The Largest Storable Number
When talking about large numbers — Graham’s Number, the Ackermann function, or busy beavers — it’s common to show that the number is literally larger than astronomical. That is, even if you could impart a decimal digit onto each atom in the observable universe, you’d run out of atoms long before you made a dent in the number.
On the other hand, Numberphile had the largest known prime number printed out. It took three large volumes, but you can easily hold this number in your hands. I don’t mean some mathematical shorthand with exponents or power towers, I don’t mean as digital information on a hard disk, I mean every individual digit is printed out and visible to the naked eye.
In a few years, the largest prime number might take up five volumes instead of three. It could still reasonably be printed. What about a number that took a room full of bookshelves to store? Or perhaps —
I chose 1980 because computer storage can’t compete with books, but we still have cars, elevators, and electric lights.
As for the size of this number, you could cheat by making every digit a nine. After all, that would be of greater magnitude than if you had a non-nine anywhere. But this seems unfair, so instead we’re going for the largest order of magnitude, i.e. the most digits — it doesn’t matter what they are.
If you’re coming down from writing digits into atoms, then covering earth’s surface area with 16TB hard drives doesn’t seem too implausible. That’s why I framed the challenge such that a person has to be able to read the digit within an hour — we’re dealing with a city of numbers, not a planet. Plus, any global project still seems too implausible.
But imagine — a city where every building it packed full of books of numbers, organized by some massive scheme that allows one to find any digit quickly. Simply using digits would be too long, but if you used English words, you might be able to identify locations in a way that fits on a sign.
So, if there’s no traffic, you travel at at least 60mph in this city, and you could do so for at least half of the hour before you need to start zeroing in. So let’s say there’s a thirty-mile radius to fill with numbers. (Apologies for the customary units; when estimating I need to use the units I grew up with. I’d be interested if any non-Americans want to go through this exercise in metric and see if they come up with a different answer.)
But anyway, it’s a big city’s worth of numbers. And no one has to live there, it can be magicked into existence, you just need one person in the middle to start driving to the correct building.
In 1980, the largest buildings were the Sears (now Willis) Tower, and the World Trade Center. The Sears tower has 4.5 million square feet of floor space; the Twin Towers had 10 million between them. So let’s say we have skyscrapers with 5 million square feet of floor space each, filling up that land area we calculated.
Each floor of the Twin Towers had an acre of rentable space, and they were pretty boxy, so let’s use an acre as an estimate for the footprint of these skyscrapers. That thirty-mile radius works out to 1.8 million acres. Figure that you have to have roads and power lines, so maybe 80% of that is actually filled with buildings. So that’s 1.4 million acres, or conveniently, 1.4 million skyscrapers. Or, 7 trillion square feet — enough for everyone in the world to have one thousand square feet to themselves.
Or is it? The area of Manhattan is 33.77 square miles. Using methodology similar to what’s on display here, someone determined that there are 60 thousand buildings in Manhattan. That works out to 1,776 per square mile. We figured 498 buildings per square mile, but the Twin Towers had a footprint much larger than most NYC brownstones, even when you account for the factor of three and a half or so. I guess 80% was a bit high . So let’s go with five trillion square feet.
Now let’s work from the other side. I could guess the area of a printed digit and how thin paper is, but since Numberphile printed out a large number, let’s use that. The number he printed has about 22 million digits, and each of the three volumes seems to be about an inch an a half thick, so 4.5 inches total. So foot of books could hold 58 million digits.
A foot worth of bookshelf can accommodate, well, a foot of books right? It’s probably a little less due to dividers and panels and such, not to mentioned wasted space in back. But that’s just one level. You can space shelves every, oh, fifteen inches vertically, up to about seven feet if you have footstools lying around. So call it five levels, which means each horizontal foot of bookshelf can store about 55 inches worth of books, or 266 million digits.
So if you wanted to stop now, a large bookshelf or two could hold about a billion digits.
Pressing on, we need to convert horizontal feet of bookshelves to square feet in a building. There may be wasted space in the back or front of the shelf, but more importantly you’ll need aisle-ways and elevators to get to everywhere. I think this is a killer — we need the numbers to be accessible, not just stored. I’m not an architect, but I think saying that 30% of available floor space is shelves is reasonable. You may be able to do better with those moving shelves on tracks, but I’m not sure they had those in 1980. Although they could have, with motorized tracks and physical buttons even.
30% of 5 trillion is 1.5 trillion, so we have that many horizontal feet of shelves, each one with 266 million digits. So, multiplying, that’s 4 × 10²⁰ digits. (If you use 7 trillion square feet, the four becomes a five. Meh.)
How big is this number? It’s perhaps the number of grains of sand on earth, or perhaps a thousandth of that number. Oh, and Earth has 10⁵⁰ atoms, so our estimate is, at least by that consideration, quite plausible.
I suspect the biggest uncertainty is how far you could drive or otherwise travel at high speed before you had to get out and search a particular building. I conservatively allocated only half of the hour to this. If you increased the time or speed, the area you can reach grows quadratically. Even a few seconds can add another skyscraper worth of books — about 400 trillion digits.
But is this actually reasonable to build? I said a few bookshelves could store a billion digits, so one library could hold a trillion digits, plus or minus an order of magnitude. I think one large building (but not a skyscraper) is the largest space any person or small group would dedicate to the task.
So that’s the engineering side of things. Now it’s up to the mathematical community to find an interesting number of this size.