Introduction to  Algebra. Basics, history and it’s branches.

Algebra is one of the broad divisions of mathematics. Number theory, geometry and analysis etc. are other major departments of mathematics. Generally, algebra is the study of mathematical symbols and the rules for manipulating these symbols.  Algebra is the subject of almost all mathematics in a formula. Many things come under algebra, from solving elementary equations to studying abstract concepts like studying groups, rings and fields. Advanced abstract part of algebra is called abstract algebra.

 

Early algebra is considered indispensable not only for mathematics, science, engineering but also medicine and economics. Early algebra differs from arithmetic in that it uses letters instead of straight numbers that are either unknown or can hold multiple values. 

Frequently asked questions:

1.What are the basics of algebra?

2.What exactly is algebra?

3.What is algebra and examples?–

Algebra  is based on solving the equation of variables and constants and deriving the values ​​of variables. The development of algebra resulted in the development of coordinate geometry and calculus, which greatly increased the usefulness of mathematics. This accelerated the development of science and technology.

The great mathematician Bhaskaracharya II has said –

The former prophet person, person, usually questioning person.

Jyantu Shakya Mandhimirnintant: Yasmantasyadvichchmi Vis Kriya Ch.

That is, people with mental intelligence cannot solve questions with the help of mathematics (arithmetic), they can solve questions with the help of latent mathematics (algebra). In other words, algebra simplifies the solution to difficult problems of arithmetic.

Algebra generally refers to the science in which numbers are represented by letters. But the operation marks remain the same, which are used in arithmetic. Suppose we want to write that the area of ​​a rectangle is equal to the product of its length and width, then we will represent this fact as follows-

X = l x f

The development of the modern notation of algebra began a few centuries ago; But the problem of means of equations is very old. 2000 years before Christ, people used to solve equations with speculation. By 300 years before Christ, our ancestors started writing equations in words and knew their solutions by geometry method.

Algebra, in the broadest sense, is a branch of mathematics in which the properties of numbers and their interrelations are interpreted by common symbols. These symbols are mostly the letters (a, b, c, …, x, y, z) and operation signs (+, -, *, …) and the relational sign (=,>, <.. .). For example, x2 + 3x = 28 means, ‘x is a number whose triple is added to its square, the result is 24. Algebraic symbols and numbers are used not only in mathematics but in various branches of science. In a broader sense, algebra involves the following topics:

 

Equation, polynomial, continued fraction, series, number sequence, determinant, form, new type of numbers, such as numerology, matrix.

 The branches and fields of algebra

Today, algebra is not only comprised of equations, polynomials, polynomials, infinite product, number sequence, quadratic or form, new types of numbers such as numerology, arithmetic etc. are studied in many cases.

Algebra can often be divided into the following categories –

 

Elementary algebra 

It is often taught in schools under the name of ‘algebra’. The ‘group theory’ taught at the university level can also be called elementary algebra.

Abstract algebra –

 Sometimes it is also called ‘modern algebra’. Under this, algebraic structures like groups, rings, fields etc. are taught under it.

Linear algebra – 

In this, the properties of vector space are studied. The matrix also falls under this.

Universal algebra –

 In this, the common properties of all types of algebraic structures are studied.

Algebraic number theory –

In this, the properties of numbers are studied by algebraic method. The number theory itself sowed the seed of Amurtikaran in algebra.

Algebraic geometry –

 It uses abstract algebra on geometric problems.

Algebraic combinatorics – 

Under this, abstract algebra is used to solve combinatorics (permutation) questions.

 History

The topic was named algebra based on a book called ‘Brahmasphutasiddhanta’ written by the Indian mathematician Brahmagupta around 624 AD. It is calculated from the seeds, ie, the basic components. Later in the 12th century Bhaskar also wrote an important treatise on algebra. Around 625 AD Muhammad Ibnmusa al-Khwarizmi named one of his treatises in Baghdad as “Aljabr and al-Muqamab”. ‘Aljabra’ is an Arabic word and ‘combat’ is Persian, and both mean equation or related. The name of this subject was named algebra in Europe after the name of this important book,. For this in Chinese, tuman-yun (meaning divine ingredient), Japanese.

 

 algebra brief introduction

history in summary

In short, the subject range in the development of algebra expanded from these levels:

 

(1) circa 1,600 BC Number-related riddles have to be solved in the period from AD to 245 AD, without the help of any symbolism;

(2) Drawing the square of the given area by geometric method;

(3) development of macro symbol system;

(4) more rational interpretation of the equations till 400–1200 AD;

(5) the rendering of the theory for the means of quadratic and quadratic equations in the 16th century;

(4) Development of a clear and convenient symbol system and

(4) Development of abstract algebra from 1800 AD.

 numbers

Numbers that are used to count objects are called natural numbers. Other numbers are called artificial numbers. The study of artificial numbers starts in arithmetic itself, but there is sufficient knowledge of fractions only. In algebra, it is necessary to think of the negative numbers, irrational, spherical, mixed numbers etc.

 algebraic expression

2a means a + a, i.e. double of a. Broadly, if m is a positive whole number, then ma means m times the a. ma is also called the product of m and a.

a2 means a.a ; a3 means a.a.a. Broadly, if m is a positive whole number then am means

a.a.a ….. m times.

In am, m is called exponent and a is called base. Later, the meanings of ma and am are expanded and are also given in those situations when m is any number of minus, different, irrational etc.

Symbols of common numbers are one or more letters and the product of a number is called a term, eg 3a2b, -4a, x (ie 1x) etc. The sum of several terms is called an algebraic expression. The expression of the aforesaid three terms is 3a2b – 4a + x. The term alone is called monomial, the expression of two terms is binomial, the three term is called trinomial. An expression with more than one term is called polynomial. One product is obtained by multiplying two or more terms. Each term to be multiplied is called the factor of the term containing the product.

Although the coefficient of a factor of a term is the product of the remaining factors, such as the coefficient of a3 in 3a3b2 can be called 3b2, but the practice is to consider the product of the starting factors as the coefficient of the product of the remaining segments. Thus the coefficient of b2 is 3a3, the coefficient of a3b2 is 3. If the coefficient is numerical, it is called numerical coefficient. The expression can be used as a term after closing it in parentheses.

 initial operations

In addition to the general operations on polynomials, addition, subtraction, multiplication and division – the factor of addition factorization, involution, square root determination, the least common multiple and the greatest common factor of two or more polynomials . Ratios and factors are used in broad terms for all types of numbers.

equation

There are mainly three types of parity:

 

(1) 3 + 2 = 5 is the relation of numbers.

(2) x + 2x = 3x is the relation that is true for all values ​​of x; It is called identity.

(3) x + 3 = 2 is the parity that is true for only one value of x (literally −1); This is called an equation. Often, the symbol (≡) is used in place of the sign = in the common identity to distinguish it from its equation. The solution of the quadratic and quadratic equations was given by Diophantus in about 250 AD (see Diophantic equation). In India, Aryabhata gave the solution of quadratic equation fundamentally in 7 AD.

 initial categories

In the medieval era, there was considerable interest towards the study of arithmetic, geometric, etc. For this reason, the compilation (finding the sum) of these series is an interesting subject of elementary algebra. Take two formulas for example:

 

1 +2 +3 + …. up to m terms = m (m + 1)

Up to 12 +22 +32 + …. m terms = m (m + 1) (2m + 1)

The study of geometric hierarchy leads us to the study of infinite series. Then important concepts like boundary etc. become necessary and differential and integration become perceptible.

 

 The importance of elementary algebra

Using symbols, rather than arithmetic, to achieve much broader results with less labor is the achievement of algebra. That is why algebra is called ‘short hand’ of the language. According to the French mathematician Bertaid (1822-1900), the operations and algebraic functions in algebra are studied independent of the numbers on which they are applicable. This is the specialty of this science. The study of algebra is essential in the practice of science. In the form of formulas, the inevitability of algebra is immediately apparent.

 Generalization and Abstract algebra

Algebra is generalized arithmetic and the process of generalization continues in the progressive development of algebra. In early algebra itself, ab, am, am. The meanings of an, (am) n etc. are broadened to make a definite mean for all values ​​of a, b, m, n. All this was possible due to the (1) square root of the imagination. Unfortunately this zodiac is considered ‘imaginary’ and the first letter of its English translation (i) is symbolized by it. When i was found so much more useful for problem solving in the 16th and 17th centuries, attention was paid to its nature. While not considered a number, it was abstractly regarded as a symbol of some arbitrary operations on numerology and its geometric interpretation was tangentially ‘rotate to a right angle’. These interpretations inspired why other symbols such as i were not discovered. In the same effort, in 1773, Hamilton invented the quaternions i and j in the context of three-dimensional rotation, and stated that ij = -ji, which was a very important discovery, because algebra was always ab = ba. Now mathematicians discovered many types of ‘very complex numbers’ and operation symbols. In the end, the question arose as to why a particular type of algebra should be created by taking any symbols in place of ordinary numbers and determining the rules for combining them.

 

In this way vector algebra and matrix (or array) algebra were created. The generalization of the basic operations of algebra gives many types of algebraic systems. These systems have different rules regarding the combination of ingredients from which other components are formed. Since the study of these systems does not have any special significance as to what the elements actually are, rather they have precedence over the rules. Therefore these systems are called abstract algebra.

 

The following concepts are necessary for an operation * to give some examples of abstract systems –

(1) Closure: If a set has two elements a and b, then a * b is also an element of the same set.

(2) Commutativity: a * b = b * a *

(3) Associativity: If a, b, c are components of a set, then (a * b) * c = a * (b * c)

4) Existence of identity: The set must have such a component that a * e = e * a = a

(5) Existence of inverse: In a set, any element corresponding to any component a has a-1 such that a * a-1 = a-1 * a = e

(6) Distribution rules against first operation and second operation: a (b * c) = (a b) * (a c) and (b * c) a = (b a) * (c a)

A group (or association) towards a set of operations is called when it has properties 1, 3, 4, 5. If property 2 is even, then it is called sequence exchange, or abeli group.

A set of two operations * and ¢ is called a ring when the five properties are 1 to 5 against the first, 1, 3 against the second and 4 against both.

Such a ring is called a field in which the properties of the second operation are 2 and 4, and the inverse of each component other than the universal (ie a * a-1) of the first operation is the inverse of the second operation. 

example

To addition and multiplication operations

(1) The set of all whole numbers including zeros is ring.

(2) is the set field of all rational numbers, or of real numbers, or of composite numbers.

Many new algebraic systems have emerged in an attempt to solve specific problems in other branches of mathematics. The Lee group was invented in an attempt to classify differential equations. Similarly, some problems of topology gave rise to homological algebra. Around 1750, Boole developed symbolic algebra, which is now used in telephone circuits and digital electronics.