is the branch of mathematics concerning the study of the rules of operations and
relations, and the constructions and concepts arising from them, including terms, polynomials,
equations and algebraic structures.
Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of
the main branches of pure mathematics.
Elementary algebra, often part of the curriculum in secondary education, introduces the concept
of variables representing numbers. Statements based on these variables are manipulated using
the rules of operations that apply to numbers, such as addition.
This can be done for a variety of reasons, including equation solving. Algebra is much broader
than elementary algebra and studies what happens when different rules of operations are used
and when operations are devised for things other than numbers.
Addition and multiplication can be generalized and their precise definitions lead to structures such
as groups, rings and fields, studied in the area of mathematics called abstract algebra.
Learn More about Number System
Algebraic Numbers are the real number for which exist a polynomial equation with integer
coefficients such that the particular real numeral is the answer. It is any number, which is a root of
non-zero polynomial with rational coefficients.
Algebraic numbers that contain the entire natural numbers , all rational numbers , a few irrational
numbers , and complex numbers .Elementary algebra
Elementary algebra is the most basic form of algebra. It is taught to students who are presumed
to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only
numbers and their arithmetical operations (such as +, -, x, /) occur. In algebra, numbers are
often denoted by symbols (such as a, x, or y). This is useful because:
It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and
thus is the first step to a systematic exploration of the properties of the real number system.
It allows the reference to "unknown" numbers, the formulation of equations and the study of how
to solve these. (For instance, "Find a number x such that 3x + 1 = 10" or going a bit further "Find
a number x such that ax+b=c". This step leads to the conclusion that it is not the nature of the
specific numbers that allows us to solve it, but that of the operations involved.)Polynomials
A polynomial is an expression that is constructed from one or more variables and constants, using
only the operations of addition, subtraction, and multiplication (where repeated multiplication of
the same variable is standardly denoted as e
xponentiation with a constant nonnegative integer
Read More on Different types of Numbers
For example, x2 + 2x - 3 is a polynomial in the single variable x.
An important class of problems in algebra is factorization of polynomials, that is, expressing a
given polynomial as a product of other polynomials. The example polynomial above can be
factored as (x - 1)(x + 3). A related class of problems is finding algebraic expressions for the roots
of a polynomial in a single variable.Abstract algebra
Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of
numbers to more general concepts.
Sets: Rather than just considering the different types of numbers, abstract algebra deals with the
more general concept of sets: a collection of all objects (called elements) selected by property,
specific for the set. All collections of the familiar types of numbers are sets. Other examples of sets
include the set of all two-by-two matrices, the set of all second-degree polynomials (ax2 + bx + c),
the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic
groups which are the group of integers modulo n.
Binary operations: The notion of addition (+) is abstracted to give a binary operation,
notion of binary operation is meaningless without the set on which the operation is defined. For
two elements a and b in a set S, a
b is another element in the set; this condition is called
closure. Addition (+), subtraction (-), multiplication (x), and division (/) can be binary operations
when defined on different sets, as is addition and multiplication of matrices, vectors, and
Iden tity elements: The numbers zero and one are abstracted to give the notion of an identity
element for an operation. Zero is the identity element for addition and one is the identity element
For a general binary operator
the identity element e must satisfy a
e = a and e
a = a. This
holds for addition as a + 0 = a and 0 + a = a and multiplication a x 1 = a and 1 x a = a. Not all set
and operator combinations have an identity element.For example :-
the positive natural numbers (1, 2, 3, ...) have no identity element for addition.Properties of Algebraic Umbers
* All algebraic numbers are determinable and therefore definable.
* The situate of algebraic numbers are countable.
* The imaginary number denoted by i, which is algebraic.
* All rational numbers are algebraic, but the irrational number may or may not be algebraic.Example :-
Set more purely, if you have a polynomial like:
2x2-4x+2 = 0
Then x is algebraic.
This is because:
It is a non-zero polynomial
x is a root (ie x gives the result of zero to the function 2x2-4x+2)
the coefficients are rational numbers