Square Roots of a Complex Number - Advanced (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 - Advanced

Previously we looked at how to find the square roots of a complex number in Cartesian form (a+bi). We can also find square roots using polar (r space open parentheses cos space theta plus straight i blank sin space theta close parentheses) and exponential form (r e to the power of straight i theta end exponent).

How do I find a square root of a complex number in polar/exponential form?

  • Let w equals r e to the power of straight i theta end exponent be a square root of z equals n e to the power of straight i straight alpha end exponent
    • w cross times w equals z
    • open parentheses r e to the power of straight i theta end exponent close parentheses squared equals n e to the power of straight i straight alpha end exponent  
  • Applying rules of indices: 
    • r squared e to the power of 2 i theta end exponent space equals space n e to the power of straight i alpha end exponent
  • Comparing the coefficients of e (moduli) and powers of e (arguments) we can state:
    • n space equals space r squared
      • r space equals space square root of n
    • alpha space equals space 2 theta
      • theta space equals space alpha over 2
  • A square root of z equals n e to the power of straight i straight alpha end exponent is w equals square root of n e to the power of straight alpha over 2 straight i end exponent
      • Square root the modulus
      • Halve the argument

How do I find the second square root?

  • The second square root is the first one multiplied by -1
    • w equals square root of n e to the power of straight alpha over 2 straight i end exponent and negative w equals negative square root of n e to the power of straight alpha over 2 straight i end exponent
  • We can write the second one in polar or exponential form too
  • Adding 2π to the argument of a complex number still gives the same complex number
    • So we could also say that n e to the power of straight i left parenthesis alpha plus 2 pi right parenthesis end exponent equals r squared e to the power of 2 straight i theta end exponent
    • Therefore alpha plus 2 pi equals 2 theta is another possibility
      • theta equals alpha over 2 plus pi
  • So the two square roots of (n e to the power of straight i alpha end exponent) are:
    • z subscript 1 equals square root of n blank e to the power of alpha over 2 straight i end exponent
    • z subscript 2 equals square root of n blank e to the power of open parentheses alpha over 2 plus straight pi close parentheses straight i end exponent
  • You should notice that the two square roots are π radians apart from each other
    • This is always true when finding square roots
  • And if you were to write them in cartesian form they would be negatives of one another
    • E.g. a+bi and -a-bi
    • This is also always true when finding square roots
  • This approach can be extended to find higher order roots (e.g. cube roots) by knowing that the nth roots will be fraction numerator 2 straight pi over denominator straight n end fraction radians apart from each other, however this is beyond the specification of this course

Examiner Tip

  • The square roots will be negatives of each other when written in cartesian form, and the two square roots will be π radians apart when written in polar form. These two facts can help you find the roots quicker and/or check your answers.
  • If your calculator is able to work with complex numbers, you should also square the square-roots you found to check that you get the original number.

Worked example

8-3-3-example-square-roots-of-complex-number-advanced-part-1

8-3-3-example-square-roots-of-complex-number-advanced-part-2

Examiner Tip

  • The square roots will be negatives of each other when written in cartesian form, and the two square roots will be π radians apart when written in polar form. These two facts can help you find the roots quicker and/or check your answers.
  • If your calculator is able to work with complex numbers, you should also square the square-roots you found to check that you get the original number.

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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.