Algebra: It is a kind of arithmetic where we use unknown quantities along with
numbers. These unknown quantities are represented by letters of the English
alphabet such as X, Y, A, B, etc. or symbols. The use of letters helps us to
generalize the formulas and rules that you write and also helps you to find the
unknown missing value.
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 student will learn in their academics. Also, there are a number of
formulas and identities present in algebra.
The basics of algebra are: Addition and subtraction of algebraic expressions, Multiplications and division of algebraic
expression,
Solving equations Literal equations and formulas
Solving equations Literal equations and formulas
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. We 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.
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.
Advanced Algebra:
This is the intermediate level
Algebra or we can say prerequisite
of Elementary 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
- Matrices
- 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
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-
1.
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.
2.
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.
3.
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 operation addition,
whereas 1 is called the identity element for the operation multiplication.
4.
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 purpose of multiplication the inverse form is
written as “a-1”.
5.
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 for 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
- Relations
- 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.
BODMAS RULE:
BODMAS is an acronym and it
stands for Bracket, of, Division, Multiplication,
Addition and Subtraction. It
explains the order of operations to solve an expression. According to BODMAS rule,
if an expression contains brackets ((), {}, []) we have to first solve or
simplify the bracket followed by of (powers and roots etc.), then division,
multiplication, addition and subtraction from left to right. Solving the
problem in the wrong order will result in a wrong answer.
Check the examples below to
have a better understanding of using the BODMAS rule.
Simplify:
1800÷10{ (12−6)+(24−12) }
=1800÷10{6+12} =180{18}
=180×18 =3240
Important Formulae:
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(a+b)2=a2+2ab+b2
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(a−b)2=a2−2ab+b2
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(a+b)(a–b)=a2–b2
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(x+a)(x+b)=x2+(a+b)x+ab
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(x+a)(x–b)=x2+(a–b)x–ab
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(a+b)3=a3+b3+3ab(a+b)
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(a–b)3=a3–b3–3ab(a–b)
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am × an =
am+n
am ÷ an =
am- n
(am)n =
amn
(ab)m = ambm
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