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

Roots of complex numbers

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

• A complex number has two square roots, and , that are also complex

  • To find them, let where

  • and form simultaneous equations from the relationship

    • by expanding

    • and equating the real and imaginary parts with

• E.g. find the square roots of

  • Let be one of the square roots of

  • This means

  • Expand the left-hand side

    • 

    • so

  • Equate the real parts on both sides

    • 

  • Equate the imaginary parts on both sides

    • 

  • Solve these two equations simultaneously

    • e.g. make the subject of to get

    • Substitute into to get

    • Rearrange to which gives or (but )

    • so which gives

  • The two square roots are and

How do I use de Moivre’s theorem to find the n^th roots of a complex number?

• De Moivre's theorem to find the ^th roots of a complex number, , is

  • 

    • Letting gives each ^th root

• In exponential (Euler's) form, this is

  • 

    • where

• e.g. find the fourth roots of

  • 

    • gives

    • gives

    • gives

    • gives

How are complex roots related to polygons?

• If you plot the complex roots of of a complex number on an Argand diagram

  • they form a regular -sided polygon

• e.g. in the example above

  • , , and

    • form a square

    • whose vertices lies on a circle of radius