What is the Largest Number?

before we discuss what exactly is the largest number we need to set a few conditions in place so that things don't go crazy okay condition number one the largest number must be useful in some way even if it has a very very very very tiny amount of use that would be fine as long as there is some use to such number second of all the number must not be arbitrary for example 1 million and how did you get to that 1 million I know you said it but how did you get to that 1 million so the number must not be arbitary third of all you can't say the largest number anyone could think of plus one that is not allowed and that's cheating okay fourth of all and most important of all no Infinity the number must be finite now that last point I'd like to stay on for a bit because Infinity is a very interesting concept it behaves much more differently than your normal numbers to explain how that is let me give you something called Hilbert's Paradox of the Grand Hotel let's say that you are the manager of a hotel with an infinite number of rooms and your hotel is fully booked there's an infinite number of guests let's say that someone shows up and would like to check in in your hotel how do you solve this problem do you tell them that it's fully booked well you don't have to because your hotel is very special how do you solve this problem you tell the guest in room number one to move to room number two and you tell the guest in room number two to move to room number three and you tell the guest in room number three to move to room number four you can keep doing this throughout the entirety of the hotel and now you have room number one empty and you can tell the the guest to check in that that means Infinity is equals to infinity + 1 yeah very weird but it gets weirder let's say that an infinite number of guests show up and would like to check in in your hotel do you tell them that it's fully booked no again your hotel is very special how do you solve this problem you tell the guest in room number one to move to room number two and you tell the guest in room number two to move to room number four and you tell the guest in room number three to move to room number six and then you tell the guests in room number four to move to room number eight and you keep doing this throughout the entirety of the hotel and now you have an infinite number of rooms and you can accommodate an infinite number of guests that mean that means infinity plus infinity equals infinity you see the problems with infinity good we can move on then so with that said let's try and find the largest number first we need an arbitrary number I know I've said that the largest number cannot be an arbitrary number but this is only a comparison point we're going to try and find use useful numbers and see whether they are larger than the arbitrary number clear good so let's pick a Google a gole is our comparison point a gole is 10 to the power of 100 or it is one with 100 zeros next to it okay so is the number of atoms in your body larger than a gole the number of atoms in your body are around seven Octan so no it is not larger then a go Google is it the number of atoms in Earth that would be about 133 quilon and no again not larger than a Google how about the number of atoms in the observable universe that would be about 100 quadrillion Vilan nope still not bigger than a Google this is the number of atoms in the observable universe and we can't even be a Google however there are other areas of mathematics that could be useful that could beat a Google one way is through this crazy Rubik's Cube over here which has this many ways to arrange that is a number that is larger than a Google so we beat a Google can we beat a number even bigger than a Google can we beat a Google Plex which is 10 to the power of a Google can we do that let's see how about the number of different possible universes calculated by and where they ended up with 10 to the power of 10 to the power of 16 nope not larger than a Google Plex how about once they assume that if we were not limited as observers to distinguish more universes how about that number that number is 10 to the power of 10 to the power of 10 to the power of seven and that number is larger than a Google Plex so let's pick our next arbitrary number how about a Google plexion which is 10 to the power of a Google Plex can we find a useful number larger than it now Beyond this point I will not get into too much detail about why the number that I will mention are useful all you have to know is that there is some usefulness to them if you'd like to know why have put links in the description anyway with that let's try and beat a Google Plex which is 10 to the power of a Google Plex so how about this guy's numbers here there was one that was 10 to the power of 10 to the power of 10 to the power of 34 so no it doesn't be a Google plexion but then there was 10 to the power of 10 to the power of 10 to the power of 963 so we beat a Google plexion Beyond this point I'm not going to mention any arbitrary comparison Point like a Google Google Plex or Google Plex because the numbers that I will mention useful numbers still useful numbers are so big so large so ridiculously huge that you can't even write them down that's how big they are about to get so with that said we have next up grams number so what exactly is grams number GS number is the upper bound to a certain solution in something called Ramsey Theory this bound is between 11 and GRS number the solution to this problem Falls in that range as I said I'm not going to get too technical about why this number is useful if you want to understand it link is in the description G's number is so big it is difficult to explain in what with you using normal mathematical notation so we need to learn a new form of mathematical notation called not up AR notation in order to express how ridiculously big gr's number really is so what exactly is not up arrow notation let's begin with a three three okay and put an up Arrow that's not up or notation and another three uh you will find out that this is very simple this is three to the power of 3 the answer to that 27 nothing too complicated about it so far so far uh let's now put two arrows three Arrow Arrow three that would be three Arrow three Arrow three that's three to the power of three to the power of three and the result of that is something like 7.6 trillion as you can see that we went from 27 to 7.6 trillion with only adding one additional Arrow um but things after this point get start to get a bit crazy so let's take three now and put three arrows between it and another three that's three Arrow Arrow Arrow three cool now this means this is three Arrow Arrow three Arrow Arrow three that's three Arrow Arrow 7.6 trillion and this means this is 3 to the power of three to the power of three to the power of three to the power of three to the power of three to the power of 3 to the power of three and you keep saying this 7.6 trillion times do you see how the number just exploded from just having one Arrow to three in order to find G's number you have to begin with three Arrow Arrow Arrow Arrow three that's four arrows so three Arrow Arrow Arrow Arrow three we already know that three Arrow Arrow Arrow three with three arrows is an unbelievably large number so you can see that the number that you'll get from having four arrows this is the reaction that you would have to it speechless but gr's number is is even much much larger than that okay so what you need to do is you need to take the result of three four arrows three and you call that G1 I would suggest you would sit on a toilet right now because things are about to get insane not just crazy so what happens next what happens next uh you you take G1 which is the result of three Arrow Arrow Arrow three four arrows of three and then call that G1 write down another three are you sitting down and you put next to it a G1 number of arrows and once you finish you write down another three big big really big you're not done you're not done you you still have to uh progress a bit more then what you need to do is take the result of that and you call it G2 cool you take G2 again you write a three and then a G2 number of arrows between it another three okay you take the result of that and you call it G3 you take G3 write down a three G3 number of arrows three you take the result of that call it G4 and so on and so forth you keep going until you end up with g64 and g64 is Gams number you can see how just big Gra number really is it's ridiculously big really really really big and what's surprising about this is that there are even larger useful numbers than Graham's number now Beyond this point Beyond this point I have to confess here my understanding of those numbers of numbers larger than graes number stars to collapse I don't get them I don't understand them so I will have to trust my friends our friends at googology Wiki which is a Wiki dedicated entirely to very very big numbers and uh yeah so let's go into that be prepared for it be prepared let's go one of those numbers is 33 33 is so big so massive that I feel like I'm going to suffocate from how big 33 really is you know at least with Way's number we could visualize how we got to the result through no up AR notation but no up AR notation is not enough to express how large 33 really is there is something called chained arrow notation which is like not up arrow notation but on crack you could get a number that's close to G's number using this notation through writing this down this is a number close to Gram's number but even chain Aron notation 33 would laugh at it 33 is that large it is that big now I tried I tried to find a simple explanation of 33 and as I've said before it is in unbelievably difficult to comprehend numbers after grams number 33 is so large I've looked up I really looked up people that try that try to explain what 33 really is in in layman terms but I'm convinced I'm really convinced that it is impossible to really explain what three really is it's an unbelievable number and what's really unbelievable is that there is a number even larger than tree three and this is where we arrive at loader's number loader's number is Big it's large it's stupidly large it's horrendously big it is much larger than 33 it is much much larger than Graham's number you see at this point all I have for you to give you some sense of how big numbers are at this point is just to explain it to you through words you can't even imagine the process for getting these numbers at least I could tell you how loader's number came to be it was the result of a contest that was held the condition was to write a c program in 512 characters or less that will generate the largest possible output the winner of which is a guy called W loader hence why it's called loaders number but even then there's a larger number than that and this is where we arrive at triu number rise number is bigger than anything else I've mentioned so far it is very very very large that's how big it really is so how did ru's number came to be well there was this contest an MIT contest that pitted two people against each other Adam Elga and Austine rayu there was even a poster for the competition um now the point of this competition was to find the largest number possible but according to certain rules first you could not use primitive semantic vocabulary for example you couldn't say the smallest number bigger than any number named by a human so far that is not allowed you also had to explain any sort of unusual notation to the audience for example I explained to you earlier something called Nots up oron notation which is probably something that is new to you so if they used any kind of new notation they had to explain it to the audience they also had to fit this expression that would name the largest number in a regularized chalkboard and finally no Infinity so as you can see the rules are kind of similar to what I mentioned earlier in the video now that we know the rules this is what happened during the competition rayu went first he approached the board and he wrote one not the largest number he was just trying to start okay and then Elga came in and he retaliated by writing 9999 9 N9 and so on and so forth then rayu countered by writing one one one one one one one he actually got a larger number because one takes less space than a nine so he could write more of them on the board and then Elga came back with a brilliant idea instead of the ones he started putting exclamation marks all the way until he was left with only two ones so now you had an 11 with a lot of exclamation marks next to it now of course exclamation marks indicate that this is a factorial operation so the number that you get at the end of it is is pretty big things started to get complicated afterwards and weird notations started to get used until rayu one how with the following expression the smallest number bigger than any finite number named by an expression in the language of set theory with a Google symbols or yes now that is not the final expression if you would like to find the final expression I would suggest you read an article written by R himself link is in the description where he takes that expression and dissects it down to what gives birth to ru's number now you might have this thought swimming about that says hold on a minute didn't you mention in the beginning of the video that the largest number must be useful in some way it must not be arbitrary Ray's number came as a result of a contest that wanted to find the largest number how is it useful and how is it not arbitrary to that here is my argument you can be convinced by it and you cannot be convinced by it it's up to you at the end of the day now that contest was actually an exercise to show how how philosophy and Mathematics actually intersect with one another so there's your usefulness right there and when it comes to be to it not being arbitrary you have to remember that the contestants they had to work with conditions in order to find the largest number so in this case the number isn't exactly arbitrary and it is useful however this is at the end of the day my argument at the same time they also wanted people to be entertained and that contest entertained them so there is more usefulness for you right there I don't know it's up to you it's up to you at the end of the day I just wanted to actually point that out for you Austine rayu you know the guy that came up with Ray's number is actually mainly a philosopher not a mathematician okay so it is very interesting how a guy that's involved in philosophy actually came up with a number that is so big so large like ru's number okay now I also have to point out that in an interview rayu mentioned that after the contest he wanted people to get out and have this idea their had that how can I come up with a number that is larger than ru's number and you probably can still do that go ahead read the article that rayu has written and if you can understand how Ray's come how Ray's number came to be try and come up with a number that is larger than ru's number but for now ru's number is the largest number and with that that has been my take what exactly is the largest number thank you very much

Channel: Sharkee

Views: 1,548,690

Keywords: Numbers, Math, mathematics, Googol, Googolplex, Googolplexian, largest number, biggest number, scariest number, skewes number, skewes, infinity, hilberts paradox of the grand hotel, paradox of the grand hotel, paradox of the hotel, grahams number, graham number, Tree 3, TREE 3, Loaders number, Rayos number, Agustin Rayo, adam elga

Id: e0xJwdcpATM

Length: 19min 10sec (1150 seconds)

Published: Sat Oct 26 2013