Largest number

What’s the largest number? If you said the googolplex, you’re off… by a lot. A lot.

Aster_Tataricus.JPG: Pascalou petitderivative work: Nevit Dilmen [CC BY-SA 3.0], via Wikimedia Commons
Okay, obviously the set of natural numbers is infinite. (Unless, of course, you subscribe to ultrafinitism… but that’s a post for another time.) So there’s no real “largest” number. And as soon as you claim one, some smartass will just present their own number, which is your number plus one.

For the purposes of today’s comparison, we’ll employ this definition: the largest number used in a published mathematical proof. In 1977, Martin Gardner wrote about Graham’s number, and at the time no larger number had been used.

I cannot adequately explain the scale of Graham’s number. You may have heard of a googol, which is 1 followed by a hundred zeros. That’s more than all the atoms in the universe. And if you’ve heard of a googol you’ve probably heard of the googolplex, which is a 1 followed by a googol zeroes. Graham’s number is so, so much larger than that.

Say we wanted to represent Graham’s number with digits. And say each digit was as small as possible – one Planck volume (or 4.2217×10−105 m3). That’s as small as volume gets, by the way… but I’ll have to save an explanation of Planck units for another time as well.

Anyway, you have each digit taking up this tiniest of tiny spaces. And say we take the entire universe, all that vast empty space, and fill it up with digits. Even then, you couldn’t write Graham’s number. Why not? Because the number of digits needed to write Graham’s number is itself too large to fit in the universe. And the number of digits needed to write that number is also too large. And on and on and on… for a Graham’s number of times.

So yeah, it’s big. Really big. Bigger than that. It’s big. But it’s still not the biggest.

Since 1977, larger numbers have appeared in mathematical proofs. But I have to stop somewhere otherwise we’ll be here all day. I’ll leave you with one last number: the Generalist’s Number. It is defined as the largest number in existence plus one. Yeah, that’s right, I went there.

[Thanks to Alistair S. for suggesting this topic.]

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