Mathematics that I like. Graham's unimaginable number The largest number in the world of graham

epigraph
If you look long into the abyss,
you can have a good time.
Mechanical Soul Engineer

As soon as a child (and this happens somewhere around three or four years old) understands that all numbers are divided into three groups "one, two and many", he immediately tries to find out: how much is a lot, how much differs from a lot , and can there be so many that there is no more. Surely you played an interesting (for that age) game with your parents, who will name the most more, and if the ancestor was no more stupid than a fifth grader, then he always won, answering "two million" for every "million", and "two billion" or "a billion plus one" for every "billion".

By the first grade of school, everyone knows - numbers infinite set, they never end and there is no largest number. To any million trillion billion you can always say "plus one" and win. And a little later comes (should come!) The understanding that long strings of numbers in themselves do not mean anything. All these trillions and billions only make sense when they serve as a representation of a certain number of objects or describe a certain phenomenon. There is no difficulty in inventing a long number, which is nothing but a set of long-sounding numbers, there are already an infinite number of them. Science, to some extent figuratively, is engaged in looking for completely specific combinations of numbers in this boundless abyss, adding to some physical phenomenon, such as the speed of light, Avogadro's number, or Planck's constant.

And the question immediately arises, what is the largest number in the world that means something? In this article, I will try to talk about a digital monster called the Graham number, although strictly speaking, science knows even more numbers. Graham's number is the most publicized one, one might say "heard of" by the general public, because it is quite simple in explanation and yet large enough to turn one's head. In general, here it is necessary to declare a small disclaimer (Russian warning). It may sound like a joke, but I'm not joking. I'm speaking quite seriously - meticulous picking in such mathematical depths, combined with the unrestrained expansion of the boundaries of perception, can (and will) have a serious impact on the worldview, on the positioning of the individual in society, and, ultimately, on the general psychological condition picking, or, let's call a spade a spade - opens the way to shiz. It is not necessary to read the following text too carefully, it is not necessary to imagine the things described in it too vividly and vividly. And don't say later that you weren't warned!

Fingers:

Before moving on to monster numbers, let's practice on cats first. Let me remind you that to describe large numbers (not monsters, but just large numbers), it is convenient to use scientific or so-called. exponential notation.

When they talk, say, about the number of stars in the Universe (in the Observable Universe), no idiot bothers to calculate how many of them there are in the literal sense with an accuracy of last star. It is believed that approximately 10 21 pieces. And this is a lower estimate. This means that the total number of stars can be expressed as a number that has 21 zeros after one, i.e. "1,000,000,000,000,000,000,000".

This is how a small part of them (about 100,000) looks like in globular cluster Omega Centauri.

Naturally, when we are talking about such scales, the actual numbers do not play a significant role in the number, after all, everything is very conditional and approximate. Maybe the actual number of stars in the universe is "1,564,861,615,140,168,357,973", or maybe "9,384,684,643,798,468,483,745". And even "3 333 333 333 333 333 333 333", why not, although it is unlikely, of course. In cosmology, the science of the properties of the universe as a whole, such trifles are not fooled. The main thing is to imagine that approximately this number consists of 22 digits, from which it is more convenient to consider it as a unit with 21 zeros, and write it as 10 21. The rule is general and very simple. What figure or number stands in place of the degree (printed small print above 10 here), so many zeros after one will be in this number, if you paint it in a simple way, with signs in a row, and not in a scientific way. Some numbers have "human names", for example 10 3 we call "thousand", 10 6 - "million", and 10 9 - "billion", and some do not. Let's say 1059 doesn't have a common name. And 10 21, by the way, has it - it's a "sextillion".

Everything that goes up to a million is intuitively understandable to almost anyone, because who does not want to become a millionaire? Then some problems start. Although a billion (10 9) is also known to almost everyone. You can even count up to a billion. If only after being born, literally at the moment of birth, begin to count once a second "one, two, three, four ..." and do not sleep, do not drink, do not eat, but only count-count-count tirelessly day and night, then when 32 years hits, you can count up to a billion, because 32 revolutions of the Earth around the Sun take about a billion seconds.

7 billion is the number of people on the planet. Based on the foregoing, it is absolutely impossible to count them all in order during a human life, you will have to live more than two hundred years.

100 billion (10 11) - how many or so people have lived on the planet throughout its history. McDonald's sold 100 billion hamburgers by 1998 in its 50 years of existence. 100 billion stars (well, a little more) are in our galaxy Milky Way and the sun is one of them. The same number of galaxies is contained in the observable universe. There are 100 billion neurons in the human brain. And the same number of anaerobic bacteria live in each reader of these lines in the caecum.
A trillion (1012) is a number that is rarely used. It is impossible to count up to a trillion, it will take 32 thousand years. A trillion seconds ago, people lived in caves and hunted mammoths with spears. Yes, a trillion seconds ago mammoths lived on Earth. There are about a trillion fish in the oceans of the planet. Our neighboring Andromeda Galaxy contains about a trillion stars. A human is made up of 10 trillion cells. Russia's GDP in 2013 amounted to 66 trillion rubles (in 2013 rubles). From Earth to Saturn, 100 trillion centimeters and the same number of letters in total have been printed in all the books ever published.

A quadrillion (1015, one million billion) is the total number of ants on the planet. Normal people do not pronounce this word out loud, well, admit it when you last time did you hear "a quadrillion something" in the conversation?

Quintillion (10 18, billion billion) - so many possible configurations exist when assembling a 3x3x3 Rubik's cube. So is the number of cubic meters of water in the world's oceans.
Sextillion (10 21) - we have already met this number. The number of stars in the Observable Universe. The number of grains of sand in all the deserts of the Earth. The number of transistors in all existing electronic devices of mankind, if Intel did not lie to us.

10 sextillion (1022) is the number of molecules in a gram of water.

10 24 is the mass of the Earth in kilograms.

10 26 is the diameter of the Observable Universe in meters, but it is not very convenient to count in meters, the generally accepted boundaries of the Observable Universe are 93 billion light years.
Science does not operate with dimensions larger than the Observable Universe. We know for sure that the Observable Universe is not the whole-all-the whole universe. This is the part that we, at least theoretically, can see and observe. Or may have seen in the past. Or we will be able to see sometime in the distant future, remaining within the framework of modern science. From the rest of the Universe, even at the speed of light, signals will not be able to reach us, which makes these places, from our point of view, as if non-existent. How big is that big universe nobody really knows. Maybe a million times more than Foreseeable. Or maybe a billion. Or maybe even endless. I say, this is no longer science, but guesswork on the coffee grounds. Scientists have some guesses, but this is more fantasy than reality.

For visualization cosmic scale it is useful to study this picture by expanding it to full screen.

However, even in the Observable Universe, you can cram much more of something else than meters.

1051 atoms make up the planet Earth.

10 80 approximate number elementary particles in the Observable Universe.

10 90 is the approximate number of photons in the Observable Universe. There are almost 10 billion times more of them than elementary particles, electrons and protons.
10 100 is a googol. This number does not physically mean anything, just round and beautiful. The company that set itself the goal of indexing the googol of links (a joke, of course, this is more than the number of elementary particles in the universe!) in 1998 took the name Google.

10,122 protons will be needed to fill the Observable Universe to the eyeballs, tightly like that, proton to proton, back to back.

10,185 Planck volumes are occupied by the Observable Universe. Smaller values than the Planck volume (a cube of dimensions of the Planck length of 10 -35 meters) our science does not know. Surely, as with the Universe, there is something even smaller, but scientists have not yet come up with sane formulas for such trifles, only sheer speculation.

It turns out that 10,185 or so is the largest number that, in principle, can mean something in modern science. In a science that can feel and measure. It is something that exists, or could exist, if it so happened that we knew everything there was to know about the universe. The number consists of 186 digits, here it is:

100 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

Science does not end here, of course, but then free theories, conjectures, and even just pseudo-scientific chess and rut go on. For example, you have probably heard about the inflationary theory, according to which, perhaps, our Universe is only a part of a larger Multiverse, in which these universes are like bubbles in an ocean of champagne.

Or heard about string theory, according to which there can be about 10,500 configurations of string vibrations, which means the same number of potential universes, each with its own laws.
The further into the forest, the less theoretical physics and science in general remains in the growing numbers, and behind the columns of zeros, an ever more pure, unclouded queen of sciences begins to peep. Mathematics is not physics, there are no restrictions and there is nothing to be ashamed of, take a walk, soul, write zeros in formulas even until you drop.
I will only mention the googolplex known to many. A number that has a googol of digits, ten to the power of a googol (10 googol), or ten to the power of ten to the power of one hundred (10 10 100 (the editor does not allow you to make another iteration of the degree, have to be a picture, or I will put a slash (/)

.
10 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
I will not write it down in numbers. Googleplex means absolutely nothing. A person cannot imagine a googolplex of anything, it is physically impossible. To write down such a number, you will need the entire observable universe, if you write with a "nano-pen" directly in vacuum, in fact, in the Planck cells of the cosmos. Let's translate all matter into ink and fill the Universe with one solid numbers, then we get a googolplex. But mathematicians (terrible people!) are only warming up with the googolprex, this is the lowest bar from which real scumbags start for them. And if you think that googolplex to the extent of googolplex is what we're talking about, you have no idea HOW wrong.
There are a lot of googolplexes interesting numbers, having this or that role in mathematical proofs, whether long or short, let's go straight to the Graham number, named after (well, of course) the mathematician Ronald Graham. First I will tell you what it is and what it is for, after which I will figuratively and on my fingers ™ describe what it is in size, and then I will write the number itself. More precisely, I will try to explain what I wrote.
The Graham number appeared in a work devoted to solving one of the problems in Ramsey theory, and "Ramsey" here is not an imperfect participle, but the surname of another mathematician, Frank Ramsey. The task, of course, is quite far-fetched from a philistine point of view, although not very confused, even easily understandable.
Imagine a cube, all the vertices of which are connected by line segments of two colors, red or blue. Connected and colored randomly. Some have already guessed that we will talk about a branch of mathematics called combinatorics.
6

Can we contrive and choose the configuration of colors in such a way (and there are only two of them - red and blue), so that when coloring these segments, it does NOT turn out that all segments of the same color connecting four vertices lie in the same plane? In this case, they do NOT represent such a figure:
7

You can think for yourself, twist the cube in your imagination before your eyes, doing this is not so difficult. There are two colors, the cube has 8 vertices (corners), which means there are 28 segments connecting them. You can choose the coloring configuration in such a way that we won’t get the above figure anywhere, there will be multi-colored lines in all possible planes.
What if we have more dimensions? What if we take not a cube, but a four-dimensional cube, i.e. tesseract ? Can we pull off the same trick as with 3D?
8

I won’t even begin to explain what a four-dimensional cube is, everyone knows? A four-dimensional cube has 16 vertices. And there is no need to puff your brain and try to imagine a four-dimensional cube. This is pure mathematics. I looked at the number of dimensions, substituted it into the formula, got the number of vertices, edges, faces, and so on. So a four-dimensional cube has 16 vertices and 120 segments connecting them. The number of coloring combinations in the four-dimensional case is much greater than in the three-dimensional case, but even here it is not very difficult to calculate, divide, reduce, and the like. In short, find out that in four-dimensional space, you can also contrive with the coloring of the segments of the hypercube so that all lines of the same color connecting 4 vertices will not lie in the same plane.
In five dimensions? And in the five-dimensional, where the cube is called a penteract or a pentacub, it is also possible.
And in six dimensions.
And then there are difficulties. Graham could not mathematically prove that a seven-dimensional hypercube could perform such an operation. Both eight-dimensional and nine-dimensional, and so on. But the given "and so on", it turned out, does not go to infinity, but ends with some very large number, which was called the "Graham number".
That is, there is some minimum dimension of the hypercube, under which the condition is violated, and it is no longer possible to avoid a combination of coloring segments that four points of the same color will lie in the same plane. And this minimum dimension is exactly greater than six and exactly less than the Graham number, this is the mathematical proof of the scientist.
And now the definition of what I described above in a few paragraphs, in a dry and boring (but capacious) language of mathematics. It is not necessary to understand, but I cannot not bring it.
Consider an n-dimensional hypercube and connect all pairs of vertices to get a complete graph with 2n vertices. Let us color each edge of this graph either red or blue color. At what the smallest value n each such coloring necessarily contains a single-colored complete subgraph with four vertices, all of which lie in the same plane?
In 1971, Graham proved that this problem has a solution, and that this solution (the number of dimensions) lies between the number 6 and some larger number, which was later (not by the author himself) named after him. In 2008, the proof was improved, the lower bound was raised, now the desired number of dimensions lies already between the number 13 and the Graham number. Mathematicians do not sleep, the work is going on, the scope is narrowing.
Many years have passed since the 70s, were found math problems in which numbers and more Graham appear, but this first monster number so impressed contemporaries who understood what scale it was about that in 1980 it was included in the Guinness Book of Records as "the largest number ever involved in a rigorous mathematical proof " on that moment.
Let's try to figure out how big it is. The largest number that can have any physical meaning 10 185 , and if the entire Observable Universe is filled with a seemingly endless set of tiny numbers, we get something commensurate with googolplex.
9

Can you imagine this community? Forward, backward, up, down, as far as the eye can see and as far as the eye can see Hubble telescope, and even how much it lacks, to the most distant galaxies and looking beyond them - numbers, numbers, numbers much smaller than a proton. Such a universe, of course, will not be able to exist for a long time, it will immediately collapse into a black hole. Do you remember how much information can theoretically fit into the universe? I did tell.
The number is really huge, breaks the brain. It's not exactly equal to googolplex, and it doesn't have a name, so I'll call it "dochulion". Just figured out why not. The number of Planck cells in the Observable Universe, and a number is written in each cell. The number contains 10,185 digits and can be represented as 10 10 185 .
dochulion = 10 10 185
Let's open the doors of perception a little wider. Remember the inflation theory? That our Universe is just one of many bubbles in the Multiverse. And if you imagine a dochulion of such bubbles? Let's take a number as long as everything that exists and imagine the Multiverse with a similar number of universes, each of which is filled with numbers to the eyeballs - we get a dochulion of dochulions. Can you imagine this? How you swim in the non-existence of a scalar field, and all around are universes-universes and numbers-numbers-numbers in them ... I hope that such a nightmare (though, why a nightmare?) Will not torment (and why torment?) an overly impressionable reader at night.
For convenience, we call such an operation "flip". Such a frivolous interjection, as if they took the Universe and turned it inside out, then it was inside in numbers, and now, on the contrary, we have as many universes outside as there were numbers, and each box is full, full of numbers. As you peel a pomegranate, you bend the crust like that, the grains turn out from the inside, and the grenades are again in the grains. It also came up on the go, why not, because it worked with dohulion.
What am I getting at? Is it worth it to slow down? Come on, hoba, and one more flip! And now we have as many universes as there were digits in the universes, the number of which was equal to the dochulion of digits that filled our Universe. And immediately, without stopping, flip again. Both the fourth and the fifth. Tenth, thousandth. Keep up with the thought, still picture the picture?
Let's not waste time on trifles, spread the wings of imagination, accelerate to the fullest and flip flip flips. We turn each universe inside out as many times as there were pre-Hulion universes in the previous flip, which flipped from the year before last, which ... uh ... well, are you following? Somewhere like that. Now let our number become, say, "dochouliard".
dohouliard = flip flips
We don't stop and keep flipping dohulions of dohouliards as long as we have the strength. Until it gets dark in the eyes, until you want to scream. Here everyone is brave Pinocchio for himself, the stop word will be "brynza".
So. What is it all about? Huge and infinite dochulions of flips and dohouliards of universes of full digits are no match for Graham's number. They don't even scratch the surface. If Graham's number is presented in the form of a stick, traditionally stretched across the entire Foreseeable Universe, then what we have stuck together here will turn out to be a notch of thickness ... well ... how can I put it this way, to put it mildly ... unworthy of mention. Here, I softened it as best I could.
Now let's digress a little, take a break. We read, we counted, our eyes were tired. Let's forget about Graham's number, we still have to crawl and crawl before it, defocus our eyes, relax, meditate on a much smaller, downright miniature number, which we will call g 1 , and write it down with only six characters:
g1 = 33
The number g 1 is equal to "three, four arrows, three". What does it mean? This is the notation called Knuth's arrow notation.
For details and details, you can read the Wikipedia article, but there are formulas, I will briefly retell it in simple words.
One arrow means ordinary exponentiation.
22 = 2 2 = 4
33 = 3 3 = 27
44 = 4 4 = 256
1010 = 10 10 = 10 000 000 000
Two arrows mean, understandably, exponentiation.
23 = 222 = 2 2/ 2 = 2 4 = 16
33 = 333 = 3 3/ 3 = 3 27 = 7,625,597,484,987 (more than 7 trillion)
34 = 3333 = 3 3/ 3/ 3 = 3 7 625 597 484 987 = number with about 3 trillion digits
35 = 33333 = 3 3/ 3/ 3/ 3 = 3 3/ 7 625 597 484 987 = 3 to the power of 3 trillion digits - googolplex already sucks
In short, "number arrow arrow another number" shows what the height of the degrees is (mathematicians say "b tower") is built from the first number. For example, 58 means a tower of eight fives and is so large that it cannot be calculated on any supercomputer, even on all computers on the planet at the same time.
In short, "number arrow arrow another number" shows how high the degrees (mathematicians say "tower") are built from the first number. For example, 58 means a tower of eight fives and is so large that it cannot be calculated on any supercomputer, even on all computers on the planet at the same time.

Let's move on to the three arrows. If the double arrow showed the height of the tower of degrees, then the triple one would seem to indicate "the height of the tower of the height of the tower"? What-there! In the case of a triple, we have the height of the tower of the height of the tower of the height of the tower (there is no such concept in mathematics, I decided to call it "towerless"). Something like this: 11

That is, 33 forms a towerless triplets, 7 trillion pieces high. What are 7 trillion triples stacked on top of each other and called "towerless"? If you carefully read this text and did not fall asleep at the very beginning, you probably remember that there are 100 trillion centimeters from Earth to Saturn. The three shown on the screen in twelfth font, this one - 3 - is five millimeters high. So the towerless triplets will stretch from your screen... well, not to Saturn, of course. Even the Sun will not reach, only a quarter of an astronomical unit, about the same as from Earth to Mars in good weather. I draw your attention (do not sleep!) that the towerless is not a number from Earth to Mars, it is a tower of degrees so high. We remember that five triples in this tower cover the googolplex, the calculation of the first decimeter of triples burns all the fuses of the planet's computers, and the remaining millions of kilometers of degrees are already useless, they simply openly mock the reader, counting them is useless and impossible.
12

Now it is clear that 34 = 3333 = 337 625 597 484 987 = 3 towerless, (not 3 to the degree of towerlessness, but "three arrows arrow towerlessness" (!)), it is towerless towerlessness will not fit either in length or height into the Observable Universe , and won't even fit in the supposed multiverse.
Words end at 35 = 33333, and interjections end at 36 = 333333, but you can practice if you are interested.
Let's move on to the four arrows. As you may have guessed, here the turret sits on the turret, drives without a turret, and even with a turret that without a turret, it doesn’t matter. I’ll just silently give a picture that reveals the scheme for calculating four arrows, when each next number of the tower of degrees determines the height of the tower of degrees, which determines the height of the tower of degrees, determines the height of the tower of degrees ... and so on until self-forgetfulness.

It is useless to calculate it, and it will not work. The number of degrees here does not lend itself to meaningful accounting. This number cannot be imagined, it cannot be described. No analogies on the fingers™ are applicable, there is simply nothing to compare the number with. We can say that it is huge, that it is grandiose, that it is monumental and looks beyond the event horizon. That is, to give him some verbal epithets. But visualization, even free and figurative, is impossible. If with three arrows it was still possible to say at least something, to draw a towerless from Earth to Mars, somehow to compare with something, then there simply cannot be analogies. Try to imagine a thin tower of threes from Earth to Mars, next to it another one almost the same, and another, and another ... The endless field of towers stretches into the distance, into infinity, towers are everywhere, towers are everywhere. And, what is most offensive, these towers do not even have anything to do with the number, they only determine the height of other towers that need to be built in order to get the height of the towers, to get the height of the towers ... in order to get the number itself after an unimaginable amount of time and iterations.
That's what g 1 is, that's what 33 is.
Rested? Now, from g 1, with renewed vigor, we return to the assault on the Graham number. Have you noticed how the escalation grows from arrow to arrow?
33 = 27
33 = 7 625 597 484 987
33 = tower, from Earth to Mars.
33 = a number that can neither be imagined nor described.
And imagine what a digital nightmare is going on when the shooter is five? When are there six? Can you imagine the number when the shooter will be one hundred? If you can, let me bring to your attention the number g 2 , in which the number of these arrows turns out to be equal to g 1 . Remember what g 1 is, right?

Everything that has been written so far, all these calculations, degrees and towers that do not fit in the multiverses of the multiverses, were needed only for one. To show the NUMBER OF ARROWS in the number g 2 . There is no need to count anything, you can just laugh and wave your hand.
I will not hide, there is also g 3 , which contains g 2 arrows. By the way, is it still clear that g 3 is not g 2 "to the power" of g 2 , but the number of towerless towers that determine the height of the towerless towers that determine the height ... and so on down the whole chain until the heat death of the Universe? This is where you start crying.
Why cry? Because absolutely true. There is also a number g 4 , which contains g 3 arrows between triples. There is also g 5 , there is g 6 and g 7 and g 17 and g 43 ...
In short, there are 64 of these g. Each previous one is numerically equal to the number of arrows in the next one. The last g 64 is Graham's number, from which everything seemed to start so innocently. This is the number of dimensions of the hypercube, which will definitely be enough to correctly color the segments in red and blue. Maybe less, this is, so to speak, the upper limit. It is written as follows:

What is the largest number in the world that means something? In this article, I will try to talk about a digital monster called the Graham number,

Writes sly2m.livejournal.com

Source:

If you peer into the abyss for a long time, you can have a good time.
Mechanical Soul Engineer

Graham Number on Fingers™

As soon as a child (and this happens somewhere around three or four years old) understands that all numbers are divided into three groups “one, two and many”, he immediately tries to find out: how much is a lot, how much differs from a lot, and can there be so many that there is no more. Surely you played an interesting (for that age) game with your parents, who will name the largest number, and if the ancestor was no more stupid than a fifth grader, then he always won, for every “million” answering “two million”, and for “billion” - "two billion" or "a billion plus one".

Already by the first grade of school, everyone knows that there are an infinite number of numbers, they never end, and there is no largest number. To any million trillion billion you can always say "plus one" and win. And a little later comes (should come!) The understanding that long strings of numbers in themselves do not mean anything. All these trillions of billions only make sense when they serve as a representation of a certain number of objects or describe a certain phenomenon. There is no difficulty in inventing a long number that is nothing but a set of long-sounding numbers, there are already an infinite number of them. Science, to some extent figuratively, is engaged in looking for very specific combinations of numbers in this boundless abyss, adding to some physical phenomenon, such as the speed of light, Avogadro's number or Planck's constant.

And the question immediately arises, what is the largest number in the world that means something? In this article, I will try to talk about a digital monster called the Graham number, although strictly speaking, science knows even more numbers. Graham's number is the most publicized one, one might say "heard of" by the general public, because it is quite simple in explanation and yet large enough to turn one's head. In general, here it is necessary to declare a small disclaimer (Russian warning). It may sound like a joke, but I'm not joking. I'm talking quite seriously - meticulous picking in such mathematical depths, combined with the unrestrained expansion of the boundaries of perception, can (and will) have a serious impact on the worldview, on the positioning of the individual in society, and, ultimately, on the general psychological state of the picker, or, we will call things in their proper names - opens the way to shiz. It is not necessary to read the following text too carefully, it is not necessary to imagine the things described in it too vividly and vividly. And don't say later that you weren't warned!

When they talk, say, about the number of stars in the Universe (in the Observable Universe), no idiot will bother to calculate how many of them there are in the literal sense, up to the last star. It is believed that approximately 10²¹ pieces. And this is a lower estimate. This means that the total number of stars can be expressed as a number that has 21 zeros after one, i.e. "1,000,000,000,000,000,000,000".

This is what a small part of them (about 100,000) looks like in the globular cluster Omega Centauri.

Naturally, when it comes to such scales, real numbers do not play a significant role in the number, everything is very conditional and approximate. Maybe the actual number of stars in the universe is "1,564,861,615,140,168,357,973", or maybe "9,384,684,643,798,468,483,745". And even “3 333 333 333 333 333 333 333”, why not, although it is unlikely, of course. In cosmology, the science of the properties of the universe as a whole, such trifles are not fooled. The main thing is to imagine that approximately this number consists of 22 digits, from which it is more convenient to consider it as a unit with 21 zeros, and write it down as 10²¹. The rule is general and very simple. What figure or number stands in place of the degree (printed in small print on top of 10), so many zeros after the unit will be in this number, if you paint it in a simple way, with signs in a row, and not in a scientific way. Some numbers have "human names", for example 10³ we call "thousand", 10⁶ - "million", and 10⁹ - "billion", and some do not. Let's say 10⁵⁹ doesn't have a common name. And 10²¹, by the way, has it - it's a "sextillion".

Everything that goes up to a million is intuitively understandable to almost anyone, because who does not want to become a millionaire? Then some problems start. Although a billion (10⁹) is also known to almost everyone. You can even count up to a billion. If only after being born, literally at the moment of birth, begin to count once a second “one, two, three, four ...” and not sleep, do not drink, do not eat, but only count-count-count tirelessly day and night, then when 32 years hits, you can count up to a billion, because 32 revolutions of the Earth around the Sun take about a billion seconds.

7 billion is the number of people on the planet. Based on the foregoing, it is absolutely impossible to count them all in order during a human life, you will have to live more than two hundred years.

100 billion (10¹¹) - how many or so people have lived on the planet throughout its history. McDonald's sold 100 billion hamburgers by 1998 in its 50 years of existence. There are 100 billion stars (well, a little more) in our Milky Way galaxy, and the Sun is one of them. The same number of galaxies is contained in the observable universe. There are 100 billion neurons in the human brain. And the same number of anaerobic bacteria live in each reader of these lines in the caecum.

A trillion (10¹²) is a number rarely used. It is impossible to count up to a trillion, it will take 32 thousand years. A trillion seconds ago, people lived in caves and hunted mammoths with spears. Yes, a trillion seconds ago mammoths lived on Earth. There are about a trillion fish in the oceans of the planet. Our neighboring Andromeda Galaxy contains about a trillion stars. A human is made up of 10 trillion cells. Russia's GDP in 2013 amounted to 66 trillion rubles (in 2013 rubles). From Earth to Saturn, 100 trillion centimeters and the same number of letters in total have been printed in all books ever published.

A quadrillion (10¹⁵, million billion) is the number of ants on the planet. Normal people don’t pronounce this word out loud, well, admit it, when was the last time you heard “a quadrillion of something” in a conversation?

Quintillion (10¹⁸, billion billion) - how many possible configurations exist when assembling a 3x3x3 Rubik's Cube. So is the number of cubic meters of water in the world's oceans.

Sextillion (10²¹) - we have already met this number. The number of stars in the Observable Universe. The number of grains of sand in all the deserts of the Earth. The number of transistors in all existing electronic devices of mankind, if Intel did not lie to us.

10 sextillion (10²²) is the number of molecules in a gram of water.

10²⁴ is the mass of the Earth in kilograms.

10²⁶ is the diameter of the Observable Universe in meters, but it is not very convenient to count in meters, the generally accepted boundaries of the Observable Universe are 93 billion light years.

Science does not operate with dimensions larger than the Observable Universe. We know for sure that the Observable Universe is not the whole-all-the whole Universe. This is the part that we, at least theoretically, can see and observe. Or may have seen in the past. Or we can see sometime in the distant future, remaining within the framework of modern science. From the rest of the Universe, even at the speed of light, signals will not be able to reach us, which makes these places, from our point of view, as if non-existent. How big is that big universe, no one really knows. Maybe a million times more than Foreseeable. Or maybe a billion. Or maybe even endless. I say, this is no longer science, but guesswork on the coffee grounds. Scientists have some guesses, but this is more fantasy than reality.

To visualize the cosmic scale, it is useful to study this picture, expanding it to full screen.

However, even in the Observable Universe, you can cram much more of something else than meters.

10⁵¹ atoms make up the planet Earth.

10⁸⁰ is the approximate number of elementary particles in the Observable Universe.

10⁹⁰ is the approximate number of photons in the Observable Universe. There are almost 10 billion times more of them than elementary particles, electrons and protons.

10¹⁰⁰ - googol. This number does not physically mean anything, just round and beautiful. The company that set itself the goal of indexing the Google of links (a joke, of course, this is more than the number of elementary particles in the universe!) in 1998 took on the name Google.

10¹²² protons will be needed to fill the Observable Universe to the eyeballs, tightly like that, proton to proton, back to back.

The Observable Universe occupies 10¹⁸⁵ Planck volumes. Less than the Planck volume (a cube of Planck length 10⁻³⁵ meters) our science does not know. Surely, as with the Universe, there is something even smaller there, but scientists have not yet come up with sane formulas for such trifles, only sheer speculation.

It turns out that 10¹⁸⁵ or so is the largest number that could possibly mean anything in modern science. In a science that can feel and measure. It is something that exists, or could exist, if it so happened that we knew everything there was to know about the universe. The number consists of 186 digits, here it is:

Or heard about string theory, according to which there can be about 10⁵⁰⁰ configurations of string vibrations, which means the same number of potential universes, each with its own laws.

The further into the forest, the less theoretical physics and science in general remains in the growing numbers, and behind the columns of zeros, an ever more pure, unclouded queen of sciences begins to peep. Mathematics is not physics, there are no restrictions and there is nothing to be ashamed of, take a walk, soul, write zeros in formulas even until you drop.

I will only mention the googolplex known to many. A number that has a googol of digits, ten to the power of a googol, or ten to the power of ten to the power of one hundred

I will not write it down in numbers. Googleplex means absolutely nothing. A person cannot imagine a googolplex of anything, it is physically impossible. To write down such a number, you will need the entire observable universe, if you write with a “nano-pen” directly in vacuum, in fact, in the Planck cells of the cosmos. Let's translate all matter into ink and fill the Universe with one solid numbers, then we get a googolplex. But mathematicians (terrible people!) are only warming up with the googolprex, this is the lowest bar from which real goodies start for them. And if you think that googolplex to the extent of googolplex is what we're talking about, you have no idea HOW wrong.

Behind the googolplex there are many interesting numbers that have one or another role in mathematical proofs, how long and short, let's go straight to the Graham number, named after (well, of course) the mathematician Ronald Graham. First I will tell you what it is and what it is for, after which I will figuratively and on my fingers ™ describe what it is in size, and then I will write the number itself. More precisely, I will try to explain what I wrote.

Graham's number appeared in a work devoted to solving one of the problems in Ramsey theory, and "Ramsey" here is not an imperfect participle, but the surname of another mathematician, Frank Ramsey. The task, of course, is quite far-fetched from a philistine point of view, although not very confused, even easily understandable.

Imagine a cube, all the vertices of which are connected by lines-segments of two colors, red or blue. Connected and colored randomly. Some have already guessed that we are talking about a branch of mathematics called combinatorics.

Will we be able to contrive and choose the configuration of colors in such a way (and there are only two of them - red and blue), so that when coloring these segments, it will NOT work out that all segments of the same color connecting four vertices lie in the same plane? In this case, they do NOT represent such a figure:

You can think for yourself, twist the cube in your imagination before your eyes, doing this is not so difficult. There are two colors, the cube has 8 vertices (corners), which means there are 28 segments connecting them. You can choose the coloring configuration in such a way that we will not get the above figure anywhere, there will be multi-colored lines in all possible planes.

What if we have more dimensions? What if we take not a cube, but a four-dimensional cube, i.e. tesseract? Can we pull off the same trick as with 3D?

In five dimensions? And in the five-dimensional, where the cube is called a penteract or a pentacub, it is also possible.
And in six dimensions.

And then there are difficulties. Graham could not mathematically prove that a seven-dimensional hypercube could perform such an operation. Both eight-dimensional and nine-dimensional, and so on. But the given “and so on” turned out not to go to infinity, but ends with some very large number, which was called the “Graham number”.

That is, there is some minimum dimension of the hypercube, under which the condition is violated, and it is no longer possible to avoid a combination of coloring segments that four points of the same color will lie in the same plane. And this minimum dimension is exactly greater than six and exactly less than the Graham number, this is the mathematical proof of the scientist.

And now the definition of what I described above in a few paragraphs, in a dry and boring (but capacious) language of mathematics. It is not necessary to understand, but I cannot not bring it.

Consider an n-dimensional hypercube and connect all pairs of vertices to obtain a complete graph with 2n vertices. Color each edge of this graph either red or blue. What is the smallest value of n for which each such coloring necessarily contains a single-colored complete subgraph with four vertices, all of which lie in the same plane?

In 1971, Graham proved that this problem has a solution, and that this solution (the number of dimensions) lies between the number 6 and some larger number, which was later (not by the author himself) named after him. In 2008, the proof was improved, the lower bound was raised, now the desired number of dimensions lies already between the number 13 and the Graham number. Mathematicians do not sleep, the work is going on, the scope is narrowing.

Many years have passed since the 70s, mathematical problems were found in which numbers and more Graham appear, but this first monster number so impressed contemporaries who understood what scale it was about that in 1980 it was included in the Guinness Book of Records as “the most the largest number ever to participate in a rigorous mathematical proof" at that time.

Let's try to figure out how big it is. The largest number that can have some physical meaning is 10¹⁸⁵, and if the entire Observable Universe is filled with a seemingly endless set of meager numbers, we get something commensurate with a googolplex.

Can you imagine this community? Forward, backward, up, down, as far as the eye can see and as far as the Hubble telescope can, and even as far as it is not enough, to the most distant galaxies and looking beyond them - numbers, numbers, numbers much smaller than a proton. Such a universe, of course, will not be able to exist for a long time, it will immediately collapse into a black hole. Do you remember how much information can theoretically fit into the universe?

The number is really huge, breaks the brain. It is not exactly equal to googolplex, and it does not have a name, so I will call it "dochulion". Just figured out why not. The number of Planck cells in the Observable Universe, and a number is written in each cell. The number has 10¹⁸⁵ digits and can be represented as

Let's open the doors of perception a little wider. Remember the inflation theory? That our Universe is just one of many bubbles in the Multiverse. And if you imagine a dochulion of such bubbles? Let's take a number as long as everything that exists and imagine the Multiverse with a similar number of universes, each of which is filled with numbers to the eyeballs - we get a dochulion of dochulions. Can you imagine this? How you float in the non-existence of a scalar field, and all around are universes-universes and numbers-numbers-numbers in them ... I hope that such a nightmare (though, why a nightmare?) Will not torment (and why torment?) an overly impressionable reader at night.

For convenience, we call such an operation "flip". Such a frivolous interjection, as if they took the Universe and turned it inside out, then it was inside in numbers, and now, on the contrary, we have as many universes outside as there were numbers, and each box is full, full of numbers. As you peel a pomegranate, you bend the crust like that, the grains turn out from the inside, and the grenades are again in the grains. It also came up on the go, why not, because it worked with dohulion.

What am I getting at? Is it worth it to slow down? Come on, hoba, and one more flip! And now we have as many universes as there were digits in the universes, the number of which was equal to the dochulion of digits that filled our Universe. And immediately, without stopping, flip again. Both the fourth and the fifth. Tenth, thousandth. Keep up with the thought, still picture the picture?

Let's not waste time on trifles, spread the wings of imagination, accelerate to the fullest and flip flip flips. We turn each universe inside out as many times as there were pre-Hulion universes in the previous flip, which flipped from the year before last, which ... uh ... well, are you following? Somewhere like that. Let now our number become, suppose, "dochouliard".

Dohouliard = flip flips

We don't stop and keep flipping dohulions of dohouliards as long as we have the strength. Until it gets dark in the eyes, until you want to scream. Here, each brave Pinocchio for himself, the stop word will be “brynza”.

So. What is it all about? Huge and infinite dochulions of flips and dohouliards of universes of full digits are no match for Graham's number. They don't even scratch the surface. If Graham's number is presented in the form of a stick, traditionally stretched across the entire Foreseeable Universe, then what we have stuck together here will turn out to be a notch of thickness ... well ... how can I put it this way, to put it mildly ... unworthy of mention. Here, I softened it as best I could.

Now let's digress a little, take a break. We read, we counted, our eyes were tired. Let's forget about Graham's number, we still have to crawl and crawl before it, unfocus our eyes, relax, meditate on a much smaller, downright miniature number, which we will call g₁, and write it down in just six characters:
g₁ = 33

The number g₁ is "three, four arrows, three". What does it mean? This is the notation called Knuth's arrow notation.

One arrow means ordinary exponentiation.

44 = 4⁴ = 256

1010 = 10¹⁰ = 10,000,000,000

Two arrows mean, understandably, exponentiation.

In short, "number arrow arrow another number" shows how high the degrees (mathematicians say "tower") are built from the first number. For example, 58 means a tower of eight fives and is so large that it cannot be calculated on any supercomputer, even on all computers on the planet at the same time.

Let's move on to the three arrows. If the double arrow showed the height of the tower of degrees, then the triple arrow, it would seem, would indicate “the height of the tower of the height of the tower”? What-there! In the case of a triple, we have the height of the tower the height of the tower the height of the tower (there is no such concept in mathematics, I decided to call it “towerless”). Something like this:

That is, 33 forms a towerless triplets, 7 trillion pieces high. What are 7 trillion triples stacked on top of each other and called "towerless"? If you carefully read this text and did not fall asleep at the very beginning, you probably remember that there are 100 trillion centimeters from Earth to Saturn. The triple shown on the screen in the twelfth font, this one - 3 - is five millimeters high. So the towerless triplets will stretch from your screen... well, not to Saturn, of course. Even the Sun will not reach, only a quarter of an astronomical unit, about the same as from Earth to Mars in good weather. I draw your attention (do not sleep!) that the towerless is not a number from the Earth to Mars, it is a tower of degrees of such a height. We remember that five triples in this tower cover the googolplex, the calculation of the first decimeter of triples burns all the fuses of the planet's computers, and the remaining millions of kilometers of degrees are already useless, they simply openly mock the reader, it is useless to count them.

Now it is clear that 34 = 3333 = 337 625 597 484 987 = 3 towerless, (not 3 to the degree of towerlessness, but “three arrows arrow towerlessness” (!)), she’s towerless towerlessness will not fit either in length or height into the Observable Universe , and won't even fit in the supposed multiverse.

Words end at 35 = 33333, and interjections end at 36 = 333333, but you can practice if you are interested.

Let's move on to the four arrows. As you may have guessed, here the turret sits on the turret, drives without a turret, and even with a turret that without a turret, it doesn’t matter. I’ll just silently give a picture that reveals the scheme for calculating four arrows, when each next number of the tower of degrees determines the height of the tower of degrees, which determines the height of the tower of degrees, determines the height of the tower of degrees ... and so on until self-forgetfulness.

That's what g₁ is, that's what 33 is.

Rested? Now from g₁ with new forces we return to the assault on the Graham number. Have you noticed how the escalation grows from arrow to arrow?

33 = 7 625 597 484 987

33 = tower, from Earth to Mars.

33 = a number that can neither be imagined nor described.

And imagine what a digital nightmare is going on when the shooter is five? When are there six? Can you imagine the number when the shooter will be one hundred? If you can, let me bring to your attention the number g₂, in which the number of these arrows turns out to be equal to g₁. Remember what g₁ is, right?

I will not hide, there is also g₃, which contains g₂ arrows. By the way, is it still clear that g₃ is not g₂ “to the power” of g₂, but the number of towerless towers that determine the height of the towerless towers that determine the height ... and so on down the whole chain until the heat death of the Universe? This is where you start crying.

Why cry? Because absolutely true. There is also the number g₄, which contains g₃ arrows between triples. There is also g₅, there is g₆ and g₇ and g₁₇ and g₄₃...

In short, there are 64 of these g. Each previous one is numerically equal to the number of arrows in the next one. The last g₆₄ is Graham's number, from which everything seemed to start so innocently. This is the number of dimensions of the hypercube, which will definitely be enough to correctly color the segments in red and blue. Maybe less, this is, so to speak, the upper limit. It is written as follows:

and write like this.

There are numbers that are so incredibly, incredibly large that it would take the entire universe to even write them down. But here's what's really maddening... some of these incomprehensibly large numbers are extremely important to understanding the world.

When I say "the largest number in the universe," I really mean the largest meaningful number, the maximum possible number that is useful in some way. There are many contenders for this title, but I warn you right away: there is indeed a risk that trying to understand all this will blow your mind. And besides, with too much math, you get little fun.

Googol and googolplex

Edward Kasner

We could start with two, very likely the biggest numbers you've ever heard of, and these are indeed the two biggest numbers that have commonly accepted definitions in English language. (There is a fairly precise nomenclature used for numbers as large as you would like, but these two numbers are not currently found in dictionaries.) Google, since it became world famous (albeit with errors, note. in fact it is googol) in the form of Google, was born in 1920 as a way to get children interested in big numbers.

To this end, Edward Kasner (pictured) took his two nephews, Milton and Edwin Sirott, on a New Jersey Palisades tour. He invited them to come up with any ideas, and then the nine-year-old Milton suggested “googol”. Where he got this word from is unknown, but Kasner decided that or a number in which one hundred zeros follow the one will henceforth be called a googol.

But young Milton didn't stop there, he came up with an even bigger number, the googolplex. It's a number, according to Milton, that has a 1 first and then as many zeros as you can write before you get tired. While the idea is fascinating, Kasner felt a more formal definition was needed. As he explained in his 1940 book Mathematics and the Imagination, Milton's definition leaves open the perilous possibility that the occasional buffoon might become a superior mathematician to Albert Einstein simply because he has more stamina.

So Kasner decided that the googolplex would be , or 1, followed by a googol of zeros. Otherwise, and in a notation similar to that with which we will deal with other numbers, we will say that the googolplex is . To show how mesmerizing this is, Carl Sagan once remarked that it was physically impossible to write down all the zeros of a googolplex because there simply wasn't enough room in the universe. If the entire volume of the observable universe is filled with fine dust particles approximately 1.5 microns in size, then the number various ways the location of these particles will be approximately equal to one googolplex.

Linguistically speaking, googol and googolplex are probably the two largest significant numbers (at least in English), but, as we will now establish, there are infinitely many ways to define “significance”.

Real world

If we talk about the largest significant number, there is a reasonable argument that this really means that you need to find the largest number with a value that actually exists in the world. We can start with the current human population, which is currently around 6920 million. World GDP in 2010 was estimated to be around $61,960 billion, but both of these numbers are small compared to the roughly 100 trillion cells that make up the human body. Of course, none of these numbers can compare with full number particles in the universe, which is generally considered to be about , and this number is so large that our language does not have a corresponding word.

We can play around with measurement systems a bit, making the numbers bigger and bigger. Thus, the mass of the Sun in tons will be less than in pounds. A great way to do this is to use the Planck units, which are the smallest possible measures for which the laws of physics still hold. For example, the age of the universe in Planck time is about . If we go back to the first Planck time unit after the Big Bang, we will see that the density of the Universe was then . We're getting more and more, but we haven't even reached a googol yet.

The largest number with any real world application—or, in this case, real world application—is probably , one of the latest estimates of the number of universes in the multiverse. This number is so large that the human brain will literally be unable to perceive all these different universes, since the brain is only capable of roughly configurations. In fact, this number is probably the largest number with any practical meaning, if you do not take into account the idea of the multiverse as a whole. However, there are still much larger numbers lurking there. But in order to find them, we must go into the realm of pure mathematics, and there is no better place to start than prime numbers.

Mersenne primes

Part of the difficulty is coming up with a good definition of what a “meaningful” number is. One way is to think in terms of primes and composites. A prime number, as you probably remember from school mathematics, is any natural number(note not equal to one) that is only divisible by and itself. So, and are prime numbers, and and are composite numbers. This means that any composite number can eventually be represented by its prime divisors. In a sense, the number is more important than, say, because there is no way to express it in terms of the product of smaller numbers.

Obviously we can go a little further. , for example, is actually just , which means that in a hypothetical world where our knowledge of numbers is limited to , a mathematician can still express . But the next number is already prime, which means that the only way to express it is to directly know about its existence. This means that the largest known prime numbers play an important role, but, say, a googol - which is ultimately just a collection of numbers and , multiplied together - actually does not. And since prime numbers are mostly random, there is no known way to predict that an incredibly large number will actually be prime. To this day, discovering new prime numbers is a difficult task.

Mathematicians Ancient Greece had a concept of prime numbers at least as early as 500 BC, and 2000 years later people still knew what primes were only up to about 750. Euclid's thinkers saw the possibility of simplification, but until the Renaissance, mathematicians could not really put it into practice. These numbers are known as Mersenne numbers and are named after the 17th century French scientist Marina Mersenne. The idea is quite simple: a Mersenne number is any number of the form . So, for example, and this number is prime, the same is true for .

Mersenne primes are much faster and easier to determine than any other kind of prime, and computers have been hard at work finding them for the past six decades. Until 1952, the largest known prime number was a number—a number with digits. In the same year, it was calculated on a computer that the number is prime, and this number consists of digits, which makes it already much larger than a googol.

Computers have been on the hunt since then, and currently the th Mersenne number is the largest prime number, known to mankind. Discovered in 2008, it is a number with almost millions of digits. This is the largest known number that cannot be expressed in terms of any smaller numbers, and if you want to help find an even larger Mersenne number, you (and your computer) can always join the search at http://www.mersenne. org/.

Skewes number

Stanley Skuse

Let's go back to prime numbers. As I said before, they behave fundamentally wrong, which means that there is no way to predict what the next prime number will be. Mathematicians have been forced to turn to some rather fantastic measurements in order to come up with some way to predict future primes, even in some nebulous way. The most successful of these attempts is probably the function that counts prime numbers, which he came up with in late XVIII century legendary mathematician Carl Friedrich Gauss.

I'll spare you the more complicated math - anyway, we still have a lot to come - but the essence of the function is this: for any integer, it is possible to estimate how many primes there are less than . For example, if , the function predicts that there should be prime numbers, if - prime numbers less than , and if , then there are smaller numbers that are prime.

The arrangement of primes is indeed irregular, and is only an approximation of the actual number of primes. In fact, we know that there are primes less than , primes less than , and primes less than . This is excellent score, to be sure, but this is always only an estimate ... and, more specifically, an estimate from above.

In all known cases up to , the function that finds the number of primes slightly exaggerates the actual number of primes less than . Mathematicians once thought that this would always be the case, ad infinitum, and that this certainly applies to some unimaginably huge numbers, but in 1914 John Edensor Littlewood proved that for some unknown, unimaginably huge number, this function will begin to produce fewer primes, and then it will switch between overestimation and underestimation an infinite number of times.

The hunt was for the starting point of the races, and that's where Stanley Skuse appeared (see photo). In 1933, he proved that the upper limit, when a function that approximates the number of primes for the first time gives a smaller value, is the number. It is difficult to truly understand, even in the most abstract sense, what this number really is, and from this point of view it was the largest number ever used in a serious mathematical proof. Since then, mathematicians have been able to reduce the upper bound to a relatively small number, but the original number has remained known as the Skewes number.

So, how big is the number that makes even the mighty googolplex dwarf? In The Penguin Dictionary of Curious and Interesting Numbers, David Wells describes one way in which the mathematician Hardy was able to make sense of the size of the Skewes number:

"Hardy thought it was 'the largest number ever to serve any particular purpose in mathematics' and suggested that if chess were played with all the particles of the universe as pieces, one move would consist of swapping two particles, and the game would stop when the same position was repeated a third time, then the number of all possible games would be equal to about the number of Skuse''.

One last thing before moving on: we talked about the smaller of the two Skewes numbers. There is another Skewes number, which the mathematician found in 1955. The first number is derived on the grounds that the so-called Riemann Hypothesis is true - a particularly difficult hypothesis in mathematics that remains unproven, very useful when it comes to prime numbers. However, if the Riemann Hypothesis is false, Skewes found that the jump start point increases to .

The problem of magnitude

Before we get to a number that makes even Skewes' number look tiny, we need to talk a little about scale because otherwise we have no way of estimating where we're going. Let's take a number first - it's a tiny number, so small that people can actually have an intuitive understanding of what it means. There are very few numbers that fit this description, since numbers greater than six are no longer separate numbers and become “several”, “many”, etc.

Now let's take , i.e. . Although we can't really intuitively, as we did for the number , figure out what , imagine what it is, it's very easy. So far everything is going well. But what happens if we go to ? This is equal to , or . We are very far from being able to imagine this value, like any other very large one - we are losing the ability to comprehend individual parts somewhere around a million. (True, crazy a large number of It would take time to actually count to a million of anything, but the point is that we are still able to perceive this number.)

However, although we cannot imagine, we are at least able to understand in general terms what 7600 billion is, perhaps by comparing it to something like US GDP. We have gone from intuition to representation to mere understanding, but at least we still have some gap in our understanding of what a number is. This is about to change as we move one more rung up the ladder.

To do this, we need to switch to the notation introduced by Donald Knuth, known as arrow notation. These notations can be written as . When we then go to , the number we get will be . This is equal to where the total of triplets is. We have now vastly and truly surpassed all the other numbers already mentioned. After all, even the largest of them had only three or four members in the index series. For example, even the Super Skewes number is "only" - even with the fact that both the base and the exponents are much larger than , it is still absolutely nothing compared to the size of the number tower with billions of members.

Obviously, there is no way to comprehend such huge numbers... and yet, the process by which they are created can still be understood. We could not understand the real number given by the tower of powers, which is a billion triples, but we can basically imagine such a tower with many members, and a really decent supercomputer will be able to store such towers in memory, even if it cannot calculate their real values .

It's getting more and more abstract, but it's only going to get worse. You might think that a tower of powers whose exponent length is (moreover, in a previous version of this post I made exactly that mistake), but it's just . In other words, imagine that you were able to calculate the exact value of a power tower of triples, which consists of elements, and then you took this value and created a new tower with as many in it as ... which gives .

Repeat this process with each successive number ( note starting from the right) until you do this once, and then finally you get . This is a number that is simply incredibly large, but at least the steps to get it seem to be clear if everything is done very slowly. We can no longer understand numbers or imagine the procedure by which they are obtained, but at least we can understand the basic algorithm, only in a sufficiently long time.

Now let's prepare the mind to actually blow it up.

Graham's (Graham's) number

Ronald Graham

This is how you get Graham's number, which ranks in the Guinness Book of World Records as the largest number ever used in a mathematical proof. It is absolutely impossible to imagine how big it is, and it is just as difficult to explain exactly what it is. Basically, Graham's number appears when dealing with hypercubes, which are theoretical geometric shapes with more than three dimensions. The mathematician Ronald Graham (see photo) wanted to find out what was the smallest number of dimensions that would keep certain properties of a hypercube stable. (Sorry for this vague explanation, but I'm sure we all need at least two math degrees to make it more accurate.)

In any case, the Graham number is an upper estimate of this minimum number of dimensions. So how big is this upper bound? Let's get back to a number so large that we can understand the algorithm for obtaining it rather vaguely. Now, instead of just jumping up one more level to , we'll count the number that has arrows between the first and last triples. Now we are far beyond even the slightest understanding of what this number is or even of what needs to be done to calculate it.

Now repeat this process times ( note at each next step, we write the number of arrows equal to the number obtained at the previous step).

This, ladies and gentlemen, is Graham's number, which is about an order of magnitude above the point of human understanding. It is a number that is so much more than any number you can imagine - it is far more than any infinity you could ever hope to imagine - it simply defies even the most abstract description.

But here's the weird thing. Since Graham's number is basically just triplets multiplied together, we know some of its properties without actually calculating it. We can't represent Graham's number in any notation we're familiar with, even if we used the entire universe to write it down, but I can give you the last twelve digits of Graham's number right now: . And that's not all: we know at least the last digits of Graham's number.

Of course, it's worth remembering that this number is only an upper bound in Graham's original problem. It is possible that the actual number of measurements required to fulfill the desired property is much, much less. In fact, since the 1980s, it has been believed by most experts in the field that there are actually only six dimensions - a number so small that we can understand it on an intuitive level. The lower bound has since been increased to , but there is still a very good chance that the solution to Graham's problem does not lie near a number as large as Graham's.

To infinity

So there are numbers bigger than Graham's number? There are, of course, for starters there is the Graham number. As for the significant number... well, there are some fiendishly difficult areas of mathematics (in particular, the area known as combinatorics) and computer science, in which there are numbers even larger than Graham's number. But we have almost reached the limit of what I can hope can ever reasonably explain. For those who are reckless enough to go even further, additional reading is offered at your own risk.

Well, now an amazing quote that is attributed to Douglas Ray ( note To be honest, it sounds pretty funny:

“I see clumps of vague numbers lurking out there in the dark, behind the little spot of light that the mind candle gives. They whisper to each other; talking about who knows what. Perhaps they do not like us very much for capturing their little brothers with our minds. Or maybe they just lead an unambiguous numerical way of life, out there, beyond our understanding.''

In order to somehow imagine the scale of the number, let's analyze its record in more detail.

1 . So, in mathematics there is the concept of "hyperoperator" to determine the level of arithmetic operations. Thus, addition is a first-level hyperoperator, and a second-level hyperoperator is multiplication, which is repeated addition. That is, a multiplier is a number that tells us how many times it is necessary to add the multiplied value. For example: 3 3 = 3 + 3 + 3 = 9. The next hyperoperator is exponentiation, x n = X^n, which is essentially a repeated multiplication. Example: 3 3 \u003d 3 3 3 \u003d 27. Writing 3 3 in Knuth notation will look like 33. Here, for clarity, it should be said that the first digit in the expression 33 is the value with which we perform the action, and the number of arrows between digits is an arithmetic operation; in this case, one arrow means exponentiation. The second digit means the power to which the first digit should be raised (how many times to multiply by itself). Accordingly, the expression 74 means seven to the fourth power. In other words, 7 must be multiplied by 7 four times.

2 . The fourth level hyperoperator is tetration, repeated exponentiation. In Knuth's entry, there are two arrows between the numbers. Example: 33 \u003d 3 3 \u003d 3 3 3 \u003d 3 27 \u003d 7 625 597 484 987. That is, the second digit in the presence of two arrows means that so many times you need to raise the first number to the power of itself. In other words, it shows us the height of the power tower from the first digit. For example, the entry 58 means a tower of eight fives piled on top of each other like cubes.

Those whose brain is completely swollen with fat or is only occupied with thoughts about how to find chan, pump up your elf or get rid of acne, you should remember that expressions are calculated in tetration top down, or from right to left. Simply put, 3 3 3 equals fucking not 27 3 , but just the same 3 27 . Now you see, my furry little friend, that tetration is already a pretty powerful notation, allowing you to write numbers 100500 times larger than 100500 itself in a short expression. But that's not all, because it is not a powerful enough hyper-operator to calculate Graham's number.

3 . Let's move on: the fifth level hyperoperator is pentation (repeated tetration). Three arrows between numbers. This is where the fuck up begins, from which people who are not professional mathematicians spit on all this crap and no longer try to understand it. But you're not like them, are you? If you thought that the pentation of the number 3 is decomposed into 3 to the power of 7 625 597 484 987, then you are mistaken. You have no idea how wrong you are. For 3 to the power of 7 625 597 484 987 is only 34. And pentation is 33 = 3(33) = 3(7 625 597 484 987) = 33…( number of exponentiations - 7,625,597,484,987 times)…3. That is, a power tower of triples is obtained with a height of more than seven and a half trillion floors! In other words, the second digit in the presence of three arrows means how high the tetration tower of the first digit will be. For clarity: 34 can be written as 3 3 3 3, or 3 (3 (3 3)). And here the main thing is to understand that this tower of tetrations is not a tower of degrees, here the escalation is much more rapid. 34 = 3 3 3 3 = 7 625 597 484 987 3 3.
Finally got it, bitch! 34 equals 3 in tetration of the number that results from calculating the power tower from the number 3 with a height of 7,625,597,484,987 floors. Accordingly, if 34 is written as a power tower of triples, then the number of floors in this tower will be equal to the number that will be obtained when calculating a power tower with a height of 7,625,597,484,987 floors. Introduced? I didn’t present, of course, such quantities cannot be comprehended with a swoop.

If you still began to slowly not understand what the hell is going on here, then re-read paragraph 2.

4 . And the last hyperoperator we need is hexation. As you may have guessed, four arrows between threes. This is, accordingly, a repeated penation. The second digit in the presence of four arrows means how high the "pentation" tower will be. 33 \u003d 3 (33) \u003d 333 ... 33, where the number of tetrations is the result of calculating the pentation 33. If you don’t understand anything again, then re-read points 3 and 2.
If we move to the very end of this unthinkable chain of tetrations and start calculating it, then the second triple from the end will be equal to 7 625 597 484 987 in the tetration. And the result of the tetration of the third triple from the end will be the number obtained by pentation of the triple in the previous paragraph. And in front of us are more googolplexes and googolplexes of repeated tetrations of the number 3. It’s already useless to try to comprehend something, somehow cover the result ... And here you may ask: “Is this really Graham's number? Wow, how huge!” But no, it's not Graham's number. It was only a mathematical hint, and it is negligible, immeasurably small compared to Graham's number.

Therefore, hexation is just adding one pissing arrow to the pentation, but the result turns out to be larger by an unimaginable number of orders. And now, in fact, the calculation of the Graham number. The number three in the examples was used for a reason, because Graham's number is essentially the multiplied triples. So, let's call the result of our hexation (33) G1. This will be the first calculation step. Only the first. And the next step speeds up the progression so that adding one, ten, MILLION arrows between the numbers is marking time. The second step is the calculation of G2. Now we take the result of our hexatization of the triple and write an expression where the number of superdegree arrows will be equal to this result. G2 = 3…(number of superpower arrows - G1)…3. I wonder what the hyperoperator of SUCH level is called?..

Recording not only the result, but even this hyperoperator is no longer possible without reduction. And the number resulting from its calculation (if, of course, it could be calculated) would fill both the Universe and Parallel Worlds, and subspace, and every other astral. And do not forget that in G1 the number of arrows was equal to four - and this is already a number that is not available for calculation and writing in the usual way! And in G2 this number is only the number of superdegrees. That's it. The progression is incredibly fast. And this is just the beginning. The next step is to calculate the number G3, where the number of superpower arrows will be equal to G2! In a similar way, after this, another 62 calculation steps follow, where the result of each step will be only the number of superpower arrows of the next step, and Graham's number is G64!

Vaistenu, sometimes the matan is worse than any drugs.

In order to somehow imagine the scale of the number, let's analyze its record in more detail. Some preamble is needed here, but, in general, nothing will be too complicated, we will try to describe everything as clearly as possible.

1 . So, in mathematics there is the concept of "hyperoperator" to determine the level of arithmetic operations. So, addition is a first-level hyperoperator. Hyper operator of the second level - multiplication. Multiplication is repeated addition. That is, a multiplier is a number that tells us how many times we need to add the multiplied value. For example: 3 3 = 3 + 3 + 3 = 9. The next hyperoperator is exponentiation, x n = X^n, which is essentially a repeated multiplication. Example: 3 3 \u003d 3 3 3 \u003d 27. Writing 3 3 in Knuth notation will look like 33. Here, for clarity, it should be said that the first digit in the expression 33 is the value with which we perform the action, the number of arrows between numbers - this is an arithmetic operation, in this case one arrow means exponentiation. The second digit means the power to which the first digit should be raised (how many times to multiply by itself). Accordingly, if the expression were 74, then this means seven to the fourth degree. In other words, 7 must be multiplied by 7 four times.

2 . The fourth level hyperoperator is tetration. Tetration is repeated exponentiation. In Knuth's entry, there are two arrows between the numbers. Example: 33 \u003d 3 3 \u003d 3 3 3 \u003d 3 27 \u003d 7 625 597 484 987. That is, the second digit in the presence of two arrows means that so many times you need to raise the first number to the power of itself. In other words, it shows us the height of the power tower from the first digit. For example, the entry 58 means a tower of eight fives piled on top of each other like cubes.

Those whose brain is completely swollen with fat or is only occupied with thoughts about how to find a chan, pump up their elf or get rid of acne, you should remember that expressions are calculated in tetration top down or from right to left. Simply put, 3 3 3 equals fucking not 27 3 , but just the same 3 27 . Now you see, my stupid little friend, that tetration is already a pretty powerful notation, allowing you to write numbers 100500 times larger than 100500 itself in a short expression. But that's not all, because it is not a powerful enough hyper-operator to calculate Graham's number.

3 . Let's move on: the fifth level hyperoperator is pentation (repeated tetration). Three arrows between numbers. This is where the fuck up begins, from which people who are not professional mathematicians spit on all this crap and no longer try to understand it. But you're not like them, are you? If you thought that the pentation of the number 3 is decomposed into 3 to the power of 7 625 597 484 987, then you are mistaken. You have no idea how wrong you are. For 3 to the power of 7 625 597 484 987 is only 34. And pentation is 33 = 3(33) = 3(7 625 597 484 987) = 33…( number of exponentiations - 7,625,597,484,987 times)…3. That is, a power tower of triples is obtained with a height of more than seven and a half trillion floors! In other words, the second digit in the presence of three arrows means how high the tetration tower of the first digit will be. For clarity: 34 can be written as 3 3 3 3, or 3 (3 (3 3)). And here the main thing is to understand that this tower of tetrations is not a tower of degrees, here the escalation is much more rapid. 34 = 3 3 3 3 = 7 625 597 484 987 3 3. Got it, finally? 34 equals 3 in tetration of the number that results from calculating the power tower from the number 3 with a height of 7,625,597,484,987 floors. Accordingly, if 34 is written as a power tower of triples, then the number of floors in this tower will be equal to the number that will be obtained when calculating a power tower with a height of 7,625,597,484,987 floors. Introduced? I didn’t present, of course, such quantities cannot be comprehended with a swoop.

If you still began to slowly not understand what the hell is going on here, then re-read paragraph 2.

4 . And the last hyperoperator we need is hexation. As you may have guessed, four arrows between threes. This is, accordingly, a repeated penation. The second digit in the presence of four arrows means how high the "pentation" tower will be. 33 \u003d 3 (33) \u003d 333 ... 33, where the number of tetrations is the result of calculating pentation 33. If you don’t understand anything again, then read paragraphs 3 and 2 again. If we move to the very end of this unthinkable chain of tetrations and start calculating it, then already the second triple from the end will be in tetration equal to 7 625 597 484 987. And the result of the tetration of the third triple from the end will be the number obtained by pentation of the triple in the previous paragraph. And in front of us are more googolplexes and googolplexes of repeated tetrations of the number 3. It’s already useless to try to comprehend something, to somehow cover the result ... And here you may ask: “Is this really Graham's number? Wow, how huge!” But no, it's not Graham's number. It was only a mathematical hint, and it is negligible, immeasurably small compared to Graham's number.

So it's a hexagon. This is just adding one arrow to the pentation, but the result is an unimaginable number of orders of magnitude greater. And now, in fact, the calculation of the Graham number. The number three in the examples was used for a reason, because Graham's number is essentially the multiplied triples. So, let's call the result of our hexation (33) G1. This will be the first calculation step. Only the first. And the next step speeds up the progression so that adding one, ten, MILLION arrows between the numbers is marking time. Step two, calculation of G2. Now we take the result of our hexatization of the triple, and write an expression where the number of superdegree arrows will be equal to this result. G2 = 3…(number of superpower arrows - G1)…3. I wonder what the hyperoperator of SUCH level is called?.. Recording not only the result, but even this hyperoperator is no longer possible without reduction. And the number resulting from its calculation (if, of course, it would be possible to calculate it) would fill with its numbers the Universe, and parallel worlds, and subspace, and any other astral. And do not forget that in G1 the number of arrows was equal to 4! And this is already a number that is not available for calculation and recording in the usual way! And in G2 this number is only the number of superdegrees. That's it. The progression is incredibly fast. And this is just the beginning. The next step is to calculate the number G3, where the number of superpower arrows will be equal to G2! And likewise, there are 62 more calculation steps after that, where the result of each step will be just the number of superpower arrows of the next step, and Graham's number is G64!

Vaistenu, sometimes the matan is worse than any drugs.