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

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Jamie W

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Jamie W

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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 z squared equals a plus b straight i then z equals c plus d straight i
  • We can then square (c plus d straight i) and equate it to the original complex number (a plus b straight i), as they both describe z squared:
    • a plus b straight i equals open parentheses c plus d straight i close parentheses squared
  • Then expand and simplify:
    • a plus b straight i equals c squared plus 2 c d straight i plus d squared straight i squared
    • a plus b straight i equals c squared plus 2 c d straight i minus d squared
  • As both sides are equal we are able to equate real and imaginary parts:
    • Equating the real components: a equals c squared minus d squared  (1)
    • Equating the imaginary components: b equals 2 c d  (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 fraction numerator b over denominator 2 d end fraction equals c 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 (c plus d straight i)
  • Note that one root will be the negative of the other root
    • i.e.  c plus d straight i  and  negative c minus d straight i

Worked example

8-1-3-square-root-of-complex-number-part-1

8-1-3-square-root-of-complex-number-part-2

Examiner 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!

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Jamie W

Author: Jamie W

Expertise: Maths

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.