For this section, we will venture a step further by proving a more specific result as follows

By proving the above, it gives us more tools when dealing with complex numbers especially the case when we will look at de Moivre’s theorem in the next lesson.

For the first result , we will use the method of induction. Now, when reading many mathematics textbooks on the proof of such results by induction, they usually summarize the steps since the focus is not on induction but complex numbers. Nonetheless, I will do my best to provide a succinct a proof as possible. It will also be good revision for those studying mathematical induction.

We have the proposition
We first check on the proposition when n=1. When n=1

And so the proposition is true when n=1, no surprise there.

We now assume the proposition is true for some , to show that the proposition is true for n+1, which is the same as showing

Imagine this, if you will, as our goal to show. It is the equation we want to achieve. In most cases, we will adopt the practical approach by starting on the LHS of the equation, do some manipulation to show the RHS, or vice versa. So, by starting on the LHS, we have

I know what you are thinking. You are probably wondering how we can assume that the proposition is true where all the while that is what we are trying to prove. This is where things get a little complicated if you are not too sure of the reasoning.

Here, what I am trying to prove is NOT that the proposition . What I AM trying to prove is that should  be true, and I emphasize should, it would mean that the proposition for n+1 is true, or  is true.

But earlier we know that  is true for n=1, and by the simple fact that  true implies that  true, it follows that  is true for n=1+1=2 and for n=2+1=3, or more simply n=1,2,3,…,

Moving swiftly along to prove the other reason, which will be much easier.
We are to show that .

I argue in this manner.

See that wasn’t so hard. We will now use these results to move on to something much harder, that being de Moivre’s theorem.