I don't know where to put this, so I put it here cause it is so advanced...
Anyway, for those who knows what arrow notation is, I have some questions, and for those who don't know what it is, here's a brief explanaition:
The arrow notation is used to apply an operation multiple times ( times) at variable with variable . For example, means that operation is applied times at , with itself. So since is defined as multiplication, means power since is multiplicated times with itself.
All this also means that . For example, .
This can go on forever, meaning that can grov from 1 to 2, from 2 to 3 and so on, creating new operators all the time. For example
This can create very big numbers if we continue developing the operators. This is an very interesting invention made by Donald Knuth, but what is it good for? I recently saw an application of when reading about power towers.
Cheers!
Yes, but you can still use the old traditional way of creating a large number, like
Another thing you can conclude from this is this, can also be written as:
For example we have
Set temporarily the variable
Now we have no reason to conclude from this that for all values of , we only know it is true for , but we can generalize so that it becomes true for all values of .
So now when we know have , we can set up the equation
Set temporarily the variable
Generalize this to make it true for all values of .
That is already true for all values of and doesn't have to be generalized.
But now . That means every operator will do the same, namely return .
Yet another thing is that the definitions of and doesn't directly lead to the conclusion that . It could as likely has been , that doesn't matter since , and . But , therefore it matters when , cause would have looked like and not like . And , thus it matters when n as well.