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 can be used to find powers of complex numbers

• It states that for,  

  • Where

    • z ≠ 0

    • r is the modulus, |z|, r ∈ ℝ⁺

    • θ  is the argument, arg z, θ ∈ ℝ

    • n ∈ ℝ

• In Euler’s form this is simply:

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• In words de Moivre’s theorem tells us to raise the modulus by the power of n and multiply the argument by n

• In the formula booklet de Moivre’s theorem is given in both polar and Euler’s form:

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How do I use de Moivre’s Theorem to raise a complex number to a power?

• If a complex number is in Cartesian form you will need to convert it to either modulus-argument (polar) form or exponential (Euler’s) form first

  • This allows de Moivre’s theorem to be used on the complex number

• You may need to convert it back to Cartesian form afterwards

• If a complex number is in the form  then you will need to rewrite it as before applying de Moivre’s theorem

• A useful case of de Moivre’s theorem allows us to easily find the reciprocal of a complex number:

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  • Using the trig identities cos(-θ) = cos(θ) and sin(-θ) = - sin(θ) gives

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• In general

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