Basic Concepts of Algebra:
Algebra! What is Algebra? Why we need Algebra?
A branch of math similar to arithmetic is Algebra and it uses main four arithmetic operations i.e. addition (+) subtraction (-), multiplication(*) and division (/). You must be thinking where is the difference between two? But there is a difference, Algebra introduces a new element, an element of the unknown, this unknown element is known as a variable. Variables are denoted by use of letters. These letters stand for unknown quantities and allow writing formulas and using them in a general way. Henceforth, Algebra is a language of mathematics which is useful to solve a type of mathematics problem in an easier way.
Components of Algebra:
Variable and Constant are two components of algebra.
Variable is something that can vary i.e. change. The value of a variable is not fixed.
Constant is something which is fixed and remains same.
For example: X + Y = 10
Here X and Y are variables and 10 is constant. We need to identify value of variables X and Y in such a way that we get 10. Now, there are many possible answers for this, let's have a look at some:
X + Y = 10, Solution could be:
X = 0, Y = 10: 0+10 = 10
X = 1, Y = 9: 1+9 = 10
X = 8, Y = 2 : 8 + 2 = 10
The thing to be noticed in the above example is that value of variables is not fixed, it can have different values whereas constant remains same as 10.
Both components and variables combine together to form algebraic terms and terms combine together is known as an algebraic expression. When this expression will have an equality operator it is known as an equation.
For now, let's have a look at this equation: 5A + 10 = 25
Here, A is a variable, 10 and 25 are constants and = is equal sign or fulcrum. One more thing to be noticed here is that 5A + 10 is an algebraic expression whereas 5A + 10 = 25 is an algebraic equation.
The answer of the following equation i.e. value of variable would be:
5A = 25-10
5A = 15
A =15/5
A =3
This was a simple equation. Now let's look at more complex one:
Equation: x + 45 = 3x - 45
This equation has a variable on both sides. Let's find out the solution for this:
x + 45 = 3x - 45
45 + 45 = 3x - x
90 = 2x
45 = x. Here it is the value of variable x is 45.
In the above two examples, you saw variables like A, x and 3x. In a variable 3x, 3 is known as a coefficient, (a coefficient is a number which is multiplied by a variable). Similarly, in the case of A and x, the coefficient is 1.
Note: Maximum power of variables determines the nature of algebraic equation. In the above examples, the power of all variables is unity, the equation is known as a linear equation. When the power of variable is 2 or 3 then such equations are known as Quadratic and Cubic equations.
Note: The terms with same algebraic factors are called like terms whereas the terms with different algebraic factors are called, unlike terms. In the algebraic expression 7xy-5xy-3x-4, 7xy and 5xy have same algebraic factors like 7, x and y or 5, x and y respectively, hence they are like terms whereas the factors of 5xy and 3x are different, thus they are unlike terms.
Algebraic Operations:
Like we mentioned above that basic algebraic operations are similar to arithmetic operations. Here are important Algebraic relationships which develop the basic concept of Algebra:
(c+d) 2==c2+2cd+d2, Sum of two variables in Quadratic form
(c-d)2= c2-2cd+d2 Negative Counterpart of sum of two variables in Quadratic form
c2-d2=(c+d)(c-d)
(c+d)3=c3+3c2d+3cd2+d3 Sum of two variables in cubic form
(c+d)3=c3+d3+3cd(c+d) Sum of two variables in cubic form written in different way.
(c-d)3=c3-3c2d+3cd2-c3 Negative Counterpart of sum of two variables in Cubic form
(c-d)3=c3-d3-3cd(c-d)
c3+d3=(c+d)(c2-cd+d2)
c3-d3= (c-d)(c2+cd+d2)
Algebraic Operations in three variables:
(a+b+c)2 = a2+b2+c2+2(ab+bc+ca)
(a+b+c)3 = a3+b3+c3+3(a+b)(b+c)(c+a)
a3+b3+c3-3abc =(a+b+c)×(a2+b2+c2-ab-bc-ca).
Importance of Algebra
Algebra is considered as the gatekeeper topic in maths. It is utilized by experts in various fields like electrical, architecture, and computer science. It is a language of generalization, a symbolic language which helps in describing patterns and some also consider it as basic human right because it is a part of the day to day calculation. In a progression of advanced Math, basic Algebra comes on top. It is one of the primary needs of students pursuing higher level maths. Learning's of Algebra is an imperative for advanced math topics like Calculus, Trigonometry and much more. For some students, expertise in a combination of basic and higher level algebra have proved advantageous in the path of their career.
Algebra is an important part of the culture and played a crucial role in history (Industrial Revolution is one such example where algebra was used and there are much more). It is a powerful tool to learn because it makes you think logically and gives us the ability to solve problems in maths, it helps in evolving reasoning skills and keeping organized. Knowing and mastering algebra becomes very handy while appearing for Olympiads and other competitive exams both during after school. Algebra is the tool which is used in modern technologies like mobile phones, satellites or to develop a game etc. Algebra, in short, is an opening to succeed in the 21st century, to excel in the fields of math's, to understand topics in technology sector, psychology, business, management, finances, economics, ideas discussed in field of chemistry, physics and to make wise decisions, one must know the transition from arithmetic (a part of mathematics dealing with number manipulation) to algebra (symbolic language of mathematics).
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