What's the biggest number you can think of...?

one one is a number representing an amount of things you probably have a rough idea of what one thing looks like and you probably have a rough idea of what a 100 things look like it's just 10 rows of 10 columns after all but what about a million a billion there are 8 billion people on Earth but can you pick picture all of them what's the biggest number you can think of I'll give you a minute to think but before you do remember Infinity is not a number but go ahead and Shout them out [Music] whenever if you've ever played an idol game like Adventure capitalist or Cookie Clicker you probably know some pretty big numbers for example I have two sep tend deilan cookies in cookie clicker that's a two with 54 zeros a humble deck of cards is also a source of a very large number there are only 52 cards in a deck but how many ways can we shuffle them well 52 factorial or 52 * 51 * 50 * 49 and so on this is roughly equal to 8 * 10^ the 67 or 8 with 67 zeros to give you a reference point it is estimated that the Milky Way contains roughly 2.4 * 10 67 atoms that means that there are more ways to shuffle a deck of cards than there are atoms in the entire galaxy now that's already an astronomically huge number but we can always add more zeros can't we a Google is a one with 100 zeros we can also write that like 10 ^ 100 going larger a Google Plex is a one with a Google zeros which we can write like 10 to the power of 10 to the power of 100 and of course we can keep stacking this exponent higher and higher but to reasonably write this we need a new form of notation multiplication is just repeated addition so 3 * 4 is equal to 3 + 3 + 3 + 3 and exponentiation is just repeated multiplication so 3 the power 4 is 3 * 3 * 3 * 3 or four copies of three tetration is repeated exponentiation so three tetr to four means a Power Tower of 3's four high or 3 to the^ of 3 to the power of 3 to the power of three working down from the top that equals 3^ 3^ 27 or 3 the^ of more than 7 trillion which is a number that has more than three trillion digits continuing further is difficult to visualize so we use what's called up arrow notation one up Arrow represents exponentiation and two up arrows represent tetration three up arrows represents repeated tetration or penetration we can also write up arrows like this with the superscript representing the amount of up arrows and n arrows represent B number of repeated n minus1 arrow operations so 3 hexed to 3 means 3 pentrated to 3 pentrated to three gr's number is up arrows taken to the extreme where g0 is equivalent to 3 hex to 3 already an unfathomable number and GN is equivalent to 3 and three with GN minus1 arrows in between meaning G1 one will have three hexad to three arrows and G2 has G1 up arrows and so on gr's number is G63 and believe it or not it has some sort of mathematical use as an upper Bound for the Grom Rothchild theorem for which I do not have time to get into today now to go higher let's play a game of trees in this game we will have different colored seeds which we connect together with branches to form trees and make a sequence of trees there are three rules the nth tree in the sequence cannot have more than n seeds no tree can contain any earlier [Music] trees and the nearest common ancestry rule also applies here so this tree would contain this one because these two nodes share a nearest common ancestor for the function tree of n we get n colors of seeds to work with and tree of n is equivalent to the maximum number of trees in the sequence we can make with n colors tree of one is one because we only have one color to work with and if we try to make another tree well that tree contains the first for tree of two we have two colors so we can have this Green Dot then just this red dot but if we try to go further we run into an issue no matter what we do each tree contains one of the previous however if before we used two red dots to make this tree then we can have three trees because the third tree does not contain the previous so tree of two is three now tree of three let's start off with the same single dot we have but then we can make this tree and this tree and that one and that one and that one and we can go on and on and on but not forever the number of trees in the sequence is still finite but it's unfathomably large so large in fact it puts crom's number to shame to give you a point of reference on how fantastically large this number is an extremely weak lower Bound for tree of three is this groms function iterated this amount of times that is just crazy sauce that's how large this number is but we don't have to stop at tree of three tree of four is provably finite and much much greater than tree of three in fact we can put any positive integer into the tree function and it's still finite what that means you could put tree of gr's number or tree of tree of three or tree of tree of tree of tree of groms number these numbers are so huge that we really start to lose the meaning of them and to be honest do they have any real world applications not really but they are pretty cool and if anybody ever asks you what's the biggest number well you can just say tree of tree of tree of tree of tree of tree of tree of tree of tree of tree ofe

Channel: Pizza Man Joe

Views: 17,949

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Id: ea4w1SDSb7Q

Length: 8min 28sec (508 seconds)

Published: Mon Jun 10 2024