At the time of its introduction, it was the largest specific positive integer ever to have been used in a published mathematical proof. {\displaystyle f(n)=3\rightarrow 3\rightarrow n} n n People would be mad at you. A googol, officially known as ten-duotrigintillion or ten thousand sexdecillion, is a 1 with one hundred zeros after it. f So far, all we’ve done with exponentiation is one computation—a base number and a power it’s raised to. The magnitude of this first term, g1, is so large that it is practically incomprehensible, even though the above display is relatively easy to comprehend. And so on. 1 ↑ 38 = 1. [and so it was decided that googolplex should be] a specific finite number, with so many zeros after the 1 that the number of zeros is a googol. And it’s bigger than the famous googol, 10100 (a 1 followed by 100 zeroes), which was defined in 1929 by American mathematician Edward Kasner and named by his nine-year-old nephew, Milton Sirotta. ) ⋯ And now we can replace the sun tower with the final number that it produces: g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) = 3 ↑↑↑ (sun tower) = 3 ↑↑↑ INSANITY, Alright, we’re ready for the second of our two feeding frenzies. ( 1013 (10 trillion) – This is about as big as we can get for numbers we hear discussed in the real world, and it’s almost always related to nations and dollars—the US nominal GDP in 2013 was just under $17 trillion, and its debt is currently just under $18 trillion. And then this happens again for g4. This final 3 ↑↑ (sun tower) operation is a power tower of 3’s whose height is the number you get when you multiply out the entire sun tower (and this final tower we’re building won’t even come close to fitting in the observable universe). So you know how upset I just got about this whole INSANITY thing? Abbreviated as AU, this is simply the average distance from the Earth to the Sun. 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. Graham’s number is so enormous it can’t even be described using everyday mathematics. You know how sometimes you go through life, and you’re lost but you don’t even know it, and then one day, the right person comes along and you realize what you had been looking for this whole time? ( The answer is what I call a “power tower feeding frenzy”. ( . (feat Ron Graham)", "How Big is Graham's Number? Color each of the edges of this graph either red or blue. We use Level 3 to bundle that Level 2 string into 34, or 3 ↑ 4. Also the number of references to Kim Kardashian that entered my soundscape in the last week. ↑↑↑ Nor even can the number of digits of that number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe. ↑ Remember how earlier we showed how quickly a power tower escalated: 3 = 3 Comparing this to the bacteria fact, it’s like you have 1/10th of the world’s ants crawling around inside your body. Skewes' Number is a bounding value, that tends to get honorable mention in discussions about large numbers. If I have 3 and I want to go up from there, I go 3, 4, 5, 6, 7, and so on until I get where I want to be. And here’s the thing about this second feeding frenzy—. There is a name for a number bigger than a googol but much smaller than a googolplex. The psycho festival ends when that final feeding frenzy produces it’s final number. A googleplex is significantly larger than the 48th Mersenne prime. 10185 – Back to the Planck volume (the smallest volume I’ve ever heard discussed in science). ⋯ ), where each operation in the sequence is an iteration up from the previous operation. 64 ) It's the tenth power of a googolchime and the 100th power of a googol. If you’re interested the last ten digits of Grahams Number are 2464195387, no one, not even Graham himself knows what the first digit is. 333 = 7,625,597,484,987 ↑↑↑ When we went from 1 to 1,000,000, we didn’t need powers—we could just use a short string of digits to represent the numbers we were talking about. , It is named after mathematician Ronald Graham who used the number as a simplified explanation of the upper bounds of the problem he was working on in conversations with popular science writer Martin Gardner. Here is a list of all the big numbers up till the infamous centillion. 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. 1080 – To get to 1080, you take trillion and you multiply it by a trillion, by a trillion, by a trillion, by a trillion, by a trillion, by a hundred million. On Level 5, we’re dealing with a string of Level 4 power towers—a power tower feeding frenzy. On my computer screen, that image was about 18cm x 450cm = .81 m2 in area. So what does pentation bundle together? Then how do we know it is more powerful than BB(10^100) for example? That would be a pretty good answer for most people. Since there are four arrows, it looks like we have a power tower feeding frenzy psycho festival on our hands. So: Now remember from before that 3 ↑↑↑ 3 is what turns into the sun tower. 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. b ( We will write down a sequence of numbers that we will call g1, g2, g3, … Before we dive in, why is Graham’s number even a number people talk about? n Pentation, or iterated tetration, bundles double arrow strings together into a single operation. The lower bound of 6 was later improved to 11 by Geoffrey Exoo in 2003,[4] and to 13 by Jerome Barkley in 2008. n It bundles the hyperoperation sequence itself. The entire g1 now feeds into g2 as its number of arrows. Other specific integers (such as TREE(3)) known to be far larger than Graham's number have since appeared in many serious mathematical proofs, for example in connection with Harvey Friedman's various finite forms of Kruskal's theorem. ⏟ f In other words, G is calculated in 64 steps: the first step is to calculate g1 with four up-arrows between 3s; the second step is to calculate g2 with g1 up-arrows between 3s; the third step is to calculate g3 with g2 up-arrows between 3s; and so on, until finally calculating G = g64 with g63 up-arrows between 3s. 1014 (100 trillion) – 100 trillion is about the number of letters in every published book in human history, as well as the number of bacteria in your body.2 Also in this range is the total wealth of the world ($241 trillion, which we discussed at great length in a previous post). 2 ↑ 3 = 8 Can you possibly imagine what kind of number is produced when you put a googol zeros after the 1? Then that frenzy happens and produces an even more ridiculous number, which then becomes the number of towers for the next frenzy. Note that the result of calculating the third tower is the value of n, the number of towers for g1. It’s really uncool to say the word quadrillion.3 Most people opt for “a million billion” instead. If each book had a mass of 100 grams, all of them would have a total mass of 10 93 kilograms. 1017 (100 quadrillion) – The number of seconds since the Big Bang. (In fact, [5] Thus, the best known bounds for N* are 13 ≤ N* ≤ N''. ) , and this notation also provides the following bounds on G: To convey the difficulty of appreciating the enormous size of Graham's number, it may be helpful to express—in terms of exponentiation alone—just the first term (g1) of the rapidly growing 64-term sequence. The final number the frenzy produces becomes the number of towers in the next feeding frenzy. And how high would that tower of 7 trillion-ish 3s be? Now all we have to do is count the number of 5’s and 2’s in the multiplication. Just think about it, Rayo's number is defined using a Googol (10^100) symbols and a Googolplex has a Googol (10^100) + 1 number of digits.172.58.111.14 21:44, February 2, … So how do we use Level 4 to bundle an exponential string? ↑↑↑ = power tower feeding frenzy {\displaystyle \uparrow } Okay so now let’s take a huge leap forward into a whole different territory—somewhere where the Earth’s volume is too tiny and the Big Bang too recent to use in examples. ⋯ 3333 is the same as saying 3 ↑ (3 ↑ (3 ↑ 3)). How can you have a string of power towers? 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 ( Because the number which Graham described to Gardner is larger than the number in the paper itself, both are valid upper bounds for the solution to the problem studied by Graham and Rothschild. 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. That’s why Graham’s number is a thing—it’s not just an arbitrarily huge number, it’s actually relevant in the world of math. ) 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. 1024 (1 septillion) – A trillion trillions. ↑ Not a high-powered operation. When that feeding finally finishes, the outcome becomes the number of towers in the final feeding frenzy. Multiplication is a more powerful operation than addition and you can create way bigger numbers with it. So: Remember, when you see ↑↑ it means a single power tower that’s b high, so: 3 ↑↑↑ 4 = 3 ↑↑ (3 ↑↑ (3 ↑↑ 3)) = 3 ↑↑ (3 ↑↑ 333). {\displaystyle g_{64}} 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. It’s actually one key tool: the hyperoperation sequence. So far, we’ve worked with a different symbol on each level (+, x, and a superscript)—but we don’t want to have to remember a ton of different symbols if we’re gonna be working with a bunch of different operations levels. If we wanted to multiply a number by 10, we just added a zero. , where, g So many podcasts. The next step is we need to get to g2. It helps track these multiples of 10 because the larger the number is, the more zeroes are needed. ( And again and again and again, all the way up to g64. 3 Googolplex: The world's second largest number with a name. I can only wonder in awe what secret does TREE(4) and above holds! where the number of arrows in each subsequent layer is specified by the value of the next layer below it; that is. Yet, a googolplex isn’t the largest number described to date. But just adding one will often vilolate the “Gentleman's Rule” of large number wars. The pattern we’ve seen is each new level bundles a string of the previous level together by using a b term as the length of the string. Put more simply, it is a zero digit with no non-zero digits to the right of it. ↑↑↑↑ = power tower feeding frenzy psycho festival. You process that tower down to its full expanded outcome, and that outcome becomes the height of the next tower. 3 n G2 takes G1’s answer and adds that many up arrows to make G2. 1012 (1 trillion – 1,000,000,000,000) – A million millions. Of course, we won’t even pretend to do anything with that information other than laugh at it, stare at it, and be aroused by it. [2] In 2019 this was further improved to:[3]. 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. 3 Addition is an iteration up from counting, which we can call “iterated counting”—so instead of doing 3, 4, 5, 6, 7, I can just say 3 + 4 and skip straight to 7. So for at least the first part of this post, the powers of 10 can serve nicely as orders-of-magnitude “checkpoints”. TREE(3) actually came from Kruskal’s tree theorem and it is far far bigger than Graham’s number. And again for g5. → n f AarexWikia04 - 22:58, August 1, 2016 (UTC)Really? Tetration is intense. A googolplex is the highest named number, but it is a nonsense name. There’s nothing we could possibly say about g2, so we won’t. A centillion is the number 1 followed by 303 zeros. ) according to the definition. The thing that differs between them is the height of each tower. How big is this number? Instead of saying 3 + 3 + 3 + 3, multiplication allows us to bundle all of those addition steps into one higher-operation step and say 3 x 4. ↑↑↑ Remember, a googol and its universe-filling microscopic mini-sand is only a 100-digit number. Dear Simon & Catalyst, After posting my previous comment, I realized that I needed to put Graham's number … You get it. So 222 = 2(22) = 24 = 16. First, we need to start back with a number called g1, and then we’ll work our way up. First, in terms of tetration ( [1] This was reduced in 2014 via upper bounds on the Hales–Jewett number to, which contains three tetrations. and the superscript on f indicates an iteration of the function, e.g., So: 3 ↑↑↑ 4 = 3 ↑↑ (3 ↑↑ (3 ↑↑ 3)) = 3 ↑↑ (3 ↑↑ 333) = 3 ↑↑ (3 ↑↑ 7,625,597,484,987). No one who has social skills ever says the word quintillion. 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). 3 As we get bigger and bigger today, we’ll stick with powers of 10, because when you start talking about really big numbers, what becomes relevant is the number of digits, not the digits themselves—i.e. Graham's number is an immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. You guessed it—once the laughable g2 is all multiplied out, that becomes the number of arrows in g3. ( 3 ↑ 4 = 81 Why? Kind of. Also in this range are the odds of winning the really big lotteries. Then Graham decides that for g2, he’ll just do the same thing as he did in g1, except instead of four arrows, there would be NO I CAN’T EVEN arrows. n ) Anyway, it’s the number of cubic meters of water in all the Earth’s oceans and the number of atoms in a grain of salt (1.2 quintillion). ) ↑ 3 F ↑↑ 1 Well I just tested how fast a human can reasonably write zeros, and I wrote 36 zeros in 10 seconds.7 At that rate, if from the age of 5 to the age of 85, all I did for 16 hours a day, every single day, was write zeros at that rate, I’d finish one half of a grain of sand in my lifetime. And just how long would it take to do that? ) However it is possible to calculate the last few digits of Graham's number. Lots and lots and lots of sand. The next layer has that many arrows between 3s. So there is no “next number” in that naming sequence. A trailing zero is formed when a multiple of 5 is multiplied with a multiple of 2. So there you go. So: 3 ↑↑↑↑ 4 = 3 ↑↑↑ (3 ↑↑↑ (3 ↑↑↑ 3)) = 3 ↑↑↑ (3 ↑↑↑ (sun tower)). Here’s hexation explained generally: And that’s how the hyperoperation sequence works. You also had to deal with this number in high school—602 sextillion, or 6.02 x 1023—is a mole, or Avogadro’s Number, and the number of hydrogen atoms in a gram of hydrogen. When you consider the scale of the Universe, this distance is very small indeed and therefore is used mainly to describe distances within our Solar System. Ever wonder what a number with 228 zeros after it is called? I don't know exactly how big it is, I just know it's bigger than a googleplex. So we have an INSANITY number of towers, each one being multiplied allllllllll the way down to determine the height of the next one, until somehow, somewhere, at some point in a future universe, we multiply our final tower of this second feeding frenzy out…and that number—let’s call it NO I CAN’T EVEN—is the final outcome of the 3 ↑↑↑↑ 3 power tower feeding frenzy psycho festival. It is the highest number given a name under the convention that every 3 zeros you add gets a new name (thousand, million, billion, trillion, etc.) Duh! ↑↑↑ It is a massive number, in theory requiring more information to store than the size of the universe itself. A trailing zero is a zero digit in the representation of a number which has no non-zero digits that are less significant than the zero digit. Skewes' Number is a bounding value, that tends to get honorable mention in discussions about large numbers. I’m horrified thinking about it. = 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. P.S. ⋯ ) Ok, let’s look at how trailing zeros are formed in the first place. What I wrote above is just the exponent—actually writing a googolplex out involves writing a googol zeros. ⋮ And heard of a Planck volume? About 107 billion human beings have ever lived in the history of the species. 7,625,597,484,987) is our 150km-high sun tower: Why You Should Stop Caring What Other People Think. , where It was cute. While this number can easily be written as googolplex = 10 googol = 10(10 100) using the exponential notation, it has often been claimed that the number googolplex is so large that it can never be written out in full. In 1971, Graham and Rothschild proved the Graham–Rothschild theorem on the Ramsey theory of parameter words, a special case of which shows that this problem has a solution N*. Trillion is a 1 with 12 zeros after it, and it looks like this: 1,000,000,000,000. 1023 (100 sextillion) – A rough estimate for the number of stars in the observable universe. It is named after mathematician Ronald Graham, who used the number in conversations with popular science writer Martin Gardner as a simplified explanation of the upper bounds of the problem he was working on. 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. Endless sand. The search website Google did get their name from this very large number. Though you're right, 0, 1, e, i, and π are arguably the five most important constants in modern mathematics. Extra Problems for ‘Number of Zeros’ Question Type. Sets of 3 zeros … = A "1" followed by a googol of zeros. We went slowly and steadily and we ended up at NO I CAN’T EVEN. The tower goes down 150 million kilometers. The much larger number googolplex has been defined as 1 followed by a googol zeros. That’s the way numbers get truly huge. n , For example: In each case, a is the base number and b is the length of the string being bundled. Join 604,263 other humans and have new posts emailed to you. {\displaystyle n=3\uparrow 3\uparrow 3\ \dots \ \uparrow 3} However, Graham's number can be explicitly given by computable recursive formulas using Knuth's up-arrow notation or equivalent, as was done by Graham. Definition. ( , the function f is the particular sequence For any fixed number of digits d (row in the table), the number of values possible for 3 In fact, Graham’s number is practically equivalent to zero when compared to TREE(3). n To put those chances in perspective, that’s about the number of seconds in six years. $\begingroup$ Suppose you took your example number manipulated it as follows: $$2091580284\dots384901284021=2091580284\dots384901284020+1$$ $$=2(1045290142\dots192450642010)+1$$ $$=4(522645071\dots096225321005)+1$$ $$=20(1045290014\dots019245064201)+1$$ And so one, we factor. So anyway, I said above that I had been limited in the kind of number I could even imagine because I lacked the tools—so what are the tools we need to do this? If each of your steps around the Earth were represented by a dot like those from the grids in the last post, the dots would fill a 6m x 6m square. According to physicist John Baez, Graham invented the quantity now known as Graham's number in conversation with Gardner. g A googolplex is laughable. for all n.) The function f can also be expressed in Conway chained arrow notation as Let’s start off where we left off last time—. 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. ( The length of the cycle and some of the values (in parentheses) are shown in each cell of this table: The particular rightmost d digits that are ultimately shared by all sufficiently tall towers of 3s are in bold text, and can be seen developing as the tower height increases. {\displaystyle f(n)>A(n,n)} Once I had done that, I had maxed out. For example: If you multiply 9,845,625,675,438 by 8,372,745,993,275, the result is still smaller than 829. Let’s call this tower the “sun tower,” because it stretches all the way to the sun. n Since ↑↑↑ means a power tower feeding frenzy, what we have here with 3 ↑↑↑ (sun tower) is a feeding frenzy with a multiplied-out-sun-tower number of towers. Graham's number is a tremendously large finite number that is a proven upper bound to the solution of a certain problem in Ramsey theory. Hexation. 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. addition, multiplication, etc. ) every 70-digit number is somewhere between 1069 and 1070, which is really all you need to know. On Level 3, the way to go as huge as possible is to make the base number massive and the exponent number massive. If you were to write or type Rayo's number in symbol form, would it be equivalent to writing or typing a Googolplex in full digit form? Arrows. Last week, we started at 1 and slowly and steadily worked our way up to 1,000,000. A googol is a 1 followed by 100 zeroes. We’re about to enter a whole new realm of craziness, and I’m gonna say some shit that’s not okay. And when I was envisioning my huge googolplexgoogolplex number, I was doing the very best I could using the highest level I knew—exponentiation. It was once recognized as the largest number ever used in a serious mathematical proof (a title that was superseded by Graham's Number in 1977) and is notable for being larger than a googolplex. I think it would be the gravest of grave errors to punch infinity into the calculator—and this is from someone who’s openly terrified of death. 7625597484987 ) ⋯ Now those towers are Level 3, exponential strings, the same way 3 x 3 x 3 x 3 is a Level 2, multiplication string. 3 ↑↑↑↑ 4 is a power tower feeding frenzy psycho festival, during which there are 3 total ↑↑↑ feeding frenzies, each one dictating the number of towers in the next one. Graham's number. The hyperoperation sequence is a series of mathematical operations (e.g. 1021 (1 sextillion) – Now we’re even beyond the vocabulary of the weirdos. This is a special case of a more general property: The d rightmost decimal digits of all such towers of height greater than d+2, are independent of the topmost "3" in the tower; i.e., the topmost "3" can be changed to any other non-negative integer without affecting the d rightmost digits. Graham's number, G, is much larger than N: As with these, it is so large that the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume, possibly the smallest measurable space. ⋯ {\displaystyle f(n)=3\uparrow ^{n}3} The much larger number googolplex has been defined as 1 followed by a googol zeros. Well who asked you anyway? Trailing zeros are often discussed in terms of the base-ten representation of factorials. Rayo's number is a large number named after Agustín Rayo [] which has been claimed to be the largest (named) number. Graham’s number is mind-bendingly huge. So yeah. 2 in Knuth's up-arrow notation; the number is between 4 → 2 → 8 → 2 and 2 → 3 → 9 → 2 in Conway chained arrow notation. 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. {\displaystyle f^{64}(4)} ⋯ ( c But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. the biggest prime number we know, which has an impressive 17,425,170 digits.   But when I learned about Graham’s number, I realized that not only had I not scratched the surface of a truly huge number, I had been incapable of doing so—I didn’t have the tools. Therefore, it requires 10 94 such books to print all the zeros of a googolplex (that is, printing a googol zeros). Thus Graham's number cannot be expressed even by power towers of the form ↑↑ Including you but a one-foot tall version. Because it’s a common estimate for the number of atoms in the universe. 4 We bundle those 4 one-arrow 3s into 3 ↑↑ 4. n H n 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. While this number can easily be written as googolplex = 10 googol = 10(10 100) using the exponential notation, it has often been claimed that the number googolplex is so large that it can never be written out in full. f And I think we’ve heard just about enough from octillion. A recent Mega Millions lottery had 1-in-175,711,536 odds of winning. 107 (10 million) – This brings us to a range that includes the number of steps it would take to walk around the Earth (40 million steps). In fact, Graham’s number is practically equivalent to zero when compared to TREE(3). Graham's number is connected to the following problem in Ramsey theory: Connect each pair of geometric vertices of an n-dimensional hypercube to obtain a complete graph on 2n vertices. So using ↑↑↑, or pentation, creates a power tower feeding frenzy, where as you go, each tower’s height begins to become incomprehensible, let alone the actual final value. It’s what you’d use to talk about the number of carbon atoms in a 12g sample. P.P.S If you must, another Wait But Why post on large numbers. 3 The 1980 Guinness Book of World Records repeated Gardner's claim, adding to the popular interest in this number. {\displaystyle \uparrow \uparrow } In 1977, Gardner described the number in Scientific American, introducing it to the general public. The research that lead to its creation has to do with the distribution of primes. While Graham was trying to explain a result in Ramsey theory which he had derived with his collaborator Bruce Lee Rothschild, Graham found that the said quantity was easier to explain than the actual number appearing in the proof. Up arrow notation grows fast and it can create some very big numbers. 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. People say the words million, billion, and trillion a lot. $\operatorname{SA}(0)=1$, since I can't name any numbers yet, and $1$ is the smallest number in my language. Mush9 (talk) 20:58, November 20, 2016 (UTC) The ratio of Grahams number to a googleplex literally cannot be expressed in any meaningful way. What this means is that if there were a universe with a volume of a googolplex cubic meters (an extraordinarily large space), random probability suggests that there would be exact copies of you in that universe.