CIE A Level Maths: Pure 3复习笔记8.1.3 Square Roots of a Complex Number

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:
    • Equating the imaginary components:
  • 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 d can then be used to find the corresponding values of c, so we now have both components of both square roots ()
  • Note that one root will be the negative of the other root
    • g.  and 


Worked Example



Exam Tip

  • Most calculators used at A-Level can handle complex numbers.
  • Once you have found the square roots algebraically; use your calculator to square them and make sure you get the number you were originally trying to square-root!