In other posts I’ve talked about Leo being fascinated by giant numbers, and a while back, based on his interest, I proposed a joke number called 11Z, called “eleventy zillion”. Today we put some specific mathematical meat on this notation, in two different ways, which we agreed to call “Leo Numbers” v. “Jeff Numbers”.

**Jeff’s Z Numbers**:

Both are based on Jeff’s Numbers, which is the following exponentiation ladder:

n^{Z} = n^n^n^n… n times.

For example:

3^{Z} = 3^3^3

Now, there is immediately a problem, which is which way to evaluate the ladder.

Does 3^{Z} = (3^3)^3 = 27^3 = 19,683, or is it = 3^(3^3) = 3^27 = 7,625,597,484,987

I think that the mathematical default if left-to-right, i.e., up the ladder, but I hate these defaults because they aren’t explicit, so I prefer to have an explicit notion telling you which way to evaluate the ladder.

So to clarify this, we use arrows as:

3^{Z→} = (3^3)^3 = 27^3 = 19,683– called “3 zillion up”

3^{Z←} = 3^(3^3) = 3^27 = 7,625,597,484,987 — called “three zillion down”

Now, Leo, who like scientific notation and thinks in terms of Googol and Googolplex, thinks of Z notion slightly differently.

**Leo’s Z Numbers**:

3^{Z→} = 3e19,683

and

3^{Z←} = 3e7,625,597,484,987

So, just as Z-down notation is way bigger than Z-up notation, Leo numbers are way bigger than Jeff numbers!

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