Complex Conjugation & Division (CIE A Level Maths: Pure 3)

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Jamie W

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Complex Conjugation & Division

When dividing complex numbers, we can use the complex conjugate to make the denominator a real number, which makes carrying out the division much easier.

What is a complex conjugate?

  • For a given complex number z equals a plus b straight i, the complex conjugate of z is denoted as z to the power of asterisk times, where z to the power of asterisk times equals a minus b straight i
  • If z equals a minus b straight i then z to the power of asterisk times equals a plus b straight i
  • You will find that:
    • z plus z to the power of asterisk times is always real because left parenthesis a plus b straight i right parenthesis plus left parenthesis a minus b straight i right parenthesis equals 2 a
      • For example: left parenthesis 6 plus 5 straight i right parenthesis space plus space left parenthesis 6 minus 5 straight i right parenthesis space equals space 6 plus 6 plus 5 straight i minus 5 straight i space equals space 12
    • z minus z to the power of asterisk times is always imaginary because open parentheses a plus b straight i close parentheses minus left parenthesis a minus b straight i right parenthesis equals 2 b straight i
      • For example: left parenthesis 6 plus 5 straight i right parenthesis space minus space left parenthesis 6 minus 5 straight i right parenthesis space equals space 6 minus 6 plus 5 straight i minus left parenthesis negative 5 straight i right parenthesis space equals space 10 straight i
    • z cross times z to the power of asterisk times is always real because open parentheses a plus b straight i close parentheses open parentheses a minus b straight i close parentheses equals a squared plus a b straight i minus a b straight i minus b squared straight i squared equals a squared plus b squared (as straight i squared equals negative 1)
      • For example: left parenthesis 6 plus 5 straight i right parenthesis left parenthesis 6 minus 5 straight i right parenthesis space equals space 36 space plus 30 straight i space – space 30 straight i space minus 25 straight i squared space equals space 36 space – space 25 left parenthesis negative 1 right parenthesis space equals space 61

 

How do I divide complex numbers?

  • When we divide complex numbers, we can express the calculation in the form of a fraction, and then start by multiplying the top and bottom by the conjugate of the denominator:
    • fraction numerator a plus b straight i over denominator c plus d straight i end fraction equals blank fraction numerator a plus b straight i over denominator c plus d straight i end fraction blank cross times blank fraction numerator c minus d straight i over denominator c minus d straight i end fraction
  • This ensures we are multiplying by 1; so not affecting the overall value
  • This gives us a real number as the denominator because we have a complex number multiplied by its conjugate (z z to the power of asterisk times)
  • This process is very similar to “rationalising the denominator” with surds which you may have studied at GCSE

Worked example

8-1-2-dividing-complex-numbers

Examiner Tip

  • We can speed up the process for finding z z asterisk timesby using the basic pattern of open parentheses x plus a close parentheses open parentheses x minus a close parentheses equals x squared minus a squared
  • We can apply this to complex numbers: open parentheses a plus b straight i close parentheses open parentheses a minus b straight i close parentheses equals a squared minus b squared straight i squared equals a squared plus b squared
    (using the fact that straight i squared equals negative 1)
  • So 3 plus 4 straight i multiplied by its conjugate would be 3 squared plus 4 squared equals 25

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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.