graham's number and infinity
where we learn about big numbers
Numbers. Small. Large. Numbers.
I am fascinated by numbers. I like very small ones like 1. 1 is small for many reasons but the reason I like the most is it is the number that takes up the least physical space when written down.
I also like very big ones like a million. If you counted to a million it would take you 23 days!
A billion is even bigger. That’s a thousand times a million. If you wanted to count to a billion it would take 95 years. Shit!
But those numbers are nothing compared to the big boys
First we have the beautifully named googol
(a one followed by a hundred zeros)
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)
and it’s exponentially more massive uncle the googolplex
(which is ten x googol)
10¹⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰,⁰⁰⁰
Of course you could argue that numbers go on forever, and you wouldn’t strictly be wrong. Numbers do kind of go on forever but then you are getting into the murky waters of infinity. Infinity is not a number it’s a concept. So somewhere there is a number right before infinity. Something that pushes the natural numbers (1,2,3,4,5 and on and on – this is what N means) into the philosophical concept of infinity. Isn’t that marvelous?
Lucretius (who I have mentioned many times before due to him being a clever guy) said that you can’t tell whether space is infinite or not because if you traveled to the end of the universe and threw a javelin, one of two things would happen. Either it would just keep going – in which case there is no end to the universe just more universe; or it would stop. It would hit a wall. End of the universe, right? Well no. because something must be after the wall. Either way, the universe cannot be infinite. Well, now we think maybe the universe is a shape. Like a torus (donut) or a hypersphere (big round thing).
Same with math. Whatever number you think of, you should always be able to add one to it. This is where the distinction between number and concept is important. Infinity is not a number, but numbers are infinite. But can you write a 1 with an infinite number of zeros after it?
No. there comes a point where you no longer have enough matter in the universe to write the zeros needed to accurately portray the number. As I mentioned above, a googol is a pretty fucking big number but it can be written really easily. Its a 1 with a hundred zeros. We can write that as a 1 with a little 100 after it. Like this. 1¹⁰⁰.
Easy!
10⁸: Number of Books ever published in Human History.
10¹⁰: Years since the big bang
10¹¹: Number of stars in the Milky Way.
10¹⁷: The number of seconds since the Big Bang.
10⁸⁰: Common Estimate for the number of atoms in the Universe.
10¹⁰⁰: Googol
These are just the numbers I can write down using my crap word processor - there are much bigger numbers than a googol or a googolplex. Numbers based on exponential powers. I would like to talk about a large number today called Graham’s number.
Grahams number is fucking huge. My favorite statistic is this – if you wrote a zero on every single planck length in the known universe (a planck length, you might remember, is a really, really small length - 1.6 x 10-35 meters. It’s the smallest known length because we cannot, using the physics and math we know, divide it any smaller) you would run out of matter and not even touch the number of zeros needed for Graham’s number. It’s so mindbogglingly huge that concepts like infinity mean nothing to it.
But as we discussed above infinity and natural numbers are two different things. We know Grahams Number is a Natural number. We know it ends with a seven! But like with a lot of theoretical math and physics we basically say it’s good enough. It’s close enough to infinity to be able to use it in calculations as an infinite.
To show how big Graham’s number is, let’s look at the way it’s calculated.
3↑3 means 3x3x3.
So 3↑3 is 27.
3↑↑3 means 3↑(3↑3)=3↑(33)=3↑(3×3×3)=7.6×10¹²
This is what we call an exponential increase. In this case, going from 3↑3 to 3↑↑3 is called Tetration - not to be confused with Titration which is a great way to come off anti-depressants.
7.6×1012 is 7.6 trillion by the way. If you tried to count to 1 trillion, it would take you two hundred thousand years. So already we can see using this method we are getting to big numbers.
Now, using the same method we can find
3↑↑↑3
(this is called Pentation, the next exponential step up).
3↑↑↑3 is equal to 3↑↑(3↑↑3)=3↑↑(3↑(3↑3))=3↑↑(7.6×10¹²) - a number so big that it would break every single calculator ever made. It’s massive.
Graham’s number calls this next step G1
3↑↑↑↑3=3↑↑↑(3↑↑↑3)=3↑↑↑(3↑↑(3↑↑3))=3↑↑↑(3↑↑(7.6×10¹²))
This number here is bigger than a googolplex. But for Graham it is only the start.
As an indicator of how high we are already, if you wrote the zeros in a googolplex down you would need 10⁹⁴ books. That's a shitload of books. A googolplex is fucking huge. And it’s just the start.
So we have G1. Graham does this 64 times.
Grahams number is so unimaginably large that for most people (say 99.99999%) it’s an entirely useless concept. It’s impossible to imagine how big it is. It’s really fucking big guys.
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