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Mathematics of operators, + * ^ etc, a few speculative notions...

This idea first occurred to me many years ago, but it was only today 2003/02/09 that I realised its applications to big numbers.

The idea starts off with the curious fact that mathematics seems to only have a few operators such as addition, multiplication, power, etc, and that there is a pattern. Multiplication can be done by repeated addition, so 2+2+3 is 2x3, and similarly power can be done by repeated multiplication, so 2*2*3 is 2^3, and this process can be continued, so there's another operator which is 2^2^2.

So then let's invent some notation:

Addition is the first operator, op[1]. Multiply is op[2], Power is op[3], and there exist an infinite series of these. Each can be derived by repeated application of the first.


2+2+2 = 2*3 , (or 2 op[2] 3)

2*2*2 = 2^3 , (or 2 op[3] 3)

2^2^2 = 2 op[4] 3

2 op[4] 2 op[4] 2 = 2 op[5] 3

and so on.

(I'm using "2" and "3" here as examples. The principle applies to any combination of numbers)

What happens with mathematics is you think "what if..." and it looks silly to start with, and then it sometimes opens up a whole new field of mathematics. Remember what happened when someone did that to square-root of minus one?!

So, immediately it's possible to say "What if non-integer numbers were used for n as op[n] numbers?" Of course that sounds silly, having a mathematical operator say halfway between add and multiply, but it might produce some interesting results.

Also, although it's obvious there are an infinite set of ever-increasing op numbers, what about lower than 1? What would op[0] be like? (Actually this produces consistent results, of just adding 1 for every use of the operator, but it's not very interesting).

By the way, in the way I did these things, A^B^C is A^(B^C), not (A^B)^C . This makes a difference for operators 3 and above, but not for addition or multiplication.

It was only recently I found this: Really Big Numbers, and I realised that the big number Megafuga of the Googolplex was actually googolplex op[4] googolplex. This then brings about the notion of what if we replaced the 4 with a bigger number, such as the googolplex, and get googolplex op[googolplex] googolplex. Well it would be silly. But it would be a really really big number.

Let's clarify a few things. Addition is op[1], Multiply is op[2], To-the-Power-of is op[3], and Mega Fuga is op[4], and there exist op[5], op[6], etc. (Megafuga is best explained by those friendly people at Really Big Numbers). There are also some big numbers found by the Ackermann Function, as per http://en.wikipedia.org/wiki/Ackermann_function. It seems there's also a similarity between this op[n] idea and Knuth's up-arrow format, another case of parallel creation. Please note that when I thought of the idea, I was a kid in school and wasn't aware of advanced mathematicians working on similar things. Plus, it was before 1976.

Also, it might be worth mentioning What's a Googolplex?. Another subject worth talking about, especially if all this is a bit baffling is: What about Quite Big Numbers?. Other quite good mind-stretching things which are good for improving the stretchability of your bungination are pi, the speed of light, and the earth is smoother than a snooker ball