The Largest Numbers Ever Discovered // The Bizarre World of Googology

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What is the biggest number in the universe? What is the biggest number that any human has ever conceived of, or used in a proof, or been able to write down? In this video we are going to explore the fascinating world of googology, the study of unbelievably big numbers. We will begin near the bottom with "tiny" number like a googol (10^100) and a googolplex (10^googol) before realizing we will really need to do introduce better notation. We will then see Knuth's up-arrow notation that let's us form power towers of numbers, repeated exponentiation called tetration that is much like how exponentiation is repeated multiplication and multiplication repeated addition. And it goes up and up from there! We will see Graham's number which infamously got the googology ball rolling when it was used as an upper bound for a problem in graph theory and finally we will the unimaginable Tree(3) which comes out of a simple to state problem in graph theory and amazingly results in this incredible number.

0:00 Intro to Googology
1:13 Googol and Googolplex
2:40 Towers of Exponents
3:58 Knuth's up-arrow notation
9:00 Graham's Number
11:21 Tree(3)

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[DrTrefor]

Or should I have titled it "largest numbers ever INVENTED" #mathcontroversies:D

[davidcarter]

g(65) makes Graham's number trivially small by comparison.

[kingpatty]

As a wise man once said “no matter how enormous the number you can think of, it still closer to zero than infinity.”

[dayisnow]

Took Discrete Math with you at UC in 2019. Awesome to see your channel blow up. Only math class I ever got an 'A' in lol.

[nchiley]

I think any discussion of Graham's number should include the lower bound as well. The answer to the question they are trying to answer is somewhere between 11 and Graham's number.

[conservaliberaltarian]

In the fast growing hierarchy, Graham's number uses the 1st ordinal (omega). It falls between f-omega+1(63) and f-omega+1(64). TREE(3) uses the 6th or 7th ordinal.

[danielrhouck]

3:22 Unless I miscounted, the previous number was *much* larger than this one. Sure, googolplex is unimaginably greater than 10, but it also takes more screen real estate to write, and the extra 10s you could fit in more than made up for that.

[taxicabnumber]

A googleplex does have a physical meaning. It is the type of timescale where you will start to observe significant failures of the second law of thermodynamics. Entropy doesn't *always* increase, it *almost always* increases. In a googleplex seconds / planck times / years (pick your unit, it doesn't matter much), you might see a boltzman brain spontaneously forming.

[TIOS]

Since a random integer chosen from “all integers” has a probability of 0 of being smaller than any number you’ve defined or any number that any one ever has defined or ever will define, I contend that all defined numbers are negligibly small.

[spelpotatis]

I would love to see some more videos on this. As a googologist myself, i'd like to say that it would also be worth it to check out a bit about ordinals, as that's where the true googlogy comes in. You could discuss things like the fast grwoing hierarchy (Which converts transfinite ordinals to finite numbers), ordinal collapsing functions and stuff like that (When it comes to way to produce ordinals, again, i would recommend ordinal collapsing functions, but something called bashicu maatrix system would also be really fun to see a video about, as it's a really simple way to make extremely large transfinite numbers.)

It could maybe even be fun if you could make your own little googology series where you discuss numbers that get lrger and larger each episode, but i understand if you don't do it, because it is kind of a niche subject

[loganm]

As someone who kind of abandoned the finite numbers in googology in favour of infinite ones which I found much more interesting. I’d love to see a video on transfinite ordinals and cardinals :)

[youregonnaletityeetyouaway]

i used to be really into googology. tbh i came for the ridiculous names and stayed for the interesting maths. id love to see a video on busy beaver or BEAF

[adb]

"There is no largest finite number"
Plot twist: There is no largest infinite number either. There are infinite sizes of different infinities.

[Memer]

it may look like im crying but that is just my brain melting through my eye sockets

[jonnaking]

Discussing the googolplex with my parents had led to some intense shouting matches. Lol when I tell them that a googolplex is 1 followed by a googol zeros, they can understand how that's different from a googol. They are like " One with a googol zeros would be a googol!" Then I try to explain it to my aunt and she doesn't get it either !!

[brandonmtrujillo]

Numberphile is great but thanks for making these two numbers easier to understand

[rykehuss]

When you get into stuff like TREE(3), it really becomes more about functions and how fast they grow. This is represented in a thing called fast growing hierarchy. Numbers lose meaning at this point, and googologist are more interested in creating functions that grow faster than other functions.

[aqwaa]

the worse the audio quality, the better the video. Great work man!!

[ssarmazi]

Thank you for explaining Tree(3) so well.

[LeoStaley]

Numbers just get so big, I like to think that there are actually an infinite number collatz conjecture violations, of looping sequences with arbitrarily large numbers of numbers which do not go back down to zero. We can just never find them.