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Complex Conjugation & Division (CIE A Level Maths: Pure 3)
Revision Note
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 , the complex conjugate of is denoted as , where
- If then
- You will find that:
- is always real because
- For example:
- is always imaginary because
- For example:
- is always real because (as )
- For example:
- is always real because
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:
- 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 ()
- This process is very similar to “rationalising the denominator” with surds which you may have studied at GCSE
Worked example
Examiner Tip
- We can speed up the process for finding by using the basic pattern of
- We can apply this to complex numbers:
(using the fact that ) - So multiplied by its conjugate would be
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