### Bang! Bang! Bang! Explosively Large Numbers

Typical Comic-book `bang’ mark [Image from vectorstock ].

Enormous numbers pop up in both mathematics and physics. The order of the monster group, the largest of the 26 sporadic groups, is

$\displaystyle 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000$

which is approximately ${8\times 10^{53}}$. The number of atoms in the universe is estimated to be about ${10^{80}}$. 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 ${\{1\uparrow1,\ 2\uparrow\uparrow2,\ 3\uparrow\uparrow\uparrow3,\ \dots \}}$ is the sequence of Ackermann numbers (Conway and Guy, 1996). It is easy to see that ${2\uparrow\uparrow2 = 4}$, but ${3\uparrow\uparrow\uparrow3}$ 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 ${n!}$ (n bang) grows rapidly at the argument increases. For example, the number of possible orderings of a deck of 52 cards is ${52! \approx 8\times 10^{67}}$. We will consider iterations of the factorial.

[We will consider iterations of the factorial function. The notation ${n!!}$ has already been reserved for the double factorial, that is, the product of all integers from ${1}$ to ${n}$ with the same parity as ${n}$. Thus, ${5!! = 5\times3\times1}$ and ${6!! = 6\times4\times2}$. We will not use the notation ${n!!}$ below; we need another way to denote iterated factorials.]

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

$\displaystyle n!^2 \equiv (n!)! \qquad n!^3 \equiv ((n!)!)! \quad\dots\quad n!^{k+1} \equiv (n!^k)! \quad\dots\quad \,.$

We can read ${3!^2}$ as “3 bang-bang”, ${5!^3}$ as “5 bang-bang-bang”, and so on. We can also read ${n!^k}$ as “the ${k}$-th factorial of ${n}$”.

We see that ${1 = 1! = 1!^2 = 1!^3 = \cdots }$ and ${2 = 2! = 2!^2 = 2!^3 = \cdots }$. But for larger integers, there is explosive growth. When ${n = 3}$, we have

$\displaystyle 3! = 6 \qquad 3!^2 = 720 \qquad 3!^3 = 2.6\times 10^{1746}$

so the third factorial of 3 is enormous. For ${n = 4}$, the growth is much more dramatic:

$\displaystyle 4! = 24 \qquad 4!^2 = 6.2\times 10^{23} \qquad 4!^3 = \mbox{\tt overflow}$

These values were obtained using the program Mathematica. Mathematica failed to evaluate the third factorial of 4, giving an $\tt overflow$ message. We can get an estimate of the true value of ${4!^3}$ using Stirling’s asymptotic formula:

$\displaystyle n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$

We let ${n = 4!^2}$ and ${N = 4!^3 = (4!^2)! = n!}$. We know ${n \approx 6.2\times 10^{23}}$ and we estimate ${N = n!}$ using Stirling’s formula. We have ${(n/e) \approx 2.3 \times 10^{23}}$ so that

$\displaystyle \left(\frac{n}{e}\right)^n \approx (2.3 \times 10^{23})^{2.3 \times 10^{23}} = (2.3^{2.3})^{10^{23}} \times 10^{(23\times 2.3)\times10^{23}} \approx (6.8)^{10^{23}} \times 10^{53\times10^{23}}$

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

$\displaystyle 4!^3 \sim 10^{53\times10^{23}}$

which could be described as ineffably large.

We conclude by noting a numerical coincidence: Avogadro’s Number is, by definition, equal to ${6.02214076\times 10^{23}}$, which is remarkably close to ${4!^2}$. So, the number of molecules in a mole is about 4-bang-bang.

Sources

${\bullet}$ Conway, John H. and Richard Guy, 1996: The Book of Numbers. Copernicus. ISBN: 978-0-3879-7993-9.

${\bullet}$ Wikipedia article Large Numbers.