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BASIC PRINCIPLES OF ALGEBRA

 
       Mathematics is the science of numbers, quantity and shape and arrangement. Arithmatic is part of Mathematics in which we add, multiply etc. numbers whereas Algebra is a part of Mathemaics in which numbers are represented by symbols and letters.  
       Algebra provides a tool for solving problems.
       As we use symbols +, ─ , x and ÷ for addition, subtraction, multiplication and division respectively in arithmatics, we also use these symbols in Algebra for the same purpose. But it is better to use * for multiplication in algebra in place of x because letter x is frequently used in algebra for representing unknown quantities.
DIFFERENCE IN BETWEEN ARITHMATICAL & ALGEBRIC TERMS
       In arithmatics whenever two or more digits occur together without any sign in betweenany two them, value is always find out by adding place values of digits. For example value of number 243 is two hundred + forty + three i.e. Two Hundred Forty Three. But in algebra, it is not so. Whenever two or more letters appear in a term without any sign in between any two of them, it is not sum of letters it is multiplication of values of letters. For example, term xyz, does not mean x + y + z. Even if value of x=1, value of y=2 and value of z=3, term xyz does not mean 123. Its value will be 1* 2* 3 (* sign has been used for multiplication) giving resultant value=6.
        In the above example, even if x, y and z are digits representing hundredth, tenth and unit place, number cannot be written as xyz or 123. Then number will be 100*1 + 10*2 +1*3 =123 (One Hundred Twenty Three).
        In Algebra, where there is no sign in between any two letters, multiplication sign (*) is assumed to exist there.
         Letters x, y z or α, β, γ (alpha, beta, gamma ) or θ, Φ, (theta, phi)etc. are used to represent unknown quantities. By multiplying one unknown quantitity by another unknown quantity, we get another unknown quantitity different from quantities which have been multiplied. Thus xy is different from x and y both. Where one unknown quantity cannot be added to another unknown quantity, their sum is written as first unknown quantity + second unknown quantity. For example, sum of xy and x is written as xy + x.
UNKNOWN QUANTITIES USED WITH A NUMBER (NUMBER IS A KNOWN QUANTITY):
COEFFICIENT: When a number is written with an unknown quantity without any sign in between them, principle is the same, a symbol of multiplication is understood in between them. For example 2x is 2*x or two times of x. Here 2 is known quantity and x is unknown quantity. Number used with unknown quantity is called coefficient. We know two times of any thing is obtained by adding that thing to the same thing. For example two times of 4 is 4 + 4. In the algebra 2x is two times of x and can be written as x + x. Similarlry, thrice of any thing is obtained by adding that thing three times. For example 3*4 which is thrice of 4 can be onbtained by adding 4 three times that is 4 + 4 + 4. In algebra too 3x can be obtained by adding x three times that is x + x + x.
VARIABLE: The letter that stands for an unknown quantity or number is called a variable. Here onwards, we will refer to unknown quantity by a variable.
POWER OR INDEX: When we multiply a variable by itself several times, we write the multiplication result by raising the power to the variable by the number it has been multiplied. For example x*x=x2. This is also called INDEX. In term x y4, y has got power of four. It is read as y to the power four. y 4 is equal to y*y*y*y
       Let us consider two quantities, 2x & 5x. These both quantities contain only one unknown quantity x. Multipliers of x in both quantities are constant numbers which are known quantities. Now let us consider another two quantities 2x & 5xy. In this case, first quantity has one unknown quantity x whereas second quantity has a different unknown quantity xy which is combination of two unknown quantities x & y.
TERM: In Algebra, a term is a number or variable or product of several variales with or without a coeficient. Following are separate terms:
(i) x
(ii) 3x/2
(iii) 2xy
(iv) 2
Basically a term does not contain sign of + or -. The term may or may not have a variable.
EQUIVALENT OR EXACTLY SIMILAR TERMS: Two terms may be said to be equivalent or exactly similar, if they contain same coefficiets and variables and powers, of each varial individually, in both terms are the same. In algebra, equivalent terms may be added or subtracted. On addition, they give result which is twice of the single term and on subtraction we get 0.
SIMILAR TERMS: If two terms differ only in their coefficients, then both terms are said to be similar. Powers of every variable in each term are exactly the same. In algebra, similar terms may be added or subtracted to give a single term.
EXPRESSION: An expression may consit of a signle term as well as many terms. In the expression which contains more than one term, terms are joined by a + or a - sign. Following are the examples of expressions
(i) 3x +2y;
(ii) x -y;
(iii) x + xy +3;
(iv) 3x/2 + 5y/3;
(vii) 3x
      An expression may or may not have a number as its one of the terms.
EQUATION: An equation consists of two expressions separated by = sign.
(i) 3x + 4y= 2
(ii) ax + by +c =0
(iii) x2 + xy + y2=90
(iv) x4 + y3=90xy
(v) x4 =90xy     This equation contains only terms.
(vi) x4 + y3=40     To the left of = sign there is an expression while to the right of = sign there is a term.
Expression to the left of = sign is called left hand side (in brief L.H.S.) of the equation and expression to the right of the = sign is called right hand side (in brief R.H.S.) of the equation.
PROPERTIES OF EQUATION:

1. Addition of same thing to both sides, subtraction of same thing from both sides, multiplication by same number or term to both sides and division of both sides by same number or term, does not affect the equation. Equation remains unchanged. For example, let there be an equation:

3x + 4y =7
Then equations,
(i) 3x + 4y + z=7 +z; is the same as 3x + 4y =7
(ii) 3x + 4y -4 =7-4; is the same as 3x + 4y =7
(iii) (3x + 4y)* z=7*z; is the same as 3x + 4y =7
(iv) (3x + 4y) /5=7/5 ; is the same as 3x + 4y =7
2. Any term from either side can be transferred to other side but with reversing its + sign to - sign and - sign to + sign. Equation 3x + 4y =7 can be re-written as 3x= 7 - 4y. Here term 4y has been transferred from left had side to right hand side. We notice that term 4y has sign + when it was on left hand side and now it has - sign after its transfer. A term with - sign is changed to +, a multiplication sign to division and a division sign to multiplication.
RULES OF POWERS OR INDICES:
1. When a variable having power m is multiplied by the same variale having power n then powers are added. For example:
(i) x4 * x8 gives x4+8 that is x12.
(ii) xm * xn gives xm+n
2. When a variable having power m is divided by the same variale having power n then power in the numerator is reduced by the power of the denominator. For example:
(i) x4 / x8 gives x4-8 that is x-4.
(ii) xm / xn gives xm-n
3. When power n is raised to a varial already having power m, then resultant power is the product of powers. For example (xm)n gives xm*n which is xmn. Similarly, (x4)2 gives x4*2 which is x8
INVERSE: 1 divided by anything is inverse of that thing. For example; 1 divided by x is inverse of x which is 1 /x; It is written with negative power of that thing. Thus inverse of x is x-1. Hence x-1=1/x.
Power of 1: If power of 1 is raised to any number or variable, it gives the same number. For example 41 is 4; x1 is x.
Power of 0: If power of 0 is raised to any number or variable, it gives 1 value. For example 40 is 1; x0 is 1.
Rule of power of 0 is true if base is 10.
 
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