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+ib) or (a+bi) 

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 } = -i         

[When n belongs to Z]

 Addition:- The addition of geometric representation is  same as Algebraic representation.

(a+bi) + (c+di) = (a+c) + (b+d)i 

Example :- (2+3i)+(6+7i) = (2+6) + (3+7)i = 8+10i 

Subtraction :- The subtraction is same as adding additive inverse of complex number.

                                           (a+bi) - (c+di) =  (a+bi) + (-c-di)  =  (a-c) + (b-d)i 

Example :- (4+5i) - (3+2i) = (4-3) + (5-2)i = 1-2i 

Multiplication :- The multiplication of geometric representation is  same as Algebraic representation.

(a+bi) * (c+di) = ac+adi+bci+bd(i)²  = (ac-bd)+(bc+ad)i

Example :- (9+2i) * (10+5i) = (90-10) + (45+20)i = 80 +65i

  

Division :- The division is same as multiply multiplicative inverse of complex number.

(a+bi)/(c+di) = (a+bi)*(p+qi) =  (ap-bq)+(bp+aq)i

[where p=(c/(c²+d²)) q=(-d/(c²+d²)) ]

Example :- (1+2i)/(3+4i) = (1*3/25) - (-2*4/25) + (-1*4/25)i + (2*3/25)i = (11/25) + (2/25) i

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