You CANNOT imagine this number! (Graham's Number)
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- Опубліковано 19 чер 2024
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⏱ Timestamps:
00:00 Intro
01:03 1: Standard Notation
01:30 2: Scientific Notation
03:48 3. Factorial Notation
06:23 4. UP-arrow Notation
11:17 5. Stupidly large numbers...
12:00 6. Unimaginable...
13:26 7. The Final Boss
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And yet these unimaginable numbers are still closer to zero than to infinity, i still can't wrap my head around it.
Yeah, that's a very accurate statement! :D Thanks for that :)
Depends on the metric. If you use d(x,y)=atan(y/x) then the midpoint between 0 and infinity is 1.
@@vizart2045 That is a true and interesting statement. However, when are such measures for distance used to compare points on a number-scale? I'm curious.
@@PenandPaperScience You can project the numberline to a circle, where infinity is exactly opposite to 0 and 1 is halfway between them. I am not an expert, just a mathematician with some knowledge of this.
No matter how big of a number you think of there's still an infinite number of numbers that are bigger than it
Phew!
you are amazing in mathematic. SUper.
Thank you! :)
Beyond Graham's Number, you'll need transfinite ordinals in the fast-growing hierarchy (diagonalisation of functions to create something faster than any function that can be expressed via strictly finite terms) in order to express how big they are, with the lower bound of TREE(3) being very far into this.
Wow, that's very interesting, I didn't know that.
Thanks for teaching me something :D
@@PenandPaperScience an easier way to look at the scale of Googology is via the Psi Levels (from Googology Wiki) where all numbers from 0 to 10 (some say 0 to 6) are in level 0 and then level 1 is double digits to a million. Thereafter, level 2 covers everything from there to a million digits, so pretty much every number that is practically feasible within our observable universe is barely into level 2 within this insanely growing scale, with the Googolplex we all know and love (a number that won't fit in our observable universe if written down in full) being within level 3. Remember, as you go up one level, the numbers increase at an incomprehensibly faster rate than the level before it to the point where each successive level is further apart from the previous than that from level 0. For example, level 5 is much further away from level 4 than level 4 is from 0 and so on.
Okay, now we got that out of the way, what if I told you Graham's Number (between the first transfinite ordinals omega and omega+1 in FGH) is level 12? Yeah, that's right and what's more, level 13 is reached through iterating that G(n) function that build's Graham's number, so basically G(G(64)) or even G(G(G(64))) and so on until you get a Graham's number of these iterations, which is omega+2 and hitting level 14 or 15. Then this gives a new function which iterates itself even further to reach the next level and so on until you reach omega+omega (omega*2) which diagonalises all of this (level 17). Rinse and repeat all those hyperoperations until you're at omega^2 (level 24) and this carries on until you reach omega^omega (level 31). Thereafter, you build power towers of omega (there aren't enough dimensions to express the layers of iterations happening here) until you reach one infinite high (each time you exhaust all finite numbers you diagonalise similar to how n^1, n^2, n^3, etc. becomes n^n but on crack) which becomes the next ordinal epsilon0 (level 51).
Rinse and repeat with epsilon0 similar to omega and you reach epsilon1 (infinite power tower of epsilon0) until you get epsilon-epsilon, then epsilon-epsilon-epsilon, etc. Eventually, you have an infinite nesting of epsilon (pentation on steroids) which becomes zeta0 (level 70). Repeat every hyperoperation (diagonalise through each one) with zeta the way you did with epsilon, you reach eta0 (level 80). After that, you'll need phi-numbers or else you'll run out of Greek letters (epsilon0 is phi1, zeta0 is phi2, eta0 is phi3, etc.). Hence an infinite nesting (and I'm talking diagonalising through 5 orders of hyperoperation each time) of eta0 is phi4 and then phi(n) is an infinite nesting of phi(n-1) until you have phi(phi(4)) and then you iterate the phi(phi(phi(so on...))) an infinite times (further diagonalisation) until you reach the insane gamma0 (level 100).
Letting all that sink in, I bet you're thinking "surely we'll have passed TREE(3) right?" WRONG! Even diagonalising the iterations-upon-iterations of fgamma0(n) [an incomprehensibly fast growing function] to reach TREE(3) would be much slower than reaching Graham's Number via mere COUNTING, forget addition! This is because TREE(3) is all the way up in level 120 of this Googology scale requiring the Veblen Ordinals to actually approximate its lower bound; that's right, I mean 20 levels beyond even the gamma0 ordinal. For comparison, 1 and Graham's Number are only 12 levels apart. On the other hand, TREE(3) is a whopping 108 levels ahead of Graham's Number. Meanwhile, SCG(13) at level 170 is 50 levels ahead of TREE(3) and (brace yourself for this one) then Rayo's Number is all the way at level 700.
TL;DR: If all numbers from 0 to 10 are level 0, double digits to a million in level 1 and level 2 runs up to a million digits, with level n to n+1 being greater than level n to 0, G64 is in level 12 and TREE(3) is all the way up in level 120. Basically, reaching TREE(3) using Graham's function (even iterating its iterations) will be FAAAAR slower than counting to Graham's Number and that's not even an exaggeration. In fact, it's actually a MASSIVE understatement!
@@AymanTravelTransport Are there mathematical problems that need this notation. Or is it just available in case there are solutions to mathematical problems that cannot be conviently expresed without it.
@@AymanTravelTransportwhere in the world did you get these levels from?
Tree(3) isn't "very far into it", as Graham's number is in the region of Omega+1 and tree(3) around Omega+2. And 'the Omegas' are only one family of many within the Fast Growing Hierarchies.
Graham's Number is probably the most popular super huge number after googolplex, but there are a bunch of numbers larger (however they are not used in any mathematical proof like graham's number).
TREE(3) is another kinda famous number that's bigger than G(64). It wasn't used for a mathematical proof, but it was used in an example for a proof, iirc.
SSCG(3) or SCG(13) is in a similar boat to TREE(3).
Busy Beaver numbers are much larger than all of these, and have to do with uncomputable sequences.
Rayo's Number is the grand champion of all big numbers (well, not really but it's most known)
Very interesting! Yes, I chose Graham's number because it was actually a solution to a tangible problem in combinatorics mathematicians had at the time. I might do a video on this problem :) Thanks for the additional information. ^^
Sam's number is bigger than Rayo's Number, right? what's the biggest number we got a name for?
@@they-call-me-martin3837 Sam's number is nothing. it's just a meme made by some dude several years ago that somehow snuck its way into people's brains.
Large Number Garden Number is generally considered the biggest number atm.
@@they-call-me-martin3837Sam's number doesn't exist, it's just a joke. Biggest number is generally agreed to be Large Number Gardeb Number rn
Thankyou! This is the clearest explanation of up-arrow notation I've seen on UA-cam. Great job!
You're very welcome! I'm glad I could help :))
The largest one is Rayo's Number right?
Well, I haven't heard of that one, will look it up! :D Thanks :)
@@PenandPaperScience yeah sure!
@@icebeartwo So it seems Rayo's number is indeed vastly larger than Graham's number. This is because Rayo's number defines large numbers by the number of symbols that are used to describe them (an up-arrow can also be a symbol). However, I did not find a "practical" application for this number. Still, incredibly interesting! :)
@@PenandPaperScience Yup, you are absolutely correct.
@@PenandPaperScience The practical part is that it won a competition.
Simple: 2022 - 2024 = -2. 2022 will be a factor of Graham's number of years ago within eight months.
I'm not quite sure I get what you're saying.
that's funny i was in bed and listening to the videos and never had herd this one before. My first thought was Grahams number when you started and asked us. of course i had no idea the title of this vid. So yea not all people think of small numbers.
Yeah, that's a good point :D
Though, I think you are one of the few ;)
I think I read on Wikipedia awhile ago, they have figured out the last 16 million digits of G64.
It is so funny to me that they know the *last* figures of this number, but have no clue about the first ones. :D
@@PenandPaperScience, I have also heard that the actual number of digits in G64 is just as impossible to comprehend as the number itself. Yet, even G64 is microscopically tiny when compared to TREE(3). That number is so big, they haven't calculated even what a single confirmed digit is. I would imagine it has millions and millions each of the digits 0-9, but where they are and in what order is unknown. By the way, my favorite number starts with 3.14 and goes on forever after that. Saw a video about it. They figured out the 100 TRILLIONTH digit after the decimal point is 0.
@pacman52280 The number of digits in Graham's number is effectively as large as the number itself. If you take the number of digits in G64, and then take the number of digits in thar number, and keep repeating the process, where you take the number of digits in the new number, you could do this 1,000 times and still be left with an unimaginably large number. In fact, the number of times you would need to do this process before you would reach a calculatable number is in itself too large to comprehend. The number of times you would have to do it would itself have to be reduced too many times to comprehend. Also, there's probably no way to ever find out any digit of TREE(3), since there's no formula to reach the number.
It is g1 that equals 3 ^^^^ 3, not g0, I think…
en.wikipedia.org/wiki/Graham%27s_number
Otherwise, good video.
Oh, you are of course correct! It should be g1 instead of g0. My bad. Thank you for pointing that out. :)
_3(3) is an unnecessarily large factor of Graham's number.
Tan 90 degree - 1 is the greatest number
That's infinity - 1, which is still infinity, strictly not a number :D
But good try ;)
Great video!!!
Ok, there is a little problem defining g(0), this has been discussed in the comment section already. but I might as well add a drop in the ocean.
If we define g(0) equal to 3↑↑↑↑3 then the grahm number is g(63). and if we define g(1) equal to 3↑↑↑↑3 then Grahm number is g(64).
Thanks!!
Yes, I think your are correct. Though it is not fashionable to define mathematical constants differently than what is the norm.
🚀 Those are some big numbers! :)
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There’s no such thing as g 0 it starts at g1
You are indeed correct! That's an error on my behalf, thank for pointing that out :)
there is a g0 which is literally 4
That was excellent. I'm glad I found your channel.
@@xyz.ijk. Awesome! Good to hear! :))
@@Itzthevice what is g-1
So the number of the atoms that compose the earth is 10^50... and the total atom in the universe is only 10^80?
How can this be possible? 🤔🤔
The difference between these two numbers is unfathomably large! 10^30 to be exactly. What is says is that an atom compared to the earth is still much smaller than the earth compared to the total mass in the universe.
Also, empty space does not contain that much atoms; stars and planets are really, really far away from each other.
@@PenandPaperScience Okok thanks for the explanation 😁😁
I think what you meant to ask was how can the particles in the observable universe only be 10^80 when the number of grains of sand you can fill the observable universe with is 10^90? Well the answer is because the universe is mostly empty space. The number of particles in the universe is no where near the number of particles that could fill up the universe.
Let X=3^^^3, a power tower of 3 with 7.6T layers.
g0=3^^^^3=3^^^3^^^3=3^^^X
=3^^3.......3^^3, a tower with X layers, each layer is another (progressively taller from top down) tower!!!
I have no way to imagine g1, let alone g64!!!!
Exactly! It's mind boggling! :D
posted this on r/googology
Oh cool! Thank you :)
Cool g64
I know, right! :)
F(1)=3
F(x)=3^f(x-1)
3^^^3=f(f(3))
3^^^^3=f^f(f(3))-1(3)
tree of grahams number tetrated to tree of grahams number
Mind blowing :D
But, does your number have a practical use? :)
3↑↑↑↑3 is the first number in the Graham series, not the zeroth.
Yes, you are correct, that's a mistake on my part. Thanks for pointing it out! Years ago you could add annotations to videos correcting errors, but that feature has been removed.
10:28 is this an error? How does the 3 ↑ ↑4 has the same digits with 3 ↑ ↑3?
3 ↑ ↑3 is roughly 7.6 trillion (13 digits), whereas 3 ↑ ↑4 has 3.5 trillion *digits*.
They definitely do not have the same number of digits :D
Does that answer your question?
g1 is 3 arrow arrow arrow arrow 3
You're completely right, thanks for pointing that out! (:
3, g63 flèches 4 et beaucoup plus grand que 3, g63 flèches 3.
3 four arrows 3 is G1, not G0
You are completely correct! That's my mistake. Well spotted :)
Alrigt Bet... Done I Imagined It Wanna Know Why Cause Imma Be A Theorical Scientist After Medical School
Why do you want to become a Theoretical Physicist? And why did you choose medical school?
@@PenandPaperScience Medical School Because I Need To Control The Degree That I Suffer From My Choices Of Making Complicated Medical Mistakes and Theoritical Scientist Because They Will Provide Solutions for Cool Projects Like Space Exploration, Or High Valued Products, & They Always Learning New Concepts That Will be Taught to Medical Doctors which Means I Will Be Exploring a lot of Uncharter Territory Allegedely
at 10:27 you said that 3 double up arrow 3 is a number with 3.5 trillion digits, which is wrong -- it only has 13 digits -- and at 2:15, you counted 12 zeroes and came up with 15 zeroes. as this is the first of your videos I've seen, let it be the last; it's getting in the way of my learning 😁 cool academic accent, though.
Hi there, thank you for your corrections, you are right, I am aware of them.
You will find less errors in my physics videos, as that is my field of expertise. Best of luck on your learning endeavours (:
Infinity is not a real number, so it'sn't allowed.
Correct.
I thought Scientfic notation its : 1e100(googol aka 10^100(e up arrow notation))
Grahams number is much larger than one googol. A googol is 10^100 (100 digits). A googol-plex is a 1 with googol zeros! :D
@@PenandPaperScience oh well theres googolplexian basically its a 1 followed by a googolplex zero
@@PenandPaperScience well Tree(3) seeing G belike
Oh wow a baby =)
@@thiennhanvo2591 :D Tree(3) is huge!
@@PenandPaperScience also dont forget that sscg(3) exist(scg and sscg grow at the same rate but sscg is more easy than scg)
Googolplex is the largest number thar makes any practical sense or is useful in reality. Anything higher is a theoretical function. Just my 2€
In fact, a googolplex would be a prime example of a non-useful large number: it's only significance is that it's a 1 with a googol zeros. Graham's number on the other hand is a solution to a tangible, albeit theoretical, problem in mathematics (more specifically Ramsay theory).
@@PenandPaperScience Thanks for the input...sometimes these numbers can boggle the mind!!