

• Based on the number of variables, quadratic equations, and polynomials are subsets of Elementary Algebra.

• For a polynomial problem, the typical form of representation is axⁿ + bx^n-1+ cx^n-2+…..k = 0, while for a quadratic equation, it is ax² + bx + c = 0.

Abstract Algebra 

Abstract Algebra is a branch of mathematics that focuses on Algebraic systems like groups, rings, fields, and modules, rather than on specific numerical computations.

• In abstract Algebra, we do not study specific operations like addition and multiplication but instead study general properties of basic operations, such as associativity, commutativity, distributivity, and the existence of inverses.

• Groups, sets, modules, rings, lattices, vector spaces, and other Algebraic structures are studied in abstract Algebra. 

Universal Algebra

Universal Algebra can be used to explain all other mathematical forms using Algebraic expressions in coordinate geometry, calculus, and trigonometry. In each of these areas, universal Algebra focuses on equations rather than Algebraic models.

• We can think of all other types of Algebra as being a subset of universal Algebra.

• Any real-world issue can be categorized into a particular discipline of mathematics and solved using abstract Algebra.

Linear Algebra 

Linear algebra, a branch of algebra, finds uses in both pure and practical mathematics. It deals with the linear mappings of the vector spaces. It also involves learning about lines and planes. It is the study of linear systems of equations with transformational features.

• It is used in almost all areas of mathematics.

• It deals with the representation of linear equations for linear functions in matrices and vector spaces.

Commutative Algebra 

Commutative algebra is one of the types of algebra that studies commutative rings and their ideals. Both algebraic geometry and algebraic number theory require commutative algebra.

• Rings of algebraic integers, polynomial rings, and other rings are all present.

• Numerous other areas of mathematics, such as differential topology, invariant theory, order theory, and generic topology, make use of commutative algebra. 

Also Read

• Commutative Property in Maths

Algebraic Expressions with Examples

Algebraic Expressions consist of constants and variables. We can add, subtract, multiply, and divide these expressions. An example of an algebraic expression is 5x + 6. The variables might also have values like x², x³, xⁿ, xy, x²y, etc.

Polynomials is also a term used for algebraic expressions. An equation that contains coefficients, non-negative integer exponents of variables, and variables is called a polynomial. For example, 5x³ + 4x² + 7x + 2 = 0.

Algebraic Equations

An algebraic equation shows the connection between two quantities when one or both of the values are unknown.

Given below are the different types of Algebraic equations, based on the degree of the variable:

1. Linear Equations
2. Quadratic Equations
3. Cubic Equations

Read More

• Algebraic Formula

Linear Equation

A linear equation is an equation in which the highest power of a variable is 1. They are also known as first-order equations.

• A linear equation is an equation for a straight line when seen in the coordinate system.

• Equation of Straight Line is written as y = mx + b, where m denotes the line’s slope and b denotes the y-intercept. 

Below are some of the important topics covered in Linear equations.

1. Standard Algebraic Identities
2. Algebraic expressions
3. Like and Unlike Algebraic Terms
4. Mathematical Operations on Algebraic Expressions
5. Standard Algebraic Identities
6. Factorization
7. Introduction to factorization
8. Division of Algebraic Expressions
9. Linear Equations in One Variable
10. Solve Linear Equations with Variables on Both Sides
11. Solving Equations that have Linear Expressions on one Side and Numbers on the other Side
12. Reducing Equations to Simpler Form
13. Linear Equations and their solutions
14. Graph of Linear Equations in Two Variables
15. Equations of Lines Parallel to the X-axis and Y-axis
16. Pair of Linear Equations in Two Variables
17. Number of Solutions to a System of Equations Algebraically
18. Graphical Methods of Solving a Pair of Linear Equations
19. Algebraic Methods of Solving a Pair of Linear Equations
20. Equation Reducible to a Pair of linear equations in two variables

Quadratic Equation

A quadratic equation is a type of algebraic equation that contains one or more terms in which the variable is raised to the power of 2 (i.e., a quadratic term). It is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and x is the variable.

Here are the topics that discuss quadratic equations thoroughly:

1. Quadratic Equations
2. Solution of a Quadratic equation by different methods
3. Roots of a Quadratic Equation
4. Complex Numbers
5. Algebra of Real Functions
6. Algebraic Operations on Complex Numbers
7. Argand plane and polar representation
8. Absolute Value of a Complex Number
9. Imaginary Numbers
10. Complex Conjugate
11. Compound Inequalities
12. Algebraic Solutions of Linear Inequalities in One Variable and Their Graphical Representation
13. Graphical Solution of Linear Inequalities in Two Variables
14. Solving Linear Inequalities Word Problems
15. Fundamental Principle of Counting

Cubic Equation

A three-degree equation, or a cubic equation, has a variable whose maximum power is 3. A cubic equation has the general form ax³ + bx² + cx + d = 0.

where x is a variable and a, b, c, and d are constants.

• A cubic equation can have one, two, or three solutions that are real or complex numbers, depending on the coefficients in the equation.

Polynomial

Polynomials are made by variables and coefficients, which are Algebraic expressions. Indeterminate is another name for variables. For polynomials, we can perform addition, subtraction, multiplication, and positive integer exponents, but not division by variable. For ex: 3x³-5x+8.

Following are the topics that discuss polynomials deeply.

1. Polynomials in One Variable
2. Zeroes of a polynomial
3. Remainder Theorem
4. Types of Polynomials
5. Multiplying Polynomials
6. Algebraic Identities of Polynomials
7. Geometrical meaning of the Zeroes of polynomial
8. Relationship between Zeroes and coefficients of a polynomial
9. Division Algorithm for polynomials
10. Division Algorithm Problems and Solutions

Sequence and Series

An ordered collection of numbers or other elements of mathematics that follow a pattern or rule is called a sequence.

• The position of each term of a sequence within the sequence is marked by its index or subscript.
• The series of even numbers, for example, can be written as 2, 4, 6, 8, 10, and so on.
• The total of a sequence’s terms forms a series.
• For instance, the series 2 + 4 + 6 +… + 2n gives the sum of the first n terms of the even number sequence.
• Series may be finite or infinite.
• Sequence and Series can be classified into two major categories – Arithmetic Progression and Geometric Progression. 

Arithmetic Progression 

Arithmetic Progression(A.P) is a series of numbers where each term is obtained by adding a constant /fixed value to the previous term. This continuous difference in the terms is denoted by ‘d’.

General format of an A.P is:

a, a + d, a + 2d, a + 3d, …, a + nd

where,

• a is First Term
• d is Common Difference
• n is Number of Terms

Geometric Progression

Geometric Progression(G.P) is a series of numbers where each term is obtained by multiplying the preceding term by a fixed constant value called the common ratio, denoted by r.

General form of a geometric progression is:

a, ar, ar², ar³, …, arⁿ

where,

• a is First Term,
• r is Common Ratio
• n is Number of Terms

Given below is the list of topics that will give you a better understanding of sequence and series:

1. Common difference and Nth term
2. A sum of First n Terms
3. Binomial Theorem for positive integral indices
4. Pascal’s Triangle
5. Introduction to Sequences and Series
6. General and Middle Terms – Binomial Theorem
7. Arithmetic Series
8. Arithmetic Sequences
9. Geometric Sequence and Series
10. Geometric Series
11. Arithmetic Progression and Geometric Progression
12. Special Series

Exponents 

Exponent is a mathematical operation, written as aⁿ where a is the base and n is the power or the exponent. Exponents help us simplify Algebraic expressions. Exponent can be represented in the form 

aⁿ = a*a*a*….n times. 

Logarithms 

Algebraic opposite of exponents is the logarithm. It is practical to simplify complicated Algebraic formulas using logarithms. Exponential form denoted by the formula a^x = n can be converted to logarithmic form by using formula log_an = x. 

In 1614, John Napier discovered logarithms. Nowadays, logarithms are a crucial component of modern mathematics.

Set Theory

Set theory is a branch of mathematical logic that investigates sets, which are arrays of objects informally.

• Term “set” refers to a well-defined group of unique items that are used to express Algebraic variables.
• Sets are used to depict the collection of important elements in a group.
• Sets can be expressed in set-builder or roster form.
• Sets are usually denoted by curly braces;{} for example, A = {1,2,3,4} is a collection.

Let’s learn more about the sets in the following articles:

1. Sets and their representations
2. Different kinds of Sets
3. Subsets, Power Sets, and Universal Sets
4. Venn Diagrams
5. Operations on Sets
6. Union and Intersection of sets
7. Cartesian Product of Sets

Vectors

A vector is a two-dimensional object of both magnitude and direction. It is normally represented by an arrow with a length (→) that indicates the magnitude and direction. It is denoted by the letter V.

• One of the most important aspects of Algebra is vector Algebra.
• It is a course that focuses on the Algebra of vector quantities.
• As we all know, there are two kinds of measurable quantities: scalars and vectors.
• The magnitude of a scalar quantity is the only thing that matters, while the magnitude and direction of a vector quantity are also essential.
• The vector’s magnitude is denoted by the letter |V|.

Let’s discuss the vector and its Algebra in the following articles:

1. Vector Algebra
2. Dot and Cross Product of two vectors
3. How to Find the Angle Between Two Vectors?
4. Section Formula

Relations and Functions

Relations and functions are two distinct terms that have different mathematical interpretations. One might be puzzled by the differences between them.

• Before we go even further, let’s look at a clear example of the differences between the two.
• An ordered pair is represented as (INPUT, OUTPUT): The relation shows the relationship between INPUT and OUTPUT. Whereas, a function is a relation that derives one OUTPUT for each given INPUT.

Let’s discuss more of the topic in the following articles:

1. Relations and functions
2. Types of Functions
3. Composite functions
4. Invertible Functions
5. Composition of Functions
6. Inverse Functions
7. Verifying Inverse Functions by Composition
8. Introduction to Domain and Range
9. Piecewise Function
10. Range of a Function

Matrices and Determinants

In linear Algebra, determinants, and matrices are used to solve linear equations by applying Cramer’s law to a series of non-homogeneous linear equations.

• Only square matrices are used to measure determinants. While a matrix’s determinant is empty, it’s known as a singular determinant, and when its determinant is one, it’s known as unimodular.
• The determinant of the matrix must be nonsingular, that is, its value must be nonzero, for the set of equations to have a unique solution.

Let us look at the definitions of determinants and matrices, as well as the various forms of matrices and their properties, using examples in the following articles:

1. Matrices and their Types
2. Mathematical Operations on Matrices
3. Properties of Matrix Addition and Scalar Multiplication
4. How to Multiply Matrices
5. Transpose of a matrix
6. Symmetric and Skew Symmetric Matrices
7. Elementary Operations on Matrices
8. Inverse of a Matrix by Elementary Operations
9. Invertible Matrices
10. Determinants
11. Properties of Determinants
12. Area of a Triangle using Determinants
13. Minors and Cofactors
14. Adjoint of a Matrix
15. Application of Determinants and Matrices

Permutations and Combinations

Permutation and Combination are methods for representing a collection of objects by choosing them from a list and dividing them into subsets.

• It specifies the different methods for organizing a set of data.
• Permutations are used to choose data or events from a group, while combinations are used to represent the order in which they are represented.

Mathematical Induction

For every natural number n, mathematical induction is a technique for proving a proposition, hypothesis, or formula that is assumed to be valid. The ‘Principle of Mathematical Induction‘ is a generalization of this that we can use to prove any mathematical statement.

Exponential & Logarithmic functions

Exponential Function Definition

An exponential function is a mathematical function in the form

y = f(x) = b^x

where “x” is a variable and “b” is a constant which is called the base of the function such that b > 1

Most commonly used exponential function base is the transcendental number e, and the value of e is approximately 2.71828.

Logarithmic Function Definition

If the inverse of the exponential function exists then we can represent the logarithmic function as given below: Suppose b > 1 is a real number such that the logarithm of a to base b is x if

b^x = a

Logarithm of a to base b can be written as log_b​a Thus, log_b​a = x if, b^x = a

In other words, mathematically, by making a base b > 1, we may recognize logarithm as a function from positive real numbers to all real numbers.

Properties of Exponential and Logarithmic Functions

Various properties of Exponential and Logarithmic Functions are:

• Domain of the exponential function is the set of all real numbers, i.e. R.
• Range of the exponential function is the set of all positive real numbers.
• Point (0, 1) is always on the graph of the given exponential function since it supports the fact that b⁰ = 1 for any real number b>1 .
• Exponential function is ever increasing; i.e., as we move from left to right, the graph rises above.
• Domain of the log function is the set of positive real numbers, i.e. R+.
• Range of the log function is the set of all real numbers.
• Point (1, 0) is always on the graph of the log function.

Exponential and logarithmic functions are related to each other since the inverse of exponential functions are the basis for defining logarithmic functions.

Algebra Formula

Here are some important algebraic formulas:

1. a²−b² = (a−b)(a+b)
2. (a+b)² = a²+2ab+b²
3. a²+b² = (a+b)²−2ab
4. (a−b)² = a²−2ab+b²
5. (a+b+c)² = a²+b²+c²+2ab+2bc+2ca
6. (a−b−c)² = a²+b²+c²−2ab+2bc−2ca
7. a³−b³ = (a−b)(a2+ab+b2)
8. a³+b³ = (a+b)(a2−ab+b2)
9. (a+b)³ = a³+3a²b+3ab²+b³
10. (a−b)³ = a³−3a²b+3ab²−b³
11. a⁴−b⁴ = (a−b)(a+b)(a²+b²)
12. a⁵−b⁵ = (a−b)(a⁴+a³b+a²b²+ab³+b⁴)

If “n” is a natural number:

• aⁿ−bⁿ = (a−b)(aⁿ⁻¹+aⁿ⁻²b+…+bⁿ⁻²a+bⁿ⁻¹)
• If “n” is even (n = 2k), aⁿ+bⁿ = (a+b)(aⁿ⁻¹−aⁿ⁻²b+…+bⁿ⁻²a−bⁿ⁻¹)
• If “n” is odd (n = 2k + 1), aⁿ+bⁿ = (a+b)(aⁿ⁻¹−aⁿ⁻²b+…−bⁿ⁻²a+bⁿ⁻¹)

Algebraic Operations

There are four basic mathematical operations that are used in Algebra. These are addition, subtraction, multiplication, and Division. These are discussed below:

Algebraic Addition

Ssummation of two or more algebraic terms is done in Addition. The addition of algebraic terms is indicated by “+” symbol.

• Addition of algebraic terms to yield a single value is only possible if there are like algebraic terms else the expression remains as it is.
• For Example, 3x²y + 5x²y = 8x²y as 3x²y and 5x²y are like algebraic terms while if we add 3xy² and 5x²y then it will not yield a single value instead it will remain as it is i.e. 3xy² + 5x²y.

Algebraic Subtraction

The method of finding the difference between two algebraic terms is called Subtraction of Algebraic terms. Subtraction of algebraic terms is indicated by “-” terms.

• Like Addition, Subtraction is also possible only between the like algebraic terms.
• For example, if we subtract 3x²y from 5x²y it will give a difference as 5x²y – 3x²y = 2x²y.
• But if subtract 3xy² from 5x²y it will not yield a single value as the two terms are unlike. Hence, the difference will be written as 5x²y – 3xy².

Algebraic Multiplication

Unlike Addition and Subtraction, Multiplication is possible between both like and unlike terms. Multiplication of algebraic terms is indicated by “⨯” or (a)(b).

• While performing the multiplication of algebraic terms, multiply the numerical coefficient as normal numbers and multiply the variables using the laws of exponents.

• For Example, if we multiply 4x²y with 5xyz then the product is given as (4x²y)(5xyz) = 20x³y²z.

Algebraic Division

Like Multiplication, Division of algebraic terms is possible between both like and unlike terms while keeping in mind the laws of exponents for variables and normal division for numbers.

• Division between algebraic terms is indicated by the “/” symbol. For Example, the division of 6x²y² by 3xy² is given as 6x²y²/3xy² = 2x.

Algebraic Laws

Algebraic Laws or Properties include Closure, Commutative, Associative, Distributive, and Identity Properties. These properties are defined for basic algebraic operations such as Addition, Subtraction, Multiplication, and Division.

A picture explaining algebraic laws has been attached below:

Algebraic Identities

Algebraic Identities are the expansion of terms of algebraic terms given as whole square or whole cube generally. These expansions help us to quickly solve the problems.

Some of the commonly used algebraic identities are mentioned below:

• (a + b)² = a² + 2ab +b²
• (a – b)² = a² – 2ab + b²
• a² – b² = (a + b)(a – b)
• (a + b)³ = a³ + b³ + 3a²b + 3ab²
• (a – b)³ = a³ – b³ – 3a²b + 3ab²

Algebra Solved Examples

Here we have provided some solved examples on algebra for your better understanding.

Example 1: Solve  3e^x + 6 = 120

Solution:

Given,

3e^x + 6 = 120

3e^x = 120 – 6

3e^x = 114

e^x = 114/3

e^x = 38

x = ln 38

Example 2: Solve for the value of y: 2(4-y)-3(y+3)=11

Solution:

 2(4-y) -3(y+3)=11

8-2y-3y-9=11

-5y-1=-11

-5y-1+1=-11+1

-5y=-10

Dividing both the sides by -5, 

-5y/-5=-10/-5

y=2

Example 3: Evaluate (2.3a⁵b²) × (1.2a²b²) when a = 1 and b = 0.5

Solution:

Let us simplify the given expression

=2.3a⁵b² × 1.2a²b²

=2.3 × 1.2 × a⁵ × a² × b² × b²

=2.76 × a⁵⁺² × b²⁺²

=2.76a⁷b⁴

Now let us substitute when, a = 1 and b = 0.5

For 2.76 a⁷ b⁴

= 2.76 (1)⁷ (0.5)⁴

= 2.76 × 1 × 0.0025

= 0.1725

Example 4: (2x-1)/3 – (6x-2)/5 = 1/3

Solution:

We have,

(2x-1)/3 – (6x-2)/5 = 1/3

By taking LCM for 3 and 5, which is 15

((2x-1)×5)/15 – ((6x-2)×3)/15 = 1/3

(10x – 5)/15 – (18x – 6)/15 = 1/3

(10x – 5 – 18x + 6)/15 = 1/3

(-8x + 1)/15 = 1/3

By using cross-multiplication, we get,

(-8x + 1)3 = 15

-24x + 3 = 15

-24x = 15 – 3

-24x = 12

x = -12/24 = -1/2

Verification

LHS = (2x – 1)/3 – (6x – 2)/5

= [2(-1/2) – 1]/3 – [6(-1/2) – 2]/5

= (- 1 – 1)/3 – (-3 – 2)/5

= – 2/3 – (-5/5)

= -2/3 + 1

= (-2 + 3)/3 = 1/3

RHS

Example 5: Find a number such that when 5 is subtracted from 5 times that number, the result is 4, more than twice the number.

Solution:

Let us consider the number as ‘x’

Then, five times the number will be 5x

And, two times, the number will be 2x

So,

5x – 5 = 2x + 4

5x – 2x = 5 + 4

3x = 9

x = 9/3 = 3

Also Read:

• Basic Math Formulas
• Various Algebraic Formulas used in Mathematics
• Algebraic Expressions

Algebra Basics to Advanced – FAQs

What is Algebra?

 Algebra is the area of mathematics that deals with the representation of mathematical statements as solutions to problems. To create a meaningful mathematical expression, it takes variables like x, y, and z together with mathematical operations like addition, subtraction, multiplication, and division.

What are the Branches of Algebra?

The branches of Algebra are:

1. Pre Algebra
2. Elementary Algebra
3. Abstract Algebra
4. Universal Algebra
5. Linear Algebra
6. Commutative Algebra 

Why are students required to learn Algebra?

Algebra is important for students to study because it develops problem-solving abilities, prepares them for higher-level arithmetic, helps in understanding and analyzing real-world problems, and has applications in a wide range of professions, including science, engineering, economics, and finance. A lot of college majors and jobs also require Algebra.

What are the basic operations in Algebra?

Basic operations performed in Algebra include addition, subtraction, multiplication, and division of Algebraic equations and solving the equations. 

What is Definition of Algebra?

Algebra is a branch of mathematics that deals with symbols and the rules to solve equations and it focuses on operations with variables, constants, and mathematical expressions.

What is Meaning of Algebra?

The word Algebra is derived from an Arabic word, ‘Al-jabr’ which means the ‘reunion of broken parts’. So, the meaning of Algebra is finding the unknown.

What is Abstract Algebra?

Abstract algebra, or modern algebra is the study of algebraic structures including groups, rings, fields, modules, vector spaces, lattices, and algebras.