which is approximately . The number of atoms in the universe is estimated to be about . When we consider permutations of large sets, even more breadth-taking numbers emerge.

In 1976, Donald Knuth introduced an arrow notation for large numbers:

The sequence is the sequence of Ackermann numbers (Conway and Guy, 1996). It is easy to see that , but is a power-tower of height greater than 7.6 trillion, which almost defies description. It can be difficult to grasp the magnitude of such large numbers.

** Iterated Factorials, or Multiple Bangs **

Let us consider a simple way to generate large numbers using iterated factorials.

Printers’ slang for the exclamation mark is “bang”. Perhaps this comes from comic books, where the “!” appears in speech-balloons to represent an explosion or a gun being fired, and interjections like “Pow!”, “Bash!” and “Ouch” are commonly used. The term “bang” is also popular among computer programmers.

The factorial (n bang) grows rapidly at the argument increases. For example, the number of possible orderings of a deck of 52 cards is . We will consider iterations of the factorial.

[We will consider iterations of the factorial function. The notation has already been reserved for the *double factorial*, that is, the product of all integers from to with the same parity as . Thus, and . We will not use the notation below; we need another way to denote iterated factorials.]

The iterated factorial function is written . For brevity, we introduce the new (non-standard) notation:

We can read as “3 bang-bang”, as “5 bang-bang-bang”, and so on. We can also read as “the -th factorial of ”.

We see that and . But for larger integers, there is explosive growth. When , we have

so the third factorial of 3 is enormous. For , the growth is much more dramatic:

These values were obtained using the program **Mathematica**. **Mathematica** failed to evaluate the third factorial of 4, giving an message. We can get an estimate of the true value of using Stirling’s asymptotic formula:

We let and . We know and we estimate using Stirling’s formula. We have so that

The dominant component is the second term, so we can write

which could be described as ineffably large.

We conclude by noting a numerical coincidence: **Avogadro’s Number** is, by definition, equal to , which is remarkably close to . So, the number of molecules in a mole is about **4-bang-bang**.

** Sources **

Conway, John H. and Richard Guy, 1996: *The Book of Numbers*. Copernicus. ISBN: 978-0-3879-7993-9.

Wikipedia article *Large Numbers.*