E (x) has the abstract property we were after, but it remains to be seen what sort of values it produces, if any. Being an infinite sum, it may generate only infinity; experimentation is in order.

Let's try a value of x = 0:

More telling is E (1):

Summing the above:

That value should be familiar. Carried on indefinitely, it is Euler's constant e.

E (0) = 1. E (1) = e. A little calculation shows E (2) = e²: clearly the factorial series E(x) is generating the powers of e.

In fact, E (x) is the formal definition of e^x, and e is defined by the series for E(1). Like π, e is transcendental; the only formulas for it are infinite.

And like i, e is not so much a number as a process. 2.71828... is just one of the multitude of possible numbers that emerge from the formula for e^x, a source far more structured and deep than a simple number. And like i, we can anticipate unforeseen and remarkable behavior from e when we investigate its powers.

E (x) was built to serve as a means of representing complex numbers; an ability which, if it exists, remains to be seen. But not for long: the elements are all in place to finally decipher Euler's formula, and the three realms of numbers we've seen converging are about to come together in a striking way.