Large Numbers

Let’s talk about large numbers. Not your humdrum, everyday sort of large number, like the number of stars in the galaxy (about 10^11 or 100,000,000,000) or drops of water in the ocean (about 10^25 or 10,000,000,000,000,000,000,000,000). Not even the somewhat larger numbers like atoms in the visible universe (10^80, I won’t bother to write it out) or the Googol (10^100, not to be confused with Google).

No, today I want to talk about seriously large numbers. A good first try is the Googolplex, or 10^(10^100), that is 1 followed by a googol zeroes. While a googol vastly exceeds the number of atoms in the known universe, a googolplex can’t even be written out in full within the universe, even assuming you could write each zero using only one atom!

Ok, now we’re talking, but this is just the beginning. To represent the sort of numbers I have in mind, we need a new type of notation, called Arrow Notation. Here goes.

Arrow Notation:

If a and b are positive integers, a^b is just defined as ab.

The general definition is recursive:

a^...n arrows...^b = a^..(n-1) arrows..^a^..(n-1) arrows..^a^.......^a

where the (n-1) arrows operation is executed (b-1) times. The arrow operations are executed from right to left.

Let’s illustrate by example. It’s easy to check that a^^..n arrows ..^^ 2 is just a^2 for any n and any a. The smallest value of b which gives us something interesting is 3.

So, to start off:

3^3 = 33 = 27

3^^3 = 3^3^3 = 327 = 7,625,597, 484, 987

Ok, so that’s one of our garden variety large numbers. The ^^ operation, known as the ‘tower’ operation quickly gives us much bigger numbers.

For example, 3^^4 = 3^3^3^3 = 3^7,625,597,484,987, which is a number with about 3.5 trillion digits. 3^^5 would be 3^(3^7,625,597,484,987), which means if you wrote it out in base 3, the number of digits would be 3^7,625,597,484,987 !! And so on...

But now let’s get serious. How about adding yet another arrow?

3^^^3 = 3^^(3^^3) = 3^^7,625,597, 484, 987

How big is this? Well, when we look at 3^^3, 3^^4 and 3^^5, we see the incredible impact of increasing the number to the right of the ^^ by 1. Well, we’ve just increased it by about 7.6 trillion, so it’s impossible to imagine not only the number itself, but even the number of digits in the number, or even the number of digits of the number of digits of the number , or even....hmmm, running into some serious linguistic limitations here, but you get the idea.

But ok, let’s quit trying to imagine and just add one more arrow.
Let’s look at 3^^^^3

3^^^^3 = 3^^^3^^^3 = 3^^3^^......3^^3

where the ^^ operation is done 3^^^3 times.

Take another quick read through the part where I describe 3^^^3. Now take a very deep breath.

Imagine you are doing the evaluation of the right hand side in the expression above. Remember it’s done from right to left.

So, at step 1, you get 3^^3 which is kid-stuff.

But at step 2, you already have 3^^(3^^3) which is our mind-cracking 3^^^3 !!! Now, you just have to continue for another (3^^^3 – 3) steps....

If you’re really feeling masochistic, you can try working out 3^^^^^3, but by now I hope you’ve realized the effect of adding just one extra arrow. So, I’ll go ahead instead and mention the biggest number ever used in a mathematical proof.

Graham’s Number

First brought to attention in 1977, the number was used by the mathematician Ronald Graham working in a field called Ramsey theory. Ramsey theory deals with problems of the form, “How many elements must a set have for a certain property to occur.”

So, for example, suppose you have a gathering where any two people either know each other or don’t. How many people must there be, so that you always have either three people who all know each other or three people who all don’t know each other ? The answer in this case is 6. (Prove it!).

If you make the property more complex, the size of the set increases correspondingly. Graham showed that for his problem, the desired property is always satisfied if the set has at least Graham’s number of elements.

So, what is this number? Let’s define a sequence as follows.

G1 = 3^^^^3, our humongous old friend.

Now let G2 = 3^^...^^3

where - and read this bit very carefully – the number of arrows is G1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (What I really feel like doing is adding at least a googol exclamation marks here, or maybe a googolplex.)

G3 = 3^^...^^3 where the number of arrows is G2.

Still hanging on? Ok then, so we define G3, G4, G5 etc in the same style.
Graham’s number is G64.


Infinity

So, finally, we come to the biggest of them all. “Nonsense!”, interjects the mathematician, “Infinity is not a number at all. It’s an abstract concept and a pretty tricky one at that.”

Of course, of course. But when we mere mortals think of Infinity, we do tend to think of something very, very big which goes on and on and on and on.

How big? Well, the ancients used to mention things like “stars in the heavens”, “drops of water in the ocean”, “grains of sand in the desert” etc to convey a sense of infinity. But as we saw right at the start, these concepts are easily tamed with standard mathematical notation, and turn out to be all too finite and very manageable.

If we go beyond and introduce arrow notation, we can quickly write down numbers which completely drown the imagination all the way up to Graham’s number (and way beyond of course.)

However, compared to Infinity, there is no difference between Graham’s number and zero.

In fact, think about the sequence G1, G2, etc, where Graham’s number if G64. If we take the “Graham’s number”-th number in this sequence (!!!) and subtract it from Infinity, it makes, not a small difference, not a tiny, puny, minute difference, but absolutely no difference whatsoever.

That’s what Infinity is. Treat it with respect.