The famous search engine, as well as the company that created this system and many other products, is named after the googol number - one of the largest numbers in the infinite set of natural numbers. However, the largest number is not even a googol, but a googolplex.
The googolplex number was first proposed by Edward Kasner in 1938 and represents one followed by an incredible number of zeros. The name comes from another number - googol - one followed by a hundred zeros. Typically, the number of Googol is written as 10,100, or 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.
A googolplex, in turn, is the number ten to the power of a googol. It's usually written like this: 10 10 ^100, and that's a lot, a lot of zeros. There are so many of them that if you were to count the number of zeros with individual particles in the universe, the particles would run out before the zeros in the googolplex.
According to Carl Sagan, writing this number is impossible because writing it would require more space than exists in the visible universe.
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History of the term
The googol is larger than the number of particles in the part of the Universe known to us, which, according to various estimates, number from 10 79 to 10 81, which also limits its application.
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See what "Google" is in other dictionaries:
Googolplex (from the English Googolplex) number depicted by a unit with Googol zero, 1010100. or 1010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 000 000 Like Google, ... ... Wikipedia
This article is about a number. See also the article about English. googol) number, in decimal notation represented by one followed by 100 zeros: 10100 = 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 Wikipedia 0 0 0 0 0
- (from the English Googolplex) number equal to the Gugol degree: 1010100 or 1010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 000. Like googol, the term ... ... Wikipedia
This article may contain original research. Add links to sources, otherwise it may be put up for deletion. More information may be on the talk page. (May 13, 2011) ... Wikipedia
Mogul is a dessert, the main components of which are beaten egg yolk with sugar. There are many variations of this drink: with the addition of wine, vanillin, rum, bread, honey, fruit and berry juices. Often used as a treat ... Wikipedia
Nominal names of powers of a thousand in ascending order
Nominal names of powers of a thousand in ascending order
Nominal names of powers of a thousand in ascending order
Nominal names of powers of a thousand in ascending order
Books
- World Magic. Fantastic novel and stories, Vladimir Sigismundovich Vechfinsky. The novel "Space Magic". The earthly magician, together with the fairy-tale heroes Vasilisa, Koshchey, Gorynych and the fairy-tale cat, are fighting a force that seeks to capture the Galaxy. A COLLECTION OF STORIES Where...
As a child, I was tormented by the question of what is the largest number, and I plagued almost everyone with this stupid question. Having learned the number one million, I asked if there was a number greater than a million. Billion? And more than a billion? Trillion? And more than a trillion? Finally, someone smart was found who explained to me that the question is stupid, since it is enough just to add one to the largest number, and it turns out that it has never been the largest, since there are even larger numbers.
And now, after many years, I decided to ask another question, namely: What is the largest number that has its own name? Fortunately, now there is an Internet and you can puzzle them with patient search engines that will not call my questions idiotic ;-). Actually, this is what I did, and here's what I found out as a result.
| Number | Latin name | Russian prefix |
| 1 | unus | en- |
| 2 | duo | duo- |
| 3 | tres | three- |
| 4 | quattuor | quadri- |
| 5 | quinque | quinti- |
| 6 | sex | sexty |
| 7 | September | septi- |
| 8 | octo | octi- |
| 9 | novem | noni- |
| 10 | decem | deci- |
There are two systems for naming numbers - American and English.
The American system is built quite simply. All names of large numbers are built like this: at the beginning there is a Latin ordinal number, and at the end the suffix -million is added to it. The exception is the name "million" which is the name of the number one thousand (lat. mille) and the magnifying suffix -million (see table). So the numbers are obtained - trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion and decillion. The American system is used in the USA, Canada, France and Russia. You can find out the number of zeros in a number written in the American system using the simple formula 3 x + 3 (where x is a Latin numeral).
The English naming system is the most common in the world. It is used, for example, in Great Britain and Spain, as well as in most of the former English and Spanish colonies. The names of numbers in this system are built like this: like this: a suffix -million is added to the Latin numeral, the next number (1000 times larger) is built according to the principle - the same Latin numeral, but the suffix is -billion. That is, after a trillion in the English system comes a trillion, and only then a quadrillion, followed by a quadrillion, and so on. Thus, a quadrillion according to the English and American systems are completely different numbers! You can find out the number of zeros in a number written in the English system and ending with the suffix -million using the formula 6 x + 3 (where x is a Latin numeral) and using the formula 6 x + 6 for numbers ending in -billion.
Only the number billion (10 9) passed from the English system into the Russian language, which, nevertheless, would be more correct to call it the way the Americans call it - a billion, since we have adopted the American system. But who in our country does something according to the rules! ;-) By the way, sometimes the word trilliard is also used in Russian (you can see for yourself by running a search in Google or Yandex) and it means, apparently, 1000 trillion, i.e. quadrillion.
In addition to numbers written using Latin prefixes in the American or English system, the so-called off-system numbers are also known, i.e. numbers that have their own names without any Latin prefixes. There are several such numbers, but I will talk about them in more detail a little later.
Let's go back to writing using Latin numerals. It would seem that they can write numbers to infinity, but this is not entirely true. Now I will explain why. First, let's see how the numbers from 1 to 10 33 are called:
| Name | Number |
| Unit | 10 0 |
| Ten | 10 1 |
| Hundred | 10 2 |
| One thousand | 10 3 |
| Million | 10 6 |
| Billion | 10 9 |
| Trillion | 10 12 |
| quadrillion | 10 15 |
| Quintillion | 10 18 |
| Sextillion | 10 21 |
| Septillion | 10 24 |
| Octillion | 10 27 |
| Quintillion | 10 30 |
| Decillion | 10 33 |
And so, now the question arises, what next. What is a decillion? In principle, it is possible, of course, by combining prefixes to generate such monsters as: andecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion and novemdecillion, but these will already be compound names, and we were interested in our own names numbers. Therefore, according to this system, in addition to the above, you can still get only three proper names - vigintillion (from lat. viginti- twenty), centillion (from lat. percent- one hundred) and a million (from lat. mille- one thousand). The Romans did not have more than a thousand proper names for numbers (all numbers over a thousand were composite). For example, a million (1,000,000) Romans called centena milia i.e. ten hundred thousand. And now, actually, the table:
Thus, according to a similar system, numbers greater than 10 3003, which would have its own, non-compound name, cannot be obtained! But nevertheless, numbers greater than a million are known - these are the same off-system numbers. Finally, let's talk about them.
| Name | Number |
| myriad | 10 4 |
| googol | 10 100 |
| Asankheyya | 10 140 |
| Googolplex | 10 10 100 |
| Skuse's second number | 10 10 10 1000 |
| Mega | 2 (in Moser notation) |
| Megiston | 10 (in Moser notation) |
| Moser | 2 (in Moser notation) |
| Graham number | G 63 (in Graham's notation) |
| Stasplex | G 100 (in Graham's notation) |
The smallest such number is myriad(it is even in Dahl's dictionary), which means a hundred hundreds, that is, 10,000. True, this word is outdated and practically not used, but it is curious that the word "myriads" is widely used, which means not a certain number at all, but an innumerable, uncountable number of things. It is believed that the word myriad (English myriad) came to European languages from ancient Egypt.
googol(from the English googol) is the number ten to the hundredth power, that is, one with one hundred zeros. The "googol" was first written about in 1938 in the article "New Names in Mathematics" in the January issue of the journal Scripta Mathematica by the American mathematician Edward Kasner. According to him, his nine-year-old nephew Milton Sirotta suggested calling a large number "googol". This number became well-known thanks to the search engine named after him. Google. Note that "Google" is a trademark and googol is a number.
In the famous Buddhist treatise Jaina Sutra, dating back to 100 BC, there is a number asankhiya(from Chinese asentzi- incalculable), equal to 10 140. It is believed that this number is equal to the number of cosmic cycles required to gain nirvana.
Googolplex(English) googolplex) - a number also invented by Kasner with his nephew and meaning one with a googol of zeros, that is, 10 10 100. Here is how Kasner himself describes this "discovery":
Words of wisdom are spoken by children at least as often as by scientists. The name "googol" was invented by a child (Dr. Kasner"s nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. a googol, but is still finite, as the inventor of the name was quick to point out.
Mathematics and the Imagination(1940) by Kasner and James R. Newman.
Even more than a googolplex number, Skewes' number was proposed by Skewes in 1933 (Skewes. J. London Math. soc. 8 , 277-283, 1933.) in proving the Riemann conjecture concerning primes. It means e to the extent e to the extent e to the power of 79, that is, e e e 79. Later, Riele (te Riele, H. J. J. "On the Sign of the Difference P(x)-Li(x)." Math. Comput. 48 , 323-328, 1987) reduced the Skewes number to e e 27/4 , which is approximately equal to 8.185 10 370 . It is clear that since the value of the Skewes number depends on the number e, then it is not an integer, so we will not consider it, otherwise we would have to recall other non-natural numbers - the number pi, the number e, the Avogadro number, etc.
But it should be noted that there is a second Skewes number, which in mathematics is denoted as Sk 2 , which is even larger than the first Skewes number (Sk 1). Skuse's second number, was introduced by J. Skuse in the same article to denote the number up to which the Riemann hypothesis is valid. Sk 2 is equal to 10 10 10 10 3 , that is 10 10 10 1000 .
As you understand, the more degrees there are, the more difficult it is to understand which of the numbers is greater. For example, looking at the Skewes numbers, without special calculations, it is almost impossible to understand which of these two numbers is larger. Thus, for superlarge numbers, it becomes inconvenient to use powers. Moreover, you can come up with such numbers (and they have already been invented) when the degrees of degrees simply do not fit on the page. Yes, what a page! They won't even fit into a book the size of the entire universe! In this case, the question arises how to write them down. The problem, as you understand, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who asked this problem came up with his own way of writing, which led to the existence of several, unrelated, ways to write numbers - these are the notations of Knuth, Conway, Steinhouse, etc.
Consider the notation of Hugo Stenhaus (H. Steinhaus. Mathematical Snapshots, 3rd edn. 1983), which is quite simple. Steinhouse suggested writing large numbers inside geometric shapes - a triangle, a square and a circle:
Steinhouse came up with two new super-large numbers. He named a number Mega, and the number is Megiston.
The mathematician Leo Moser refined Stenhouse's notation, which was limited by the fact that if it was necessary to write numbers much larger than a megiston, difficulties and inconveniences arose, since many circles had to be drawn one inside the other. Moser suggested that after the squares, draw not circles, but pentagons, then hexagons, and so on. He also proposed a formal notation for these polygons, so that numbers could be written without drawing complex patterns. Moser notation looks like this:
Thus, according to Moser's notation, Steinhouse's mega is written as 2, and megiston as 10. In addition, Leo Moser suggested calling a polygon with the number of sides equal to mega - megagon. And he proposed the number "2 in Megagon", that is, 2. This number became known as the Moser's number or simply as moser.
But the moser is not the largest number. The largest number ever used in a mathematical proof is the limiting value known as Graham number(Graham "s number), first used in 1977 in the proof of one estimate in Ramsey theory. It is associated with bichromatic hypercubes and cannot be expressed without a special 64-level system of special mathematical symbols introduced by Knuth in 1976.
Unfortunately, a number written in Knuth's notation cannot be translated into Moser's notation. Therefore, this system will also have to be explained. In principle, there is nothing complicated in it either. Donald Knuth (yes, yes, this is the same Knuth who wrote The Art of Programming and created the TeX editor) came up with the concept of superpower, which he proposed to write with arrows pointing up:
In general, it looks like this:
I think that everything is clear, so let's get back to Graham's number. Graham proposed the so-called G-numbers:
The number G 63 began to be called Graham number(it is often denoted simply as G). This number is the largest known number in the world and is even listed in the Guinness Book of Records. And, here, that the Graham number is greater than the Moser number.
P.S. In order to bring great benefit to all mankind and become famous for centuries, I decided to invent and name the largest number myself. This number will be called stasplex and it is equal to the number G 100 . Memorize it, and when your children ask what is the largest number in the world, tell them that this number is called stasplex.
Update (4.09.2003): Thanks everyone for the comments. It turned out that when writing the text, I made several mistakes. I'll try to fix it now.
- I made several mistakes at once, just mentioning Avogadro's number. First, several people have pointed out to me that 6.022 10 23 is actually the most natural number. And secondly, there is an opinion, and it seems to me true, that Avogadro's number is not a number at all in the proper, mathematical sense of the word, since it depends on the system of units. Now it is expressed in "mol -1", but if it is expressed, for example, in moles or something else, then it will be expressed in a completely different figure, but it will not cease to be Avogadro's number at all.
- 10 000 - darkness
100,000 - legion
1,000,000 - leodre
10,000,000 - Raven or Raven
100 000 000 - deck
Interestingly, the ancient Slavs also loved large numbers, they knew how to count up to a billion. Moreover, they called such an account a “small account”. In some manuscripts, the authors also considered the "great count", which reached the number 10 50 . About numbers greater than 10 50 it was said: "And more than this to bear the human mind to understand." The names used in the "small account" were transferred to the "great account", but with a different meaning. So, darkness meant no longer 10,000, but a million, legion - the darkness of those (million millions); leodrus - a legion of legions (10 to 24 degrees), then it was said - ten leodres, a hundred leodres, ..., and, finally, a hundred thousand legions of leodres (10 to 47); leodr leodr (10 to 48) was called a raven and, finally, a deck (10 to 49). - The topic of national names of numbers can be expanded if we recall the Japanese system of naming numbers that I forgot, which is very different from the English and American systems (I will not draw hieroglyphs, if anyone is interested, then they are):
100-ichi
10 1 - jyuu
10 2 - hyaku
103-sen
104 - man
108-oku
10 12 - chou
10 16 - kei
10 20 - gai
10 24 - jyo
10 28 - jyou
10 32 - kou
10 36-kan
10 40 - sei
1044 - sai
1048 - goku
10 52 - gougasya
10 56 - asougi
10 60 - nayuta
1064 - fukashigi
10 68 - murioutaisuu - Regarding the numbers of Hugo Steinhaus (in Russia, for some reason, his name was translated as Hugo Steinhaus). botev assures that the idea of writing super-large numbers in the form of numbers in circles does not belong to Steinhouse, but to Daniil Kharms, who, long before him, published this idea in the article "Raising the Number". I also want to thank Evgeny Sklyarevsky, the author of the most interesting site on entertaining mathematics on the Russian-speaking Internet - Arbuz, for the information that Steinhouse came up with not only the numbers mega and megiston, but also proposed another number mezzanine, which is (in his notation) "circled 3".
- Now for the number myriad or myrioi. There are different opinions about the origin of this number. Some believe that it originated in Egypt, while others believe that it was born only in Ancient Greece. Be that as it may, in fact, the myriad gained fame precisely thanks to the Greeks. Myriad was the name for 10,000, and there were no names for numbers over ten thousand. However, in the note "Psammit" (i.e., the calculus of sand), Archimedes showed how one can systematically build and name arbitrarily large numbers. In particular, placing 10,000 (myriad) grains of sand in a poppy seed, he finds that in the Universe (a sphere with a diameter of a myriad of Earth diameters) no more than 10 63 grains of sand would fit (in our notation). It is curious that modern calculations of the number of atoms in the visible universe lead to the number 10 67 (only a myriad of times more). The names of the numbers Archimedes suggested are as follows:
1 myriad = 10 4 .
1 di-myriad = myriad myriad = 10 8 .
1 tri-myriad = di-myriad di-myriad = 10 16 .
1 tetra-myriad = three-myriad three-myriad = 10 32 .
etc.
If there are comments -
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 significant 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 generally accepted definitions in the 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 of different ways in which these particles can be arranged 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 the total number of particles in the universe, which is usually considered to be about , and this number is so large that our language does not have a word for it.
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 (not equal to one) that is divisible only by 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.
The mathematicians of ancient Greece had a concept of prime numbers at least as early as 500 BC, and 2000 years later people still only knew what prime numbers were up to about 750. Euclid's thinkers saw the possibility of simplification, but until the Renaissance mathematicians couldn't really use it in 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 ever since, and the th Mersenne number is currently 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 prime number function, invented in the late 18th century by the 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 . It's a great estimate, to be sure, but it's always just 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 Skuse's 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 cease to be 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. (Admittedly, it would take an insanely long time to actually count to a million of anything, but the point is that we are still able to perceive that 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 Skuse's super 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 comes into play 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.''