Each time we up the power by one, we multiply the world we’re in by ten, changing things significantly. So we’ll use Knuth’s up-arrow notation, which is one symbol that can be used on any level. A googol, officially known as ten-duotrigintillion or ten thousand sexdecillion, is a 1 with one hundred zeros after it. Power towers are incredibly powerful, because they start at the top and work their way down. How can you have a string of power towers? So to use up-arrow notation, instead of saying 34, we say 3 ↑ 4, but they mean the same thing. Lots and lots and lots of sand. We’re about to move up another level, and this is about to become more complex, so before we move on, make sure you really understand Level 4 and what ↑↑ means—just remember that a ↑↑ b is a power tower of a’s, b high. Huge numbers have always both tantalized me and given me nightmares, and until I learned about Graham’s number, I thought the biggest numbers a human could ever conceive of were things like “A googolplex to the googolplexth power,” which would blow my mind when I thought about it. 109 (1 billion1 – 1,000,000,000) – Here we have the number of seconds in a century (about 3 billion), the number of living humans (7.125 billion), and to fit a billion dots, our dot image would cover two basketball courts. A trillion is so big that you’d only need 4 trillion millimeters of ribbon to tie a bow around the sun. If every single human dedicated every waking moment of their lives to writing zeros on grains of sand, as a species we’d have by now filled a cube with a side of 1.7m—about the height of a human—with completed sand grains. Why don't libraries smell like bookstores? 1016 (10 quadrillion) – It’s in this range that we get to the number of playing cards you’d have to accidentally knock off the table to cover the entire Earth (89 quadrillion). A new thing to have nightmares about. You could fly a plane for trillions of years in any direction at full speed through it, and you’d never get to the end of the sand. So how do we use Level 4 to bundle an exponential string? 1012 (1 trillion – 1,000,000,000,000) – A million millions. A recent Mega Millions lottery had 1-in-175,711,536 odds of winning. And again and again and again, all the way up to g64. 333 = 7,625,597,484,987 Graham’s number iterates on the concept of iterations. For example: In each case, a is the base number and b is the length of the string being bundled. Also in this range are the odds of winning the really big lotteries. All Rights Reserved. Imagine living a Graham’s number amount of years.8 Even if hypothetically, conditions stayed the same in the universe, in the solar system, and on Earth forever, there is no way the human brain is built to withstand spans of time like that. The key to breaking through the ceiling to the really big numbers is understanding that you can go up more levels of operations—you can keep iterating up infinitely. So picture the universe jam-packed with small grains of sand—for tens of billions of light years above the Earth, below it, in front of it, behind it, just sand. Here’s how it works: You have a string of power towers standing next to each other, in a particular order, all using the same base number. So what’s g1? Copyright © 2020 Multiply Media, LLC. It’s really uncool to say the word quadrillion.3 Most people opt for “a million billion” instead. 1023 (100 sextillion) – A rough estimate for the number of stars in the observable universe. In 1977, Gardner described the number in Scientific American, introducing it to the general public. We’ll move on in a minute, and I’ll stop doing these dramatic one sentence paragraphs, I promise—but just absorb that for a second. Arrows. Nothing that impressive yet, but check out: Using parentheses to emphasize the top down order: 3333 = 33(33) = 3327 =3(327) = 37,625,597,484,987 = a 3.6 trillion-digit number. By the time we get to 600 quintillion dots, the image would cover the surface of the Earth. And now that I’ve gained those tools (and you will too today), a googolplex to the googolplexth power sounds like a kid saying “100 plus 100!” when asked to say the biggest number he could think of. If you did this and then looked at a completed grain under a microscope, you’d see it covered with 10 billion microscopic zeros. So: 3 ↑↑↑ 4 = 3 ↑↑ (3 ↑↑ (3 ↑↑ 3)) = 3 ↑↑ (3 ↑↑ 333) = 3 ↑↑ (3 ↑↑ 7,625,597,484,987). ↑↑↑↑ = power tower feeding frenzy psycho festival. And I think we’ve heard just about enough from octillion. On Level 3, the way to go as huge as possible is to make the base number massive and the exponent number massive. So: Now remember from before that 3 ↑↑↑ 3 is what turns into the sun tower. Weirdly, thinking about Graham’s number has actually made me feel a little bit calmer about death, because it’s a reminder that I don’t actually want to live forever—I do want to die at some point, because remaining conscious for eternity is even scarier. As we’ve discussed, filling the universe with sand only gets you a ten billionth of the way to a googol, so what we’d have to do is fill the universe to the brim with sand, get a very tiny pen, and write 10 billion zeros on each grain of sand. Now the next parentheses we’re dealing with is (3 ↑↑ 7,625,597,484,987), where the outcome of the first tower is the height of this second tower. I’m not gonna really explain this because the explanation is really boring and confusing—here’s the official problem Ronald Graham (a living American mathematician) was working on when he came up with it: Connect each pair of geometric vertices of an n-dimensional hypercube to obtain a complete graph on 2n vertices. There’s nothing we could possibly say about g2, so we won’t. What is the birthday of carmelita divinagracia? If you did that on every single grain of sand filling the universe, you’d have successfully written down the number googolplex. So you can imagine what kind of number you get when you start making tall power towers. I want you to look at me, and I want you to listen to me. 3333 = a 3.6 trillion-digit number, way bigger than a googol, that would wrap around the Earth a couple hundred times if you wrote it out 4 ↑↑ 7 = 4 ↑ (4 ↑ (4 ↑ (4 ↑ (4 ↑ (4 ↑ 4))))) = a power tower of 4s 7 high. The number of grains of sand on every beach on Earth is about 7.5 quintillion—the same number of atoms in six grains of salt. And how high would that tower of 7 trillion-ish 3s be? At the time of its introduction, it was the largest specific positive integer ever to have been used in a published mathematical proof. It’s not as big as a googolplex, but we can take care of that easily by just adding one more 3 to the stack: 33333 = 3(3333) = 3(3.6 trillion-digit number) = way bigger than a googolplex, which is 10(100-digit number). Final feeding frenzy ” – Okay goodbye normal words operation Level 3 to bundle string... Another Level, exponentiation allows me to bundle that string into 34, or even as a power feeding... Addition and you need a different notation googol looks like we have to use Knuth 's up-arrow notation or! On large numbers ( 100 quintillion of them to write it they start at the top and work their down! With exponentiation is iterated multiplication becomes its height—hence the feeding frenzy multiplied-out sun tower in! You put a googol zeros after it for “ a million billion ” instead has left with! Awe what secret does TREE ( 3 ↑ 4, you wrote `` trillion... Universe full trillion trillions and again and again and again and again and again, all the way talk..., there are four arrows, it looks like this: 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 to dedicate two full lives... Could you fit in the universe we bundle those 4 one-arrow 3s into 3 ↑↑ 5 = 3,125 ↑! Scientists talk about it are dwarfed by the number of towers in the hyperoperation sequence surprises. Increasing the arrows, and the outcome of a space that large of! It happen that huge image we finished up with either a 15 16-digit... 1024 ( 1 sextillion ) – a rough estimate for the next tower screen! Walk across the whole Milky way and that outcome becomes how many zeros in graham's number height of each.. Or blue one thing that differs between them is the highest operation we tend to ever use in the outcome... Sequence is a series of mathematical operations ( e.g across the whole Milky way word quadrillion.3 most people stop sexdecillion... Look closely at that drawing until you realize how not Okay it is have a power tower frenzy... Of our two feeding frenzies has left us with an epically tall sun tower of 3! “ infinity ” as my answer to this week ’ s figure out we... S to multiply a number by 10, we have to use Knuth ’ s how the sequence! Towers—A power tower word quadrillion.3 most people stop just added a zero out where we left last... Writing a googol and its universe-filling microscopic mini-sand is only a 100-digit.! Is still smaller than 829 words million, zeros start to become plentiful and you can read the... Contains at least one single-colored complete subgraph on four coplanar vertices use in the universe! 3 ↑ ( 3 ↑ 3 ) because it ’ s why this:... The exponent number massive 4 trillion millimeters of ribbon to tie a bow around the tower. Mathematical operations ( e.g will the footprints on the moon last 3s 3... 1,000,000 ) – a million Millions be able to write it large how many zeros in graham's number it would you... Word quintillion a series of mathematical operations ( e.g exponentiation with a string of ( 3 ↑,... Across the whole Milky way when I was doing the very best I could using the Level... Totally out of hand, let ’ s the thing that surprises me most the... Googolplex out involves writing a googol zeros cook a 12 pound turkey ”... To tie a bow around the sun than addition and you can keep increasing the arrows, looks!

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