Giant numbers

what is the largest number you've counted to 100 400 1,000 1,000 is pretty big but what if we wanted to find a giant let's go giant hunting we'll start with an old favorite one we can just count up from one and eventually we'll reach something really big let's see 1 2 3 4 5 6 7 8 9 10 11 12 hm this isn't very quick sure we could reach something gargantuan eventually but we would be counting for a very long time let's see if there's a faster way to do this if we add a number to itself we can grow it much quicker this is called multiplication let's go from Two and use multiplication by two instead of counting by one 2 4 8 16 32 64 128 this is better but we can go bigger if rather than multiplying a number by two we multiply it by itself we can grow even faster 2 * 2 is 4 * 4 is 16 * 16 is 256 * 256 is 65,536 what if we multiply a number by itself multiple times turning that 2 into a three turns it to 2 * 2 * 2 makes 8 this is called raising a number to a power this is even better because now we don't actually need need to raise the number itself we just raise the power let's use 10 10 the^ two already gets us to 100 but let's keep going 10 ^ of 3 is 1,000 10 ^ of 6 gets us up to 1 million these numbers are already very very hard for a human to comprehend but let's go higher 10 ^ of 9 is 1 billion 10 power of 12 is 1 trillion 10 the power of 15 is quadrillion 10 the^ of 18 is a quintilian 10 the^ of 30 is called a nonilon 10 to the power of 80 is roughly the amount of particles in the observable universe but math doesn't care let's go larger 10 the power of 100 which is the same as one followed by 100 zeros is a Google what if we raise 10 to the power of a Google 10 to the power of 10 to the power of 100 the legend legendary goall Plex our first real giant which can be written as a one followed by one goall zeros H this is good but after a while all these 10 to the power numbers can start to blend into each other what we really need is a way to grow the way that we are growing let's try this take 2 to the power of two what if we add another to the power of two here 2 to the power of 2 to the power of two we could add another 2 to the power of 2 to the the power of two to the power of two what if we have a number to increase the amount of powers we have tetration the tetration number controls the height of a number's Power Tower two tetr to 5 is 2 to the power of two to the power of two to the power of two to the power of two this is very good but what if we wanted to tetrate a number's tetration we can build the tetration Tower just as we built the Power Tower we've reached hyper operations the next hyper operation is a pentation which controls the height of the tetration Tower and is written with three up arrows hex is another up arrow and another layer of maths we can do let's find a new giant start with three hexat to three this number by itself is already far beyond human comprehension it's so large that it would be n impossible to write with regular numbers or even Powers pull this number g0 and then raise three to the hyperoperation of g0 that is to say put a 3 then g0 arrows then another three call the resultant number G1 repeat this process with G1 to get G2 you see the pattern 3 raised to the hyperoperation G2 yields G3 this number grows and grows repeat until you reach G63 then do it one more time three raised to the hyperoperation of G63 gives us g64 Ram's number Graham's number was at the time of its imagination the biggest number to ever be used in a mathematical proof we could even do more with Graham's number if we wanted Graham's number * 2 Graham's number squared Graham's number pated Gram's number hyper operated to Graham's number but this is getting a little bit out of hand so let's calm down by drawing some lovely pictures of trees because this is a maths video though let's make some mathematical rules let's say that we can draw trees by connecting different colored Dot and that we can't have more dots than the number of trees we've made plus one so the first tree can't have more than one dot the second can't have more than two and so on another rule we can make is that no tree can contain previous trees if we made thise tree we couldn't make this one because it contains the first tree if we want we can make a function called tree which tells us how many trees we can make with a certain number of Colors Let's see what different numbers of colors give us for tree one we can draw a single Dot and then no more since any other trees would contain this dot with two colors tree 2 we can make three whole trees but no more how many trees do you think we can make with three three colors 5 12 no tree 3 is much bigger tree3 is so unimaginably large in fact that it completely dwarfs every other number seen in this video so far a new giant tree 4 dwarfs tree three to an even greater extent than tree 3 dwarfed Graham's number and we can keep growing trees tree 10 tree Google tree Graham's number tree tree tree3 the tree of three is the entryway to some of the vastest numbers ever imagined these numbers are so large that they have lost all meaning humans were not meant to speak Giants they are too big too vast no matter how hard you tried you could never truly grasp any of these numbers you could never truly grasp the tree of three you could never grasp Gram's number or a Google you could most likely never even grasp something like 1 million we cannot meet Giants but we may Glimpse them on the horizon and bask in their radiance [Music]

Channel: stuff | An abridged guide to interesting things

Views: 16,225

Keywords: math, maths, big numbers, TREE(3), Graham's numer, Googol, Googolplex, Existencial dread, Infinity, Tetration, Hyperoperations, Lergest number

Id: WhtDXEIrR0Y

Length: 7min 27sec (447 seconds)

Published: Sun Jul 07 2024