First, how many is ?

## arrows of integer numbers

You know these:

And these:

Look at the previous formulas again. You can see a pattern:

So, tetration is generalized like this:

Now I can answer the question.

This is proper because it is greater than and smaller than .

Next, I will make for n arrows.

It's almost same pattern:

I think this is not good, but it must be like this by the definition.

## arrows of non-integer numbers

### Old version

Next, I have to think about a half arrow.

Look at these more...

And one more...

Now I have done (but too assertive).

I will summarize all definitions:

### New version

There is a problem on the old version. It is that is not 4 when n is not an integer. Is can be 3.7683... when n is around 0.53. This is a gap of 8% and it is not ignoble.

I have thought how to fix it for a week, and I have an answer: linear interpolation.

In math language,

If I want to make always four, it must be like this.

### comparizon between old one and new one

(not be written yet)