TEACHING UDERGRADUATE MATHEMATICS (Small Review)

                  TEACHING UDERGRADUATE MATHEMATICS 

         1.     Algebra:

            Algebra (from Arabic al-jebr meaning "reunion of broken parts") 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. Algebra is a branch of mathematics that deals with relations, operations and their constructions. It is one of building blocks of mathematics and it finds a huge variety of applications in our day-to-day life.

The word “algebra” means many things. Apart from its significance as a core subject of mathematics, Algebra helps students and kids a lot in developing an overall understanding of other advanced branches of mathematics such as Calculus, Geometry, Arithmetic etc. Basic algebra is the first in a series of higher-level math classes students need to succeed in college and life. Algebraic Expression: Rhetorical Algebra, Syncopated Algebra and Symbolic Algebra. Conceptual Stages: Geometric Stages, Static equation solving stage, Dynamic functional stage, Abstract Stage.

               Because many students fail to develop a solid math foundation, an alarming number of them graduate from high school unprepared for college or work. Many end up taking remedial math in college, which makes getting a degree a longer, costlier process than it is for their more prepared classmates. And it means they’re less likely to complete a college-level math course. For middle-schoolers and their parents, the message is clear: It’s easier to learn the math now than to relearn it later.

Some area of Mathematics with the word algebra in the Name. They are following,

ü Elementary algebra.

ü Abstract algebra.

ü Linear algebra.

ü Boolean algebra.

ü Modern algebra.

ü Commutative algebra.

  ü Computer algebra.

  ü Universal algebra.

  ü Systematic algebra.

  ü Classical algebra etc.

 Algebra emerged at the end of the 16th century in Europe with the work of François Viète. The most important Area of algebra is described below.

    2.     Classical Algebra:-

                                i.            Introduction:

               The start of the classical discipline of algebra developments included several related trends, among which the following deserve special mention: the quest for systematic solutions of higher order equations, including approximation techniques; the rise of polynomials and their study as autonomous mathematical entities; and the increased adoption of the algebraic perspective in other mathematical disciplines, such as geometry, analysis, and logic. During this same period, new mathematical objects arose that eventually replaced polynomials as the main focus of algebraic study. The classical algebra uses finding roots or values of unknowns symbols instead of specific numbers, arithmetic operations to establish procedures for manipulating symbols.

The development of classical algebra can be traced to the ancient period Where Algebra Comes From.

                             ii.            Contents:

•      systems of equations

•      number and equations

•      Equations and Their Solutions.

•      Numerical Solution of Equations.

•      Introduction to cryptography

•      Polynomial equation and complex number.

                           iii.            Period:

  Classical algebra has been developed over a period of 4,000 years ago.

             3.     Abstract Algebra:-

                                i.            Introduction:

                        School algebra can be seen as a generalization of arithmetic in which the variables are numbers and the expressions and equations are formed with the four arithmetic operations. Abstract algebra is a generalization of school algebra in which the variables can represent various mathematical objects, including numbers, vectors, matrices, functions, transformations, and permutations, and in which the expressions and equations are formed through operations that make sense for the particular objects: addition and  multiplication for matrices, composition for functions, and so on.  This section provides a short sketch of abstract algebra in order to highlight ideas of structure and to present the terms, concepts, notations, and perspectives that undergird the research questions and subsequent analysis.

Abstract algebra consists of axiomatic theories that provide opportunities to consider many different mathematical systems as being special cases of the same abstract structure.  The theories are called axiomatic because the structures are defined by axioms. Group theory is “one of the oldest (and also one of the simplest) of axiomatic theories”

                             ii.            Contents:

·       Groups theory

·       Rings theory

·       Fields theory

·       Modules theory

·       Vector space

·       Functions and integers

·       Galois Theory

                           iii.            Period:

In the 19^th century algebra was no longer restricted to ordinary number systems.  Algebra expanded to the study of algebraic structures.

         4.     Modern Algebra:-

                                i.            Introduction:

                        Modern algebra, also called abstract algebra, branch of mathematics concerned with the general algebraic structure of various sets (such as real numbers, complex numbers, matrices, and vector spaces), rather than rules and procedures for manipulating their individual elements. During the second half of the 19th century, various important mathematical advances led to the study of sets in which any two elements can be added or multiplied together to give a third element of the same set. The elements of the sets concerned could be numbers, functions, or some other objects.

A definitive treatise, Modern Algebra, was written in 1930 by the Dutch mathematician Bartel van der Waerden, and the subject has had a deep effect on almost every branch of Mathematics. The study of groups, rings, fields is not only the students first introduction to modern mathematical thinking but provides some of the essential crutches. Abstract algebra emerged around the start of the 20th century, under the name modern algebra. Its study was part of the drive for more intellectual rigor in mathematics. Initially, the assumptions in classical algebra, on which the whole of mathematics (and major parts of the natural sciences) depend, took the form of axiomatic systems.

                             ii.            Contents:

Ø Group theory

Ø Modulus and vector spaces

Ø Ring theory

Ø Field and Galois theory

                           iii.            Period:

        The end of the 19th and the beginning of the 20th century saw a tremendous shift in the methodology of mathematics. Abstract algebra emerged around the start of the 20th century, under the name modern algebra.

         5.     Linear Algebra:-

                                i.            Introduction:-

                     The origins of the concepts of a determinant and a matrix, as well as an understanding of their basic properties, are historically closely connected. Both concepts came from the study of systems of linear equations. Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. The branch of mathematics that deals with the theory of systems of linear equations, matrices, vector spaces, determinants, and linear transformation. Or the part of algebra that deals with the theory of linear equations and the linear transformations.

Branch of algebra concerned with method of solving system of linear equations, more generally, the mathematics of linear transformation and vector spaces. The term "linear algebra" is also used to describe a particular type of algebra. Techniques from linear algebra are also used in analytic  geometry, engineering, physics, natural sciences, computer science, computer animation, advanced facial recognition algorithms and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models.

                             ii.            Contents:

ü Fields and Vector spaces

ü Linear transformations

ü Matrix theory

ü Eigenvalue and eigenvectors

ü Subspace, span and basis

ü Theory of Determinants

ü Linear Coding Theory

ü Linear Equations

                           iii.            Period:

  The emergence of the subject came from determinants, values connected to a square matrix, studied by the founder of calculus, Leibnitz, in the late 17^th century, Gabriel Cramer devised Cramer's Rule for solving linear systems in 1750 and  19^th century.

       6.     Conclusions:

                  I am at the end of my individual assignment. This work was given as a part of University campus curriculum. I am very grateful to the school authority for providing a golden platform for the creative work. I express my gratitude to the subject teacher Mr. Abatar subedi for his guidance and support.

              The topic was very interesting and I learned a lot of things about “algebra ”. Such works develoapp the creativity of the students and also increase their knowledge base. I hope to get such works in future days. Once again I am grateful to everyone who was involved in this work.     

    REFRENCE:-

 -        - Alan sultan and alice F. artz (2011). The mathematics that every secondary school mathematics    - teacher needs to know. Routledge

-        -  Wikipedia and various related websites.

-        -Powerpoint of Subject teacher etc.