Algebra is one among the oldest branches in the history of mathematics that deals with the number theory, geometry, and analysis. The definition of algebra sometimes states that the study of the mathematical symbols and the rules, and it involves the manipulation of these mathematical symbols. Algebra includes almost everything right from solving elementary equations to the study of the abstractions. Algebra equations are included in many chapters of Maths, which students will learn in their academics. Also, there are several formulas and identities present in algebra.
What is Algebra?
Algebra helps in solving the mathematical equations and helps to derive the unknown quantities, like the bank interest, proportions, percentages. The variables in the algebra can be used to represent the unknown quantities which are coupled in such a way to rewrite the equations.
The algebraic formulas are used in our daily life to find the distance, the volume of containers, and to figure out the sales prices as and when needed. Algebra is very helpful in stating a mathematical equation and relationship by making use of letters or other symbols representing the entities. The unknown quantities in the equation can be solved through algebra.
Some of the main topics coming under algebra include Basics of algebra, exponents, simplification of algebraic expressions, polynomials, quadratic equations, etc.
In BYJU’S, students will get the complete details of algebra, including its equations, terms, formulas, etc. Also, solve examples based on algebra concepts and practice worksheets to get a better understanding of the fundamentals of algebra. Algebra 1 and algebra 2 are the Maths courses included for students in their early and later stages of academics, respectively. Like, algebra 1 is the elementary algebra practised in classes 7,8 or sometimes 9, where basics of algebra are taught. But, algebra 2 is the advanced algebra, which is practised in high school level. The algebra problems will involve expressions, polynomials, the system of equations, real numbers, inequalities, etc. Know more algebra symbols that are used in Maths.
Branches of Algebra
As it is known that, algebra is the concept based on unknown values called variables. The important concept of algebra is equations. It follows various rules to perform arithmetic operations. The rules are used to make sense of sets of data that involves two or more variables. It is used to analyse many things around us. You will probably use the concept of algebra without realising it. Algebra is divided into different sub-branches such as elementary algebra, advanced algebra, abstract algebra, linear algebra, and commutative algebra.
Algebra 1 or Elementary Algebra
Elementary Algebra covers the traditional topics studied in a modern elementary algebra course. Arithmetic includes numbers along with mathematical operations like +, -, x, ÷. But in the field of algebra, the numbers are often represented by the symbols and are called variables such as x, a, n, y. It also allows the common formulation of the laws of arithmetic such as, a + b = b + a and it is the first step that shows the systematic exploration of all the properties of a system of real numbers.
The concepts coming under the elementary algebra includes variables, evaluating expressions and equations, properties of equalities and inequalities, solving the algebraic equations and linear equations having one or two variables, and so on.
Algebra 2 or Advanced Algebra
This is the intermediate level Algebra. This algebra has a high level of equations to solve as compared to pre-algebra. Advanced algebra will help you to go through the other parts of algebra such as:
- Equations with inequalities
- Solving system of linear equations
- Graphing of functions and linear equations
- Conic sections
- Polynomial Equation
- Quadratic Functions with inequalities
- Polynomials and expressions with radicals
- Sequences and series
- Rational expressions
- Discrete mathematics and probability
Abstract algebra is one of the divisions in algebra which discovers the truths relating to algebraic systems independent of specific nature of some operations. These operations, in specific cases, have certain properties. Thus we can conclude some consequences of such properties. Hence this branch of mathematics called abstract algebra.
Abstract algebra deals with algebraic structures like the fields, groups, modules, rings, lattices, vector spaces, etc.
The concepts of the abstract algebra are below-
- Sets – Sets is defined as the collection of the objects that are determined by some specific property for a set. For example – A set of all the 2×2 matrices, the set of two-dimensional vectors present in the plane and different form of finite groups.
- Binary Operations – When the concept of addition is conceptualized, it gives the binary operations. The concept of all the binary operations will be meaningless without a set.
- Identity Element – The numbers 0 and 1 are conceptualized to give the idea of an identity element for a specific operation. Here, 0 is called the identity element for the addition operation, whereas 1 is called the identity element for the multiplication operation.
- Inverse Elements – The idea of Inverse elements comes up with a negative number. For addition, we write “-a” as the inverse of “a” and for the multiplication, the inverse form is written as “a-1″.
- Associativity – When integers are added, there is a property known as associativity in which the grouping up of numbers added does not affect the sum. Consider an example, (3 + 2) + 4 = 3 + (2 + 4)
Linear algebra is a branch of algebra which applies to both applied as well as pure mathematics. It deals with the linear mappings between the vector spaces. It also deals with the study of planes and lines. It is the study of linear sets of equations with the transformation properties. It is almost used in all the areas of Mathematics. It concerns the linear equations for the linear functions with their representation in vector spaces and through the matrices. The important topics covered in linear algebra are as follows:
- Linear equations
- Vector Spaces
- Matrices and matrix decomposition
- Relations and Computations
Commutative algebra is one of the branches of algebra that studies the commutative rings and its ideals. The algebraic number theory, as well as the algebraic geometry, depends on the commutative algebra. It includes rings of algebraic integers, polynomial rings, and so on. There are many other areas of mathematics that draw upon commutative algebra in different ways such as differential topology, invariant theory, order theory, and general topology. It has occupied a remarkable role in modern pure mathematics.
Frequently Asked Questions on Algebra
What are the basics of algebra?
The basics of algebra are:
Addition and subtraction of algebraic expressions
Multiplications and division of algebraic expression
Literal equations and formulas
Applied verbal problems
Mention the types of algebraic equations
The five main types of algebraic equations are:
Monomial or polynomial equations
What are the branches of algebra?
The branches of algebra are: