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Exponent and Root of Complex Numbers

    what you'll learn...


Exponent of Complex Numbers

 »  Play with the forms of the complex number

    →  component form a+ib

    →  polar form r(cosθ+isinθ)

    →  exponent form reiθ

 »  Convert to the standard form of complex numbers a+ib

    →  eg: (a+ib)c+id: convert (a+ib) to polar form reiθ

    →  eg: rid : convert r to elnr

Power of i

 »  To calculate in

Remember i4=1 and i2=-1. Quickly derive the following using this. No need to memorize

    →  express n in the form n=4p+2q+r, where p,q,r natural numbers.

    →  in=(-1)q×ir


Given z1=a1+ib1 and z2=a2+ib2, Can the exponent z1z2 be written in a+ib form?
Yes. By converting to polar form `z_1 = r e^(i theta)

Given z1=a1+ib1 and z2=a2+ib2, exponent
  =r1a_2   ×r1ib2   ×eiθ1a2   ×eiθ1ib2

  ×eib2lnr1(   ×eiθ1a2   ×e-θ1b2

  =r1a_2e-θ1b2   ×ei(b2lnr1+θ1a2)

The result is in the polar form and can be converted to coordinate form.

Given z1=a1+ib1 and z2=a2+ib2, Can the root z1z2 be computed in a+ib form?
Yes. By considering root as power of (1z2)

Given z1=a1+ib1 and z2=a2+ib2, root
By following the rules of exponent of a complex number, the root can be solved.

What is (1+i)32?

The answer is '234(cos(3π8)+isin(3π8))'


Exponent and Roots of Complex number
 •  For z1z2, convert z1 to polar form reiθ
 •  For aib, convert a to elna form
 •  For z11z2, convert 1z2 to a complex number in numerator z2¯|z2|2

To find exponent and root of complex numbers, the rules of numerical expression is used to arrive at the coordinate form a+ib.

power of i

The value of i is '-1'

The value of i2 is '-1'

It is defined that i=-1. Substitute that in the following

The value of i3 is '-i'


The value of i4 is '1'


The value of i0 is '1'


The value of in, if n=4p+2q+r is '(-1)q×ir'



What is the value of i27?
The answer is '-i'

What is i20?
The answer is '1'

What is i7?
The answer is '-i'


Power of i: To calculate in, express n in the form n=4p+2q+r, where p,q,r natural numbers. Then

To calculate in, use i4=1 and i2=-1.


The outline of material to learn "complex numbers" is as follows.

Note : Click here for detailed overview of Complex-Numbers

  →   Complex Numbers in Number System

  →   Representation of Complex Number (incomplete)

  →   Euler's Formula

  →   Generic Form of Complex Numbers

  →   Argand Plane & Polar form

  →   Complex Number Arithmetic Applications

  →   Understanding Complex Artithmetics

  →   Addition & Subtraction

  →   Multiplication, Conjugate, & Division

  →   Exponents & Roots

  →   Properties of Addition

  →   Properties of Multiplication

  →   Properties of Conjugate

  →   Algebraic Identities