Graham’s Number has been regarded as the largest number ever informal mathematical proofs for a long time.
(1). This record was recognized by the Guinness Book of Records in 1980.
How big is this number? If you store the Ge Liheng number in your brain, your brain will collapse into a black hole due to excessive information density. This is because the total amount of information entropy contained in a black hole with a size similar to that of the brain is less than the information entropy required to store the Ge Liheng number, so if you really try to put the entire Ge Liheng number in your brain, then at a certain point in time Your brain will inevitably collapse into a black hole
(2). This is probably the most scientific way to «brain holes»… Who said, «learning some mathematics will not die»? Learning mathematics is really risky.
Since most people do not have an intuitive concept of black holes, I will try to use a few more methods to let you experience how abnormal the Ge Liheng number is: Imagine assuming that every atom in the observable universe is a person, and every person is from From the first second of the Big Bang, we write a 0 every second, until today, and then we add a 1 to the front of this pile of 0s. Compared with Ge Liheng’s number, this number is still negligibly small.
Imagine the number N (3) of all atoms in the observable universe, and then imagine N∧ⁿ∧ⁿ∧…There are a total of N Ns, that is, N to the N power to the N power… to the N power, repeated N times. However, this number is no different from zero for Ge Liheng numbers.
In fact, using any concept of the real world to describe the Ge Liheng number is an insult to the Ge Liheng number, because it surpasses all the possibilities that the space and time of our observable universe can show. We can look at it this way: the radius of the observable universe is about 46 billion light-years(4.35×10∧26 meters)(4), And the smallest possible length is the Planck length, which is 1.62×10∧-35 meters (5), So the observable universe should be divided into about 8.18×10∧184 cubic spaces with a Planck length as the side length. Then there are 8.18×10∧184! possible combinations of shuffling and rearranging these spaces. The Big Bang has experienced 1.32×10∧51 smallest units of time (Planck time) (6). So if the observable universe is completely shuffled and rearranged on the smallest scale every Planck time since the Big Bang, then very roughly speaking, the upper limit of the possible combination of matter in the observable universe up to now is approximately ((8.18×10∧184)!)∧1.32×10∧51. However, the Ge Liheng number is still far greater than this number. In other words, the entire space and time scale of the observable universe is too small for Ge Liheng numbers.
Seeing this, some people will probably ask: «What the hell is Ge Liheng’s number? How could it be so big? Don’t wait for a while after blowing the cowhide!» Don’t worry, I will explain next. It’s cool for you, and it’s for you.
Although there are some professional terms in the following parts, they can be understood only by knowledge of junior high school mathematics~
Ge Liheng number is generally expressed like this:
After reading it, everyone will probably say: «What is this? A bunch of 3, a bunch of arrows and a bunch of ellipses?» No way, Ge Liheng numbers can only be written like this, ordinary mathematical symbols can not express…
Let me first introduce this arrow [formula], which is the «Gartner Arrow Notation» (7), sometimes used in iterative power operations.
Tetration is also called «Tetration Tower operation», which specifically refers to the next super operation level of power. (8)
We usually use «exponential growth» to mean ultra-fast and accelerated growth that is one dimension higher than multiplication, while iterative power is one dimension higher than «exponential growth». And ↑↑ is an iterative power operation symbol: 3↑3=3∧3∧3, the first «3» indicates the base, and the second «3» indicates how many layers the power tower has. The N∧ⁿ∧ⁿ∧… we mentioned earlier is actually ↑↑.
Now let’s compare multiplication, power and iterative power:
3×3=9
3∧3=27
3↑↑3=3∧3∧3=7.63×10∧12
This is where the iterative power metamorphosis is, it is completely an overwhelming unreasonable gap.
Okay, let’s look at the bottom layer of Ge Liheng number (there are 64 layers in total).
g1=3↑↑↑↑3
This is the layer with the smallest value, but it is also the only layer that can barely be intuitively understood by the human brain.
3↑↑↑↑3=3↑↑↑(3↑↑↑3)
And 3↑↑↑3=3↑↑(3↑↑3), so
3↑↑↑↑3=3↑↑↑(3↑↑↑3)=3↑↑(3↑↑(3↑↑…(3↑↑3)))Among them, there are 3↑↑(3↑↑3)= 3↑↑(7.63×10∧12) 3 in the right bracket.
And 3↑↑(7.63×10∧12) It is a super long index tower with 7.63×10∧12 floors. What concept? The distance between the earth and the sun is 1.496×10∧13 centimetres. That is to say if the index tower 3∧3∧3∧3…∧3 is written out and a 3 is written every 2 centimetres or so, then the index tower can basically be inserted from the earth to the sun, so we call 3↑↑(3↑↑3) for short As the «Sunburst Chrysanthemum».
If written purely in the form of an exponential tower, then:
From left to right, there are a total of «burst gerbera» index towers, and each index tower represents the number of layers of the adjacent index tower on the left. That is to say, if the index towers of this index tower are written in two centimetres apart from each tower, then the sun will be exploded again, but this time it is not exploded by a pile of 3, but by a pile that contains the number itself. Unclear 3 index towers burst…
Sun: «Who did I provoke me, my day! Why make my chrysanthemum so uncomfortable?!»
Well, with the help of the poor sun, we barely understood the first level of Ge Liheng’s number, g1. In fact, g1 should be enough to meet the abnormal conditions mentioned at the beginning of the article, and the subsequent 63 layers are all overkill and corpses.
Let us recall the whole picture of Ge Liheng number:
Everyone already knows how abnormal arithmetic symbols ↑ is. Just 4↑ has produced an unreasonably large number, so now let’s take a look at the second layer of Ge Liheng’s number, which is 3↑↑↑↑3 The top layer:
g2=3↑↑……↑3, There are g1 arrows.
Notice! Not g1 3, not g1 layer index tower, or even g1 index tower, but ↑ arrow! ! arrow! ! What is madness? ! Is this illegal? There is simply a feeling of being beautiful and not writing drafts…
Every additional arrow can make a number into a stack of index towers. Here there are more g1 arrows…
Sun: «Don’t look at me, I have already called the police.»
The Ge Liheng number has 64 layers in total, and the number of arrows in each layer is derived from the previous layer. So Ge Liheng number is simply an arrow tower of an exponential tower… right… In short, it can only be understood…
So how big is the Ge Liheng number? No one knows, no one knows how many digits this number has, and no one even knows how many digits Ge Liheng’s digits have… We only know the last few hundred digits, of which the last digit is 7. So in the future, don’t use googol, googolplex, and Ge Liheng counts every second. Of course, you have to explain it clearly…
In fact, I want to say, why only build 64 floors? Why not make a g1 layer? haha. Or to go one step further, don’t you like arrows? Seeing you so many arrows is very troublesome, I simply invented another symbol, one symbol will help you stack up all so many arrows, just like the arrow stacks up an exponential tower, isn’t it cool?…
Of course, although Ge Liheng’s number is so big that the world is shattered and the earth is shattered, and the ancient Shuo is now lawless, it is still indistinguishable from zero in comparison with Infinity. Ge Liheng number is actually the upper limit of the solution of a math problem, and the lower limit is 11, so the answer to this question is between 11 and Ge Liheng number, haha. Of course, as far as mathematicians are concerned, as long as they can prove that the upper limit is not infinite, even if it is a Ge Liheng number, then this problem is half solved.
Human abstract thinking is really interesting. A number so big that it is almost meaningless in the physical universe was invented to solve mathematical problems.
Mathematicians are very cute, don’t tease them.
In addition, this article belongs to the soft science popularization of novelty hunting. I am not a relevant major, so I ask professionals to write about it…
ANNOTATION:
(1) Now this record has been broken by some other numbers, such as TREE(3), but Ge Liheng numbers are still the most famous. TREE(k) seems to be a function, TREE(1)=1, TREE(2)=3, TREE(3) is greater than Ge Liheng’s number… It’s just headstrong…
(2) My description here should not be professional and rigorous, but it is roughly what it means. In fact, even the small number of googol is enough to collapse the brain into a black hole.
How To Visualize Graham’s Number, If You Dare
(3)How Many Atoms Are There in the Universe?