Algebra

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

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.

Parts of Algebra

Introduction to Algebra

Exponents

Simplifying

Polynomials

Quadratic Equations

Branches of Algebra

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 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 affect 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, depending 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 textbook at BYJU’S.


Algebra

In this article we will learn about basics of algebra and the concepts related to it such as constants, variable, expressions etc.

Try the following problems:

algebra

It’s simple. You can solve these problems without lifting a pen and a paper.

variable

Problems such as these, where we need to find the missing number can be easily solved using algebra. ‘Algebra’, the word seems tough but it is not. Now in the problems mentioned above, instead of using blank spaces we can use ’alphabets’.

algebra

Confused!! Well don’t be. We’ll explain you how. If we write above questions as:

algebra

The x, y and z in above represent the same thing as the blank spaces and their value is unknown. In algebra, we use alphabets in place of numbers and these letters can take any value. With help of algebra we can generalize different formulas and rules.  The alphabets used in the above problems are known as ‘variables’.

algebra

The basic components of elementary algebra are as follows:

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Constants mean quantities which are non-varying. It refers to a fixed value.

For example: 5, 8, 9, 75533, 9488 etc.

Variable is a quantity whose value can change depending upon the mathematical context.

These are known so because their value can vary according to the problem and it does not have a fixed value. Any symbol can be used to represent a variable. Any letter a, b, c, d ……. can be used as a variable.

Any combination of constants, variables and operators is known as an expression.

Example: 5x, x + y + z, 9a + 2 etc.

Any expression having an equality sign is known as an equation.

Example: 5x = 4, x + y + z = 9a, 9a + 2 = 15 etc.

Mathematical operators are mathematical entities that take two values and perform calculations on them.

Example: The four basic mathematical operators are ‘+’, ‘-’, ‘×’, ‘÷’

Try to solve the following problems.

Illustration 1: Vishal is trying to build a stack of pencils as shown. For each layer he uses 2 pencils. Now, the question is how pencils will be used till 50th layer?

algebra

Solution:

algebra

Do you notice a pattern here? For every layer 2 pencils add up every time. If I want to know the number of pencils used in 25th layer, I will simply multiply the number of layer to the number of pencils used in each layer, i.e. 25 x 2 = 50. Thus, I use 50 pencils. Similarly the number of pencils used in 50th layer can easily be given as 50 x 2 = 100.

Now, if I want to generalize this pattern, then for any level ‘n’, the number of pencils required will be ‘2n’. This will directly give us the number of pencils required for any level by putting different values for n. Here, ‘n’ is the variable as its value can be chosen accordingly.

So, number of pencils in nth layer = 2n

algebra

This is just an illustration; you can solve varied problems using algebra. To learn more about algebra download BYJU’s –The Learning App.


Practise This Question

What is the sum of x3+4xy+3y2 and 3x2+5xy ?