Take note that if z has a real number, i.e. to say that b=0, then we also have , since the imaginary part is 0 and so the negative of the imaginary part is also 0.

In interest of time, we shall show the following algebraic properties of the complex conjugate without proof. They are,

of which we will rarely use the last result.

NOTE: In the video, I did mention that the conjugate is like an ‘inverse’. This is certainly not true. Because  does not give us the identity, assuming an identity of the complex plane exists. What I meant to say is that there is a conjugate for every complex number.

Moving on, we define the magnitude, or sometimes known as the absolute value of the complex number  by

Just as the name suggest, the magnitude gives a scalar value. We shall later see the geometric interpretation of this quantity. However at this point, it is vital that we recognize that in most cases, . This is simply shown that when we take the square of a complex number, we must do the standard multiplying procedures with complex numbers.  means that we find the magnitude first and then square it giving us a real value. The exception to the case is when z has no imaginary part.

We again have the following result.

To conclude the section, we shall talk about a technique we use in the complex plane which is similar to the technique in real-numbers algebra called rationalizing the denominator.

To recall, to rationalizing a denominator such as , we multiply by the number 1 expressed as . This way we can eliminate the root 2. In complex numbers, rationalizing the denominator involves removing the imaginary part in the denominator using a parallel method. We generalize the case by using any complex number z.

Which gives us the fairly useful result.