Square Roots of a Complex Number (Cambridge (CIE) A Level Maths): Revision Note

Square roots of a complex number

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 (see 1.1.5 Further Solving Quadratic Equations (Hidden Quadratics))

• 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

  • i.e.    and