Algebra is one among the branches of mathematics dealing with the number theory, geometry, and its analysis. It is sometimes referred to as the study of the mathematical symbols and the rules involving the manipulation of these mathematical symbols. Algebra includes almost everything right from solving of elementary equations to the study of the abstractions.
Algebra helps in to solve the mathematical equations and to derive the unknown quantities, like the bank interest, proportions, percentages. The letter variables in the algebra can be used to represent the unknown quantities which are coupled with the ability to rewrite the equations making it easier in finding the data for a given set of equations.
The algebraic formulas are used in daily life like finding the distance, find the volume in containers, or 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 as entities. The values of the equations of unknown quantities can be solved through algebra.
Some of the main topics coming under algebra include Basics of algebra, exponents, simplifying of algebraic expressions, polynomials, quadratic equations, etc.
Introduction to Algebra
- Basics of Algebra
- Addition And Subtraction Of Algebraic Expressions
- Multiplication Of Algebraic Expressions
- BODMAS And Simplification Of Brackets
- Substitution Method
- Solving Inequalities
- Introduction to Exponents
- Square Roots and Cube Roots
- Simplifying Square Roots
- Laws of Exponents
- Exponents in Algebra
- Associative Property, Commutative Property and Distributive Laws
- Cross Multiply
- Fractions in Algebra
- What is a Polynomial?
- Adding And Subtracting Polynomials
- Multiplying Polynomials
- Rational Expressions
- Dividing Polynomials
- Polynomial Long Division
- Rationalizing The Denominator
Branches of Algebra
Algebra is divided into different sub-branches of which some are given below-
- 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 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, logarithmic and Exponential equations, etc.
- Abstract Algebra – 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 as abstract algebra.
Abstract algebra deals with the 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 by 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 as the identity element for the operation addition, whereas 1 is called the identity element for the operation multiplication.
- Inverse Elements – The idea of Inverse elements comes up with the negative number. For Addition, we write -an as the inverse of a and for the purpose of 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 affects the sum. Consider for Example – (3 + 2) + 4 = 3 + (2 + 4)
- Linear Algebra – 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.
Commutative algebra– 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, depend on the commutative algebra.
This was just a brief discussion on Branches of Algebra. Now get model solutions for chapter Algebra in detail and step-by-step procedure to all questions in an NCERT textbooks at BYJU’S.
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