In my Analysis class today I defined the trigonometric functions and by means of the complex exponential. As usual I noted that for real we have , i.e. lies on the complex unit circle. Then I drew the following picture:
This was meant to show that the real part and the imaginary part of are what is known as and , respectively.
After the lecture a student came to me and noted that we could have started with and note that and could do the same thing. The question is: Does this work out? My initial reaction was: Yeah, that works, but you’ll get a different …
But then I wondered, if this would lead to something useful. At least for the logarithm one does a similar thing. We define for and real as , notes that this gives a bijection between and and defines the inverse function as
So, nothing stops us from defining
Many identities are still valid, e.g.
For the derivative one has to be a bit more careful as it holds
Coming back to “you’ll get a different ”: In the next lecture I am going to define by saying that is the smallest positive root of the functions . Naturally this leads to a definition of “ in base ” as follows:
Definition 1 is the smallest positive root of .
How is this related to the area of the unit circle (which is another definition for )?
The usual analysis proof goes by calculating the area of a quarter the unit circle by integral .
Doing this in base goes by substituting :
Thus, the area of the unit circle is now …
Oh, and by the way, you’ll get the nice identity
(and hence, the area of the unit circle is indeed )…