I think partly out of my amateur fascination with statistics, I geek out a lot over big numbers.

One I learned about some time ago is the Graham's number, which I can say with neither reluctance nor embarrassment, I don't fully understand. But it did give me an interesting idea for a somewhat easier to calculate function that grows at a startling rate.

Suppose you have a number S_n. The subscript n is both an index, as well as an initial value. S_n is defined as n to the nth power, to the nth power, to the nth power, and so on n times. For instance, S_2 = 2^2 = 4, but S_3 = 3^3^3 = 19683, S_4 = 4^4^4^4 = 3.40282366e38 and so on.

It becomes quite apparent that S_n grows faster than an exponential even as n grows monotonically. I read, just now, that there are ways to manufacturer even faster growing functions using a similar, but in more ... dimensions than this.

Any other big numbers that are easily expressed (other than the googol and googolplex)?

I hope you find what you're looking for so I can finally start grading my routes.

I'm trying to get a number recognised by science, but it seems there is some antagonism against me and I don't know why.

Airscape's constant (A): If a number > 0 is divided by Airscape's constant then it is halved.

A = 2

I think partly out of my amateur fascination with statistics, I geek out a lot over big numbers.

One I learned about some time ago is the Graham's number, which I can say with neither reluctance nor embarrassment, I don't fully understand. But it did give me an interesting idea for a somewhat easier to calculate function that grows at a startling rate.

Suppose you have a number S_n. The subscript n is both an index, as well as an initial value. S_n is defined as n to the nth power, to the nth power, to the nth power, and so on n times. For instance, S_2 = 2^2 = 4, but S_3 = 3^3^3 = 19683, S_4 = 4^4^4^4 = 3.40282366e38 and so on.

It becomes quite apparent that S_n grows faster than an exponential even as n grows monotonically. I read, just now, that there are ways to manufacturer even faster growing functions using a similar, but in more ... dimensions than this.

Any other big numbers that are easily expressed (other than the googol and googolplex)?

Just change the order of association in your series. That is,

S'_2 = 2^2 = 4
S'_3 = 3^(3^3) = 7.6e12
S'_4 = 4^(4^(4^4)) = ?

Jay

I didn't even think to look at that. You can tell how often nested exponentials are used in my day-to-day life.

One thing I learned from all of this is that it is very easy to produce a number that would take more than the entire universe to fully express, if one base-10 digit were placed in each Planck volume.

Well, a quick check led to this:

S_4 = 4^(4^(4^4)) = 4^(4^256) = 4^(1.3407807929942597099574024998206*(10^154)) = 4^(10^154)^1.3407...

Which is pretty fuckoff big.

If you had an i7 (about 100 gigaflops/second) processor running only this calculation, it would still take about 10^44 seconds, or about 3*10^36 years to calculate it out, assuming that the computer does one "times four" operation per flop. It might attack it logarithmically, which might speed things up quite a bit, but still, we're looking at two or three orders of magnitude in terms of speed, max. So you may as well just stop now Jay.

He's not talking about fractions, but numbers that are much greater than 1.

about 100 gigaflops/second

Damn you!!!

I think partly out of my amateur fascination with statistics, I geek out a lot over big numbers.

One I learned about some time ago is the Graham's number, which I can say with neither reluctance nor embarrassment, I don't fully understand. But it did give me an interesting idea for a somewhat easier to calculate function that grows at a startling rate.

Suppose you have a number S_n. The subscript n is both an index, as well as an initial value. S_n is defined as n to the nth power, to the nth power, to the nth power, and so on n times. For instance, S_2 = 2^2 = 4, but S_3 = 3^3^3 = 19683, S_4 = 4^4^4^4 = 3.40282366e38 and so on.

It becomes quite apparent that S_n grows faster than an exponential even as n grows monotonically. I read, just now, that there are ways to manufacturer even faster growing functions using a similar, but in more ... dimensions than this.

Any other big numbers that are easily expressed (other than the googol and googolplex)?

When in doubt and wanting the BIG just go Factorial...n!

By comparison, factorial grows at a snail's pace.

And I am aware of the definition of flops, Tristan. There's just a difference between FLOPs (FLoating-point OPerations) and FLOPS (FLoating-point Operations Per Second) which creates some ambiguity.

Memo from the pedantry department to the department of redundancy department: FLOPS = Floating Point Operations Per Second.

Perhaps... hell it's been years since i really had to use statistics..

By comparison, factorial grows at a snail's pace.

And I am aware of the definition of flops, Tristan. There's just a difference between FLOPs (FLoating-point OPerations) and FLOPS (FLoating-point Operations Per Second) which creates some ambiguity.