Square Roots of a Complex Number (CIE A Level Maths: Pure 3)

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 :
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• Then expand and simplify:
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• 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