Branches of Algebra

Algebra is one of the fundamental branches of mathematics that deals with symbols, variables, and the rules for manipulating these symbols. Algebra is divided into several important branches, each with its own focus and application. Below are the key branches:

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. 

Read More: Real-Life Applications of 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.

Read More: Applications of Linear Algebra

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

Boolean Algebra

Boolean Algebra is a branch of algebra that deals with binary variables and logical operations. It is based on two values, typically denoted as 0 (false) and 1 (true), and involves operations like AND, OR, and NOT.

• AND (conjunction): The result is 1 if both operands are 1, otherwise 0.
• OR (disjunction): The result is 1 if at least one operand is 1, otherwise 0.
• NOT (negation): The result is the inverse of the operand (i.e., 1 becomes 0, and 0 becomes 1).