Cartesian Form of Complex Numbers
Complex numbers are a set of numbers which contain both a real part and an imaginary part. The set of complex numbers is denoted as .
What is an imaginary number?
- Up until now, when we have encountered an equation such as we would have stated that there are “no real solutions” as the solutions are which are not real numbers
- To solve this issue, mathematicians have defined one of the square roots of negative one as ; an imaginary number
- We can use the rules for manipulating surds to manipulate imaginary numbers.
- We can do this by rewriting surds to be a multiple of using the fact that
What is a complex number?
- Complex numbers have both a real part and an imaginary part
- For example:
- The real part is 3 and the imaginary part is 4
- Note that the imaginary part does not include the ''
- Complex numbers are often denoted by and we can refer to the real and imaginary parts respectively using and
- In general:
- This is the Cartesian form of z
- It is important to note that two complex numbers are equal if, and only if, both the real and imaginary parts are identical.
- For example, and are not equal
Examiner Tip
- Be careful in your notation of complex and imaginary numbers.
- For example:
could also be written as , but if you wrote this could easily be confused with .
Worked example