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.

or

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 )…

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