Question

find the value of the Complex number:i²⁰⁴⁹ is

Answer 1

SOLUTION

TO DETERMINE

The value of the complex number

$\sf{ {i}^{2049} }$

CONCEPT TO BE IMPLEMENTED

Complex Number

A complex number z = a + ib is defined as an ordered pair of Real numbers ( a, b) that satisfies the following conditions :

(i) Condition for equality :

(a, b) = (c, d) if and only if a = c, b = d

(ii) Definition of addition :

(a, b) + (c, d) = (a+c, b+ d)

(iii) Definition of multiplication :

(a, b). (c, d) = (ac-bd , ad+bc )

Of the ordered pair (a, b) the first component a is called Real part of z and the second component b is called Imaginary part of z

EVALUATION

Here the given complex number is

$\sf{ {i}^{2049} }$

We simplified the number as below

$\sf{ {i}^{2049} }$

$\sf{ = {i}^{2048 + 1} }$

$\sf{ = {i}^{2048} \times i }$

$\sf{ = { ({i}^{2} )}^{1024} \times i }$

$\sf{ = { ( - 1 )}^{1024} \times i }$

$\sf{ = 1 \times i }$

$\sf{ = i }$

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