8.1.2 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