So you didn’t answer my question. My question was that from the equation i^i=. 2078… we see that e=2.718… even in our case, as well as we see it from 2^i=e^.693i .
But I found an answer to my question at https://www.mathsisfun.com/algebra/eulers-formula.html. My question was, what does e mean in Euler’s formula. The answer is, that since e^x = 1+x^1/1!+x^2/2!+x^3/3!…, and when we plug in x=xi it turns out that e^xi= cos(x)+isin(x), we make it for a rule, even though it doesn’t really fit into the regular definition of exponents, which would be growing to e for i units of time. So we never reached 2.718…, but we used the same formula that produces it.
I suggest you should add this explanation to your post, because it is a must in order to understand euler’s formula.

e is better seen as an engine of continuous change, which happens to be 2.718… (e.g., 2.718 is the consequence of e’s definition, not the starting point).

Check out https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ for a detailed walkthrough of i^i. In essence, we take our “change engine” and feed it two types of imaginary fuel, which ultimately results in a real number.

Thanks for answering.
So you’re saying that e in this case does not mean 2.718…? So why is i^i .2078…?

Hi George,

I see e as the engine of continuous growth. We can use that engine with real numbers (e^2, e^3), or imaginary numbers (e^i, e^{pi i}, etc.).

In technical terms, e is defined as:

e = lim n-> inf [1 + 1/n]^n

The number 2.71828… is what that process returns (to the best our ability to measure it), but under the hood e is shorthand for “apply 100% continuous growth for 1 unit of time”. If we change the assumptions (300% growth for 2 units of time) we can write e^{300% * 2} and run that calculation.

“Continuous change” may be a better phrase, because you only grow when the new input moves in the same direction you were already going. i constantly pushes you in a perpendicular direction, so you end up orbiting (changing your direction) vs. “growing” (increasing your magnitude). Hope that helps.