Roots of Complex Numbers (DP IB Analysis & Approaches (AA)): Revision Note

Roots of Complex Numbers

How do I find the square root of a complex number?

• The square roots of a complex number will themselves be complex:

  • i.e. if  then 

• We can then square () and equate it to the original complex number (), as they both describe :

  • 

• Then expand and simplify:

  • 

  • 

• As both sides are equal we are able to equate real and imaginary parts:

  • Equating the real components:   (1)

  • Equating the imaginary components:   (2)

• These equations can then be solved simultaneously to find the real and imaginary components of the square root

  • In general, we can rearrange (2) to make  and then substitute into (1)

  • This will lead to a quartic equation in terms of d; which can be solved by making a substitution to turn it into a quadratic

• The values of can then be used to find the corresponding values of , so we now have both components of both square roots ()

• Note that one root will be the negative of the other root

  • g. and 

How do I use de Moivre’s Theorem to find roots of a complex number?

• De Moivre’s Theorem states that a complex number in modulus-argument form can be raised to the power of n by

  • Raising the modulus to the power of n and multiplying the argument by n

• When in modulus-argument (polar) form de Moivre’s Theorem can then be used to find the roots of a complex number by

  • Taking the nth root of the modulus and dividing the argument by n

  • If then 

    • 

    • Recall that adding 2π to the argument of a complex number does not change the complex number

    • Therefore we must consider how different arguments will give the same result

  • This can be rewritten as 

• This can be written in exponential (Euler’s) form as 

  • For ,  

• The nth root of complex number will have n roots with the properties:

  • The modulus is  for all roots

  • There will be n different arguments spaced at equal intervals on the unit circle

  • This creates some geometrically beautiful results:

    • The five roots of a complex number raised to the power 5 will create a regular pentagon on an Argand diagram

    • The eight roots of a complex number raised to the power 8 will create a regular octagon on an Argand diagram

    • The n roots of a complex number raised to the power n will create a regular n-sided polygon on an Argand diagram

• Sometimes you may need to use your GDC to find the roots of a complex number

  • Using your GDC’s store function will help when entering complicated modulus and arguments

  • Make sure you choose the correct form to enter your complex number in

  • Your GDC should be able to give you the answer in your preferred form