thoughtsOperator Theory

This particular thought has been rattling around in my brain literally for years. Every few months, it pops back out, I play with it on paper, get nowhere, and file it away again. I figure it needs to get out on the Internet, then maybe I'll be able to forget it.

The thing is, it seems so basic. It seems so obvious. Someone surely must have thought of it before. It looks like a new field of mathematics. And yet, as far as I can tell, it is almost entirely useless.

I call it Operator Theory.

See, when you first teach a child multiplication, you expose the operation as compound addition. Multiplying one number by another is merely adding a number to itself a number of times.

4x5 = 4+4+4+4+4

Similarly, when you first teach a child exponents, you expose the operation as compound multiplication. Addition is to Multiplication as Multiplication is to Exponents.

4^5 = 4x4x4x4x4

Of course, later, you go on to do nifty things with the higher operations, but their base concept is still the compounding of addition. All arithmetic is based on these operations. (Subtraction and Division, obviously, are merely Addition and Multiplication backwards. In a perfect world, they would not be needed.)

The obvious question, then, is "What comes after exponents?". I think the obvious answer is: hyperexponents.

4^^5 = 4^4^4^4^4

Hyperexponents, of course, produce very large ridiculous numbers very quickly. And I really have no idea what the value of the equation would be if the second value were ever negative or zero or fractional. But still, the operation should exist. It's only logical.

And then, of course, we open ourselves up to thinking of the operator itself as a number. Arbitrarily, I have assigned the number 1 to addition, 2 to multiplication, and so on. What about the operator zero? Negative or fractional or even imaginary operators? The mind boggles.

And, indeed, my mind has boggled. I can't imagine where we go from here, how we deal with these equations. I have no sufficient mathematical training to deal with such questions. If, as I suspect, this represents a completely new branch of mathematics, it could be that sufficient training does not exist.

There are, of course, a few things I can derive.

Addition Table = Multiplication Table
Anything operated to itself is that thing next-level-operated to itself.
Two any-operation two is always four.

I would love it if someone would email me and tell me that this was all already discovered millenia ago by some ancient Greek guy or something. There is no way I thought of this first, it's too basic, too elemental.

Still, I did come up with it independantly. That's got to count for something.

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