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Representations of Complex Numbers and its Properties (1)

  OVERVIEW There has mainly three representation types of complex number in mathematics, which are a very popular Algebraic Representation , Geometric Representation , Polar Representation . We  discuss Algebraic and Geometric  Representations in this blog . Polar Representation will be discussed in next blog . ALGEBRIC   REPRESENTATION When a complex number is written as a duplet "(a,b)" form, is called Algebraic Representation .  We have discussed this extensively in previous blogs of complex numbers . GEOMETRIC   REPRESENTATION When a complex number is written as real and imaginary part called Geometric Representation. In geometric representation (a,b) is written as (a+ i b)   or (a+b i )  Power of i :-  ( i ) 4n =1 ;           ( i ) 4n+1 = i ;       ( i ) 4n+2   = -1 ;      ( i ) 4n+3 = - i  ( i ) -1 = 1/ i = i /-1 {doing multiply and divide by i }...
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Properties of Complex Numbers

  OVERVIEW A complex number satisfy 11 properties. 5 properties are concerning addition, 5 are concerning multiplication and 1 are mixed of both. {x+ i y is written as (x,y)} 1)  PROPERTIES   USING   ADDITION I)Binary Composition:-  If  Z 1  & Z 2  are complex number,   Z 1  +  Z 2     be a complex numbers. Example:- If (2,3) and (4,5) are complex numbers.  (2+4,3+5)=(6,8) is a complex numbers.(proved) II) Commutative Law:-   If  Z 1  & Z 2  are complex number,  Z 1  +  Z 2  =   Z 2   +  Z 1   Example:- If (2,3) and (4,5) are complex numbers. (2+4,3+5)=(6,8)---LHS (4+2,5+3)=(6,8)---RHS LHS=RHS (proved) III) Associative Law:-   If  Z 1  , Z 2  & Z 3  are complex number, ( Z 1  +  Z 2 )+   Z 3 =    Z 1  +(  Z 2 +   Z 3 ) Example:- If (2,3),(4,5) and (6,7) are complex numbers. {(2,...
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Basic of Complex Numbers

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  INTRODUCTION We know square of any real number is never negative. But in Mathematics and Physics we get many x 2 +c=0 (c>0) terms. In real number system we haven't any answer on these term. So, solving this problem, we introduced a new system called complex number system, where we solve these equations by using a new term i  (√(-1)). DEFINATION    OF   COMPLEX   NUMBERS A order pair of real number (a,b) which satisfy the following condition  < 1 >     (a,b)=(c,d) if and only if a=b and c=d < 2 >      (a,b)±(c,d)={(a±c),(b±d)}             < 3 >      (a,b) . (c,d)={(ac-bd),(ad+bc)}        called complex numbers. [(c,d) is a order pair of real number] NOTATION It is written by z=a+b i (a,b∈ R & i is imaginary unit) . Therefore a complex part is multiplication of imaginary unit and real part. TYPES   OF   COMP...
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