Complex Conjugation & Division (Cambridge (CIE) A Level Maths): 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:

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