De Moivre's Theorem (DP IB Analysis & Approaches (AA)): Revision Note

De Moivre's theorem

What is de Moivre’s theorem?

• De Moivre’s theorem states that for 

  • 

  • where

    • 

    • 

    • 

    • 

• In exponential (Euler’s) form this is simply

  • 

    • It confirms that you can use 'index laws' with complex exponentials

How do I find positive powers of a complex number?

• To raise a complex number to a power,

  • write in modulus-argument (polar) form

  • apply de Moivre's theorem in the form

  • simplify the real and imaginary parts

• e.g. if , find

  • 

    • using

  • which simplifies to

    • giving

How do I find negative powers of a complex number?

• works for negative powers

  • e.g.

    • recall that and

    • giving