Exponents vs. Power Functions


#1

At a high level, a power function like x^2, x^3, x^4, tracks a set number of interactions. That is, xx is two items interacting (x with x), xx*x is 3 items interacting, and so on. Even as the size of “x” changes (the part gets bigger, like the progression 2^2, 3^2, 4^2), the number of interactions always stays the same.

Exponential functions are different: here, the number of interactions is effectively infinite: interest earns interest, which earns interest, and so on (modeled by e^x). The “child” and “grand-child” interactions are on smaller and smaller amounts, so the accumulation reaches a limit, but the potential is there to overpower and fixed amount.

This starts to get really, really large over time because we essentially have an infinite number of components to track. For small sizes, these taper off and reach a limit, but as the “time” gets larger (e^1, e^2, e^3) we have more chances for all these interactions to build up. Given enough time, e^x will always grow larger than a fixed number of interactions (x^a, where a is a constant). Even x^100 will get dominated with enough time.

Another way to see it: e^x is essentially all exponents combined into one:

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + … $$

A specific term like x^2 can dominate in the short term, but eventually the division by 2 factorial doesn’t matter.


insights:

regular polynomials are fixed-complexity, exponents are infinite-complexity.

From other points of view, you can see e^x as similar to sine/cosine in that infinite cycle (Euler’s Formula describes the connection more deeply).

Don’t see equations as a static item. It’s a bundle of interactions, or limitless interactions (which are decaying) and we go from there.