In 1977, Gardner described the number in Scientific American, introducing it to the general public. You can read in the Wikipedia how the number is defined. If you’d like to support Wait But Why, here’s our Patreon. 1024 (1 septillion) – A trillion trillions. Once I had done that, I had maxed out. So a googol is 1 with just 100 zeros after it, which is a number 10 billion times bigger than the grains of sand that would fill the universe. The first tower (in blue) is a straightforward little one because the value of b is only 3: g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) = 3 ↑↑↑ (3 ↑↑ (3 ↑↑ 3)) = 3 ↑↑↑ (3 ↑↑ 333). So 222 = 2(22) = 24 = 16. 1019 (10 quintillion) – The number of millimeters from here to the closest next star (38 quintillion millimeters). We’re running out of room here on both the small and big end of things to fit these numbers into the physical world, but three more for you: 10113 – The number of hydrogen atoms it would take to pack the universe full of them. 1 ↑ 38 = 1. He could barely finish the question before Milton opened his un-nuanced mouth and declared the number googolplex, which he, in typical Milton form, described as “one, followed by writing zeroes until you get tired.”6 At this, Krasner showed some uncharacteristic restraint, ignoring Milton and giving the number an actual definition: 10googol or 1 with a googol zeros written after it. 4 ↑↑ 7 = 4 ↑ (4 ↑ (4 ↑ (4 ↑ (4 ↑ (4 ↑ 4))))) = a power tower of 4s 7 high. That’s not a misspelling! To put those chances in perspective, that’s about the number of seconds in six years. If that were the case for every single grain of sand in this hypothetical—if each were actually a bundle of 10 billion tinier grains—the total number of those microscopic grains would be a googol. With its full written-out exponent, a googolplex 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. as a power tower. Ten: 10 (1 zero) Hundred: 100 (2 zeros) Thousand: 1000 (3 zeros) Ten thousand 10,000 (4 zeros) Hundred thousand 100,000 (5 zeros) Million 1,000,000 (6 zeros) Billion 1,000,000,000 (9 zeros) Trillion 1,000,000,000,000 (12 zeros) Quadrillion 1,000,000,000,000,000 (15 zeros) Quintillion 1,000,000,000,000,000,000 (18 zeros) Sextillion 1,000,000,000,000,000,000,000 (21 zeros) Septillion … Here’s how we get there: Look closely at that drawing until you realize how not okay it is. Please stop. The psycho festival ends when that final feeding frenzy produces it’s final number. So it’s like knowing a hedgehog will sneeze once and only once in the next six years and putting your hard-earned money down on one particular second—say, the 36th second of 2:52am on March 19th, 2017—and only winning if the one sneeze happens exactly at that second. So what’s g1? Addition being “iterated counting” means that addition is like a counting shortcut—a way to bundle all the counting steps into one, more concise step. There’s no possible way to wrap your head around that number—the best we can do is try to understand how long it would take to write the number. Once we get a meter down, the number is truly far, far, far bigger than we could ever fathom. You can keep increasing the arrows, and each arrow you add dramatically explodes the scope you’re dealing with. ↑↑↑↑ = power tower feeding frenzy psycho festival. Including you exactly how you are except instead of a pinky finger on your left hand you have Napoleon’s penis there as your fifth finger. But if I multiply two eight-digit numbers together, I end up with either a 15 or 16-digit number—much bigger. Now, the thing is, this is where most people stop. It was cute. Answer: Writing Graham's number as a 1 followed by zeros, suppose you could write one of the zeros in every Planck volume in the observable universe. So let’s go through it. 1017 (100 quadrillion) – The number of seconds since the Big Bang. Because every possible arrangement of matter in a human-sized space would likely occur many, many times in a space that vast, meaning everything that could possibly exist would exist—including you. And we don’t get to our final value of 3 ↑↑↑ 4 until we multiply out this final tower. Without being able to go smaller or bigger on either end, we’ve reached the largest number where the physical world can be used to visualize it. 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). Join 604,263 other humans and have new posts emailed to you. Here’s how it looks visually: So g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3), and we have two feeding frenzies to worry about. 33333 = a number with a 3.6 trillion-digit exponent, way way bigger than a googolplex and a number you couldn’t come close to writing in the observable universe, let alone multiplying out. First, we need to start back with a number called g1, and then we’ll work our way up. Who of the proclaimers was married to a little person? And when I was envisioning my huge googolplexgoogolplex number, I was doing the very best I could using the highest level I knew—exponentiation. So we’ll use Knuth’s up-arrow notation, which is one symbol that can be used on any level. And we’ve learned that 333 = 7,625,597,484,987, so: g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) = 3 ↑↑↑ (3 ↑↑ (3 ↑↑ 3)) = 3 ↑↑↑ (3 ↑↑ 333) = 3 ↑↑↑ (3 ↑↑ 7,625,597,484,987). Why? We used dots. In the real world, exponentiation is the highest operation we tend to ever use in the hyperoperation sequence. P.S. Yes, death comes way, way too quickly, but the thought “I do want to die at some point” is a very novel concept to me and actually makes me more relaxed than usual about our mortality. 1010 (10 billion) – Now we’re up to the years since the Big Bang (13.7 billion) and the number of seconds since Jesus Christ lived (60 billion). Even if we used tiny, typed 2mm 3’s, our tower would reach the moon and back to the Earth and back to the moon forty times before finishing. Including you but a one-foot tall version. I don’t think I’ve ever heard someone say “sextillion” out loud, and I hope to keep it that way. ↑↑ = power tower every 70-digit number is somewhere between 1069 and 1070, which is really all you need to know. After popularizing the newly-named googol, Krasner could barely keep his pants on with this adorable new schtick and asked his nephew to coin another term. Now let’s move up a level and start seeing the insane power of the hyperoperation sequence: Tetration is iterated exponentiation. 1011 (100 billion) – This is about the number of stars in the Milky Way and the number of galaxies in the observable universe (100-400 billion)—so if a computer listed one observable galaxy every second since Christ, it wouldn’t be anywhere close to finished currently.
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