A Numerical Pattern?

Part 1

Let's imagine a world where we only ever knew about natural numbers(1,2,3...). We start doing some cool things with them like counting:

count(1) = 2
count(count(1)) = 3
count(count(count(1))) = 4

Lots of fun. But slow. Too slow.

So one day we came up with this idea for a repeated counting function to save us from having to count a whole bunch.

We call it Addition.

We then start adding stuff and it's cool, but then one day we want to undo adding.

We call it Subtraction.

But then some smarty pants comes along and comes up with an expression that we don't have a number for! What is 1 - 2?

So we invent negative numbers.

Part 2

One day, someone is doing a bunch of adding.

2 + 2 = 4
2 + 2 + 2 = 6
2 + 2 + 2 + 2 = 8

Lots of fun. But slow. Too slow.

So one day we came up with this idea for repeated addition function to save us from having to add a whole bunch.

We call it Multiplication.

We then start multiplying stuff and it's cool, but then one day we want to undo multiplying.

We call it Division.

But then some smarty pants comes along and comes up with an expression that we don't have a number for! What is 1 / 2?

So we invent fractional numbers.

Part 3

One day, someone is doing a bunch of multiplying.

2 * 2 = 4
2 * 2 * 2 = 8
2 * 2 * 2 * 2 = 16

Lots of fun. But slow. Too slow.

So one day we came up with this idea for repeated multiplication function to save us from having to multiply a whole bunch.

We call it a Power function.

We then start powering stuff and it's cool, but then one day we want to undo powering.

We call it a Root function.

But then some smarty pants comes along and comes up with an expression that we don't have a number for! What is the square root of -1?

So we invent imaginary numbers.

Epilogue

My questions:

1. Is there an explanation behind this pattern of repeated function -> inverted repeated function -> new number?
2. If there is a real pattern here, does it continue?