What comes after Google. The largest number in the world
Once in childhood, we learned to count to ten, then to a hundred, then to a thousand. So what is the biggest number you know? A thousand, a million, a billion, a trillion ... And then? Petallion, someone will say, will be wrong, because he confuses the SI prefix with a completely different concept.
In fact, the question is not as simple as it seems at first glance. First, we are talking about naming the names of the powers of a thousand. And here, the first nuance that many people know from American films is that they call our billion a billion.
Further more, there are two types of scales - long and short. In our country, a short scale is used. In this scale, at each step, the mantis increases by three orders of magnitude, i.e. multiply by a thousand - a thousand 10 3, a million 10 6, a billion / billion 10 9, a trillion (10 12). In the long scale, after a billion 10 9 comes a billion 10 12, and in the future the mantisa already increases by six orders of magnitude, and the next number, which is called a trillion, already stands for 10 18.
But back to our native scale. Want to know what comes after a trillion? Please:
10 3 thousand
10 6 million
10 9 billion
10 12 trillion
10 15 quadrillion
10 18 quintillion
10 21 sextillion
10 24 septillion
10 27 octillion
10 30 nonillion
10 33 decillion
10 36 undecillion
10 39 dodecillion
10 42 tredecillion
10 45 quattuordecillion
10 48 quindecillion
10 51 sedecillion
10 54 septdecillion
10 57 duodevigintillion
10 60 undevigintillion
10 63 vigintillion
10 66 anvigintillion
10 69 duovigintillion
10 72 trevigintillion
10 75 quattorvigintillion
10 78 quinvintillion
10 81 sexwigintillion
10 84 septemvigintillion
10 87 octovigintillion
10 90 novemvigintillion
10 93 trigintillion
10 96 antirigintillion
On this number, our short scale does not stand up, and in the future, the mantissa increases progressively.
10 100 googol
10 123 quadragintillion
10 153 quinquagintillion
10,183 sexagintillion
10 213 septuagintillion
10,243 octogintillion
10,273 nonagintillion
10 303 centillion
10 306 centunillion
10 309 centduollion
10 312 centtrillion
10 315 centquadrillion
10 402 centtretrigintillion
10,603 decentillion
10 903 trecentillion
10 1203 quadringentillion
10 1503 quingentillion
10 1803 sescentillion
10 2103 septingentillion
10 2403 octingentillion
10 2703 nongentillion
10 3003 million
10 6003 duomillion
10 9003 tremillion
10 3000003 miamimiliaillion
10 6000003 duomyamimiliaillion
10 10 100 googolplex
10 3×n+3 zillion
googol(from the English googol) - a number, in the decimal number system, represented by a unit with 100 zeros:
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
In 1938, the American mathematician Edward Kasner (Edward Kasner, 1878-1955) was walking in the park with his two nephews and discussing large numbers with them. During the conversation, we talked about a number with one hundred zeros, which did not have its own name. One of his nephews, nine-year-old Milton Sirotta, suggested calling this number "googol". In 1940, Edward Kasner, together with James Newman, wrote the popular science book "Mathematics and Imagination" ("New Names in Mathematics"), where he taught mathematics lovers about the googol number.
The term "googol" does not have a serious theoretical and practical value. Kasner proposed it to illustrate the difference between an unimaginably large number and infinity, and for this purpose the term is sometimes used in the teaching of mathematics.
Googolplex(from the English googolplex) - a number represented by a unit with a googol of zeros. Like googol, the term googolplex was coined by American mathematician Edward Kasner and his nephew Milton Sirotta.
The number of googols is greater than the number of all particles in the part of the universe known to us, which ranges from 1079 to 1081. Thus, the number of googolplexes, consisting of (googol + 1) digits, cannot be written in the classical “decimal” form, even if all matter in the known turn parts of the universe into paper and ink or into computer disk space.
Zillion(eng. zillion) is a common name for very large numbers.
This term does not have a strict mathematical definition. In 1996, Conway (English J. H. Conway) and Guy (English R. K. Guy) in their book English. The Book of Numbers defined a zillion of the nth power as 10 3×n+3 for the short scale number naming system.
The world of science is simply amazing with its knowledge. However, even the most brilliant person in the world will not be able to comprehend them all. But you need to strive for it. That is why in this article I want to figure out what it is, the largest number.
About systems
First of all, it must be said that there are two systems for naming numbers in the world: American and English. Depending on this, the same number can be called differently, although they have the same meaning. And at the very beginning it is necessary to deal with these nuances in order to avoid uncertainty and confusion.
American system
It will be interesting that this system is used not only in America and Canada, but also in Russia. In addition, it has its own scientific name: the system of naming numbers with a short scale. How are large numbers called in this system? Well, the secret is pretty simple. At the very beginning, there will be a Latin ordinal number, after which the well-known suffix “-million” will simply be added. The following fact will be interesting: in translation from Latin the number "million" can be translated as "thousands". The following numbers belong to the American system: a trillion is 10 12, a quintillion is 10 18, an octillion is 10 27, etc. It will also be easy to figure out how many zeros are written in the number. To do this, you need to know a simple formula: 3 * x + 3 (where "x" in the formula is a Latin numeral).
English system
However, despite the simplicity of the American system, the English system is still more common in the world, which is a system for naming numbers with a long scale. Since 1948, it has been used in countries such as France, Great Britain, Spain, as well as in countries - former colonies England and Spain. The construction of numbers here is also quite simple: the suffix “-million” is added to the Latin designation. Further, if the number is 1000 times larger, the suffix "-billion" is already added. How can you find out the number of zeros hidden in a number?
- If the number ends in "-million", you will need the formula 6 * x + 3 ("x" is a Latin numeral).
- If the number ends in "-billion", you will need the formula 6 * x + 6 (where "x", again, is a Latin numeral).
Examples
On the this stage for example, we can consider how the same numbers will be called, but on a different scale.
You can easily see that the same name in different systems ah stands for different numbers. Like a trillion. Therefore, considering the number, you still need to first find out according to which system it is written.
Off-system numbers
It is worth mentioning that, in addition to system numbers, there are also off-system numbers. Maybe among them the largest number was lost? It's worth looking into this.
- Google. This number is ten to the hundredth power, that is, one followed by one hundred zeros (10,100). This number was first mentioned back in 1938 by scientist Edward Kasner. Very interesting fact: The global search engine "Google" is named after a rather large number at that time - Google. And the name came up with Kasner's young nephew.
- Asankhiya. This is a very interesting name, which is translated from Sanskrit as "innumerable." Its numerical value is one with 140 zeros - 10140. The following fact will be interesting: this was known to people as early as 100 BC. e., as evidenced by the entry in the Jaina Sutra, a famous Buddhist treatise. This number was considered special, because it was believed that the same number of cosmic cycles are needed to reach nirvana. Also at that time, this number was considered the largest.
- Googolplex. This number was invented by the same Edward Kasner and his aforementioned nephew. Its numerical designation is ten to the tenth power, which, in turn, consists of the hundredth power (that is, ten to the googolplex power). The scientist also said that in this way you can get as large a number as you want: googoltetraplex, googolhexaplex, googoloctaplex, googoldekaplex, etc.
- Graham's number is G. This is the largest number recognized as such in the recent 1980 by the Guinness Book of Records. It is significantly larger than the googolplex and its derivatives. And scientists did say that the whole Universe is not able to contain the entire decimal notation of Graham's number.
- Moser number, Skewes number. These numbers are also considered one of the largest and they are most often used in solving various hypotheses and theorems. And since these numbers cannot be written down by generally accepted laws, each scientist does it in his own way.
Latest developments
However, it is still worth saying that there is no limit to perfection. And many scientists believed and still believe that the largest number has not yet been found. And, of course, the honor to do this will fall to them. over this project long time an American scientist from Missouri worked, his work was crowned with success. On January 25, 2012, he found the new largest number in the world, which consists of seventeen million digits (which is the 49th Mersenne number). Note: until that time, the largest number was the one found by the computer in 2008, it had 12 thousand digits and looked like this: 2 43112609 - 1.
Not the first time
It is worth saying that this has been confirmed by scientific researchers. This number went through three levels of verification by three scientists on different computers, which took a whopping 39 days. However, these are not the first achievements in such a search for an American scientist. Previously, he had already opened the largest numbers. This happened in 2005 and 2006. In 2008, the computer interrupted Curtis Cooper's streak of victories, but in 2012 he regained the palm and the well-deserved title of discoverer.
About the system
How does it all happen, how do scientists find the biggest numbers? So, today most of the work for them is done by a computer. In this case, Cooper used distributed computing. What does it mean? These calculations are carried out by programs installed on the computers of Internet users who have voluntarily decided to take part in the study. As part of this project, 14 Mersenne numbers were identified, named after the French mathematician (these are prime numbers that are divisible only by themselves and by one). In the form of a formula, it looks like this: M n = 2 n - 1 ("n" in this formula is a natural number).
About bonuses
A logical question may arise: what makes scientists work in this direction? So, this, of course, is the excitement and desire to be a pioneer. However, even here there are bonuses: Curtis Cooper received a cash prize of $3,000 for his brainchild. But that's not all. The Electronic Frontier Special Fund (abbreviation: EFF) encourages such searches and promises to immediately award cash prizes of $150,000 and $250,000 to those who submit 100 million and a billion prime numbers for consideration. So there is no doubt that a huge number of scientists around the world are working in this direction today.
Simple Conclusions
So what is the biggest number today? On the this moment it was found by an American scientist from the University of Missouri Curtis Cooper, which can be written as follows: 2 57885161 - 1. Moreover, it is also the 48th number of the French mathematician Mersenne. But it is worth saying that there can be no end to these searches. And it is not surprising if, after a certain time, scientists will provide us with the next newly found largest number in the world for consideration. There is no doubt that this will happen in the very near future.
Countless different numbers surround us every day. Surely many people at least once wondered what number is considered the largest. You can simply tell a child that this is a million, but adults are well aware that other numbers follow a million. For example, one has only to add one to the number every time, and it will become more and more - this happens ad infinitum. But if you disassemble the numbers that have names, you can find out what the largest number in the world is called.
The appearance of the names of numbers: what methods are used?
To date, there are 2 systems according to which names are given to numbers - American and English. The first is quite simple, and the second is the most common around the world. American allows you to give names big numbers so: first, the ordinal number in Latin is indicated, and then the suffix “million” is added (the exception here is a million, meaning a thousand). This system is used by Americans, French, Canadians, and it is also used in our country.
English is widely used in England and Spain. According to it, the numbers are named like this: the numeral in Latin is “plus” with the suffix “million”, and the next (a thousand times greater) number is “plus” “billion”. For example, a trillion comes first, followed by a trillion, a quadrillion follows a quadrillion, and so on.
So, the same number in different systems can mean different things, for example, an American billion in the English system is called a billion.
Off-system numbers
In addition to numbers that are written according to known systems (given above), there are also off-system ones. They have their own names, which do not include Latin prefixes.
You can start their consideration with a number called a myriad. It is defined as one hundred hundreds (10000). But for its intended purpose, this word is not used, but is used as an indication of an innumerable multitude. Even Dahl's dictionary will kindly provide a definition of such a number.
Next after the myriad is the googol, denoting 10 to the power of 100. For the first time this name was used in 1938 by an American mathematician E. Kasner, who noted that his nephew came up with this name.
Google (search engine) got its name in honor of Google. Then 1 with a googol of zeros (1010100) is a googolplex - Kasner also came up with such a name.
Even larger than the googolplex is the Skewes number (e to the power of e to the power of e79), proposed by Skuse when proving the Riemann conjecture about prime numbers(1933). There is another Skewes number, but it is used when the Rimmann hypothesis is unfair. It is rather difficult to say which of them is greater, especially when it comes to large degrees. However, this number, despite its "enormity", cannot be considered the most-most of all those that have their own names.
And the leader among the largest numbers in the world is the Graham number (G64). It was he who was used for the first time to conduct proofs in the field of mathematical science (1977).
When it comes to such a number, you need to know that you cannot do without a special 64-level system created by Knuth - the reason for this is the connection of the number G with bichromatic hypercubes. Knuth invented the superdegree, and in order to make it convenient to record it, he suggested using the up arrows. So we learned what the largest number in the world is called. It is worth noting that this number G has hit the pages famous book records.
10 to 3003 degrees
The debate about which is the most big figure in the world are ongoing. Different calculus systems offer different variants and people do not know what to believe, and what kind of figure to consider the largest.
This question has interested scientists since the time of the Roman Empire. The biggest snag lies in the definition of what is a "number" and what is a "number". At one time, people for a long time considered the largest number to be decillion, that is, 10 to the 33rd power. But, after scientists began to actively study the American and English metric systems, it was found that the largest number in the world is 10 to the power of 3003 - a million. Men in Everyday life consider that the largest number is a trillion. Moreover, this is quite formal, because after a trillion, names are simply not given, because the account starts too complicated. However, purely theoretically, the number of zeros can be added indefinitely. Therefore, to imagine even a purely visual trillion and what follows it is almost impossible.
in roman numerals
On the other hand, the definition of "number" in the understanding of mathematicians is a little different. A number is a sign that is universally accepted and is used to indicate a quantity expressed in numerical terms. The second concept of "number" means the expression of quantitative characteristics in a convenient form through the use of numbers. It follows that numbers are made up of digits. It is also important that the figure has sign properties. They are conditioned, recognizable, unchangeable. Numbers also have sign properties, but they follow from the fact that numbers are made up of digits. From this we can conclude that a trillion is not a figure at all, but a number. Then what is the biggest number in the world if it's not a trillion, which is a number?
The important thing is that numbers are used as constituent numbers, but not only that. The figure, however, is the same number if we are talking about some things, counting them from zero to nine. Such a system of signs applies not only to the Arabic numerals familiar to us, but also to the Roman I, V, X, L, C, D, M. These are Roman numerals. On the other hand, V I I I is a Roman number. In Arabic reckoning, it corresponds to the number eight.
in Arabic numerals
Thus, it turns out that counting units from zero to nine are considered numbers, and everything else is numbers. Hence the conclusion that the largest number in the world is nine. 9 is a sign, and a number is a simple quantitative abstraction. A trillion is a number, and not a number, and therefore cannot be the largest number in the world. A trillion can be called the largest number in the world, and then purely nominally, since numbers can be counted to infinity. The number of digits is strictly limited - from 0 to 9.
It should also be remembered that the numbers and numbers of different calculus systems do not match, as we saw from the examples with Arabic and Roman numbers and numerals. This is because numbers and numbers are simple concepts that a person himself invents. Therefore, the number of one system of calculation can easily be the number of another and vice versa.
Thus, the largest number is uncountable, because it can be continued to be added indefinitely from digits. As for the numbers themselves, in the generally accepted system, 9 is considered the largest number.
It is impossible to answer this question correctly, since the number series has no upper limit. So, to any number, it is enough just to add one to get an even larger number. Although the numbers themselves are infinite, they do not have very many proper names, since most of them are content with names made up of smaller numbers. So, for example, the numbers and have their own names "one" and "one hundred", and the name of the number is already compound ("one hundred and one"). It is clear that in the finite set of numbers that humanity has awarded own name must be some largest number. But what is it called and what is it equal to? Let's try to figure it out and at the same time find out how big numbers mathematicians came up with.
"Short" and "long" scale
Story modern system The names of large numbers date back to the middle of the 15th century, when in Italy they began to use the words "million" (literally - a large thousand) for a thousand squared, "bimillion" for a million squared and "trimillion" for a million cubed. We know about this system thanks to the French mathematician Nicolas Chuquet (c. 1450 - c. 1500): in his treatise "The Science of Numbers" (Triparty en la science des nombres, 1484), he developed this idea, proposing to further use the Latin cardinal numbers (see table), adding them to the ending "-million". So, Shuke's "bimillion" turned into a billion, "trimillion" into a trillion, and a million to the fourth power became a "quadrillion".
In Schücke's system, a number that was between a million and a billion did not have its own name and was simply called "a thousand million", similarly it was called "a thousand billion", - "a thousand trillion", etc. It was not very convenient, and in 1549 the French writer and scientist Jacques Peletier du Mans (1517–1582) proposed to name such “intermediate” numbers using the same Latin prefixes, but the ending “-billion”. So, it began to be called "billion", - "billiard", - "trilliard", etc.
The Shuquet-Peletier system gradually became popular and was used throughout Europe. However, in the 17th century, an unexpected problem arose. It turned out that for some reason some scientists began to get confused and call the number not “a billion” or “thousand millions”, but “a billion”. Soon this mistake quickly spread, and a paradoxical situation arose - "billion" became simultaneously a synonym for "billion" () and "million million" ().
This confusion continued for a long time and led to the fact that in the United States they created their own system for naming large numbers. According to the American system, the names of numbers are built in the same way as in the Schuke system - the Latin prefix and the ending "million". However, these numbers are different. If in the Schuecke system names with the ending "million" received numbers that were powers of a million, then in the American system the ending "-million" received the powers of a thousand. That is, a thousand million () became known as a "billion", () - "trillion", () - "quadrillion", etc.
The old system of naming large numbers continued to be used in conservative Great Britain and began to be called "British" all over the world, despite the fact that it was invented by the French Shuquet and Peletier. However, in the 1970s, the UK officially switched to the "American system", which led to the fact that it became somehow strange to call one system American and another British. As a result, the American system is now commonly referred to as the "short scale" and the British or Chuquet-Peletier system as the "long scale".
In order not to get confused, let's sum up the intermediate result:
| Number name | Value on the "short scale" | Value on the "long scale" |
| Million | ||
| Billion | ||
| Billion | ||
| billiard | - | |
| Trillion | ||
| trillion | - | |
| quadrillion | ||
| quadrillion | - | |
| Quintillion | ||
| quintillion | - | |
| Sextillion | ||
| Sextillion | - | |
| Septillion | ||
| Septilliard | - | |
| Octillion | ||
| Octilliard | - | |
| Quintillion | ||
| Nonilliard | - | |
| Decillion | ||
| Decilliard | - | |
| Vigintillion | ||
| viginbillion | - | |
| Centillion | ||
| Centbillion | - | |
| Milleillion | ||
| Milliilliard | - |
The short naming scale is currently used in the US, UK, Canada, Ireland, Australia, Brazil and Puerto Rico. Russia, Denmark, Turkey, and Bulgaria also use the short scale, except that the number is called "billion" rather than "billion". The long scale continues to be used today in most other countries.
It is curious that in our country the final transition to the short scale took place only in the second half of the 20th century. So, for example, even Yakov Isidorovich Perelman (1882–1942) in his “Entertaining Arithmetic” mentions the parallel existence of two scales in the USSR. The short scale, according to Perelman, was used in everyday life and financial calculations, and the long one was used in scientific books on astronomy and physics. However, now it is wrong to use a long scale in Russia, although the numbers there are large.
But back to finding the largest number. After a decillion, the names of numbers are obtained by combining prefixes. This is how numbers such as undecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion, novemdecillion, etc. are obtained. However, these names are no longer of interest to us, since we agreed to find the largest number with its own non-composite name.
If we turn to Latin grammar, we will find that the Romans had only three non-compound names for numbers more than ten: viginti - "twenty", centum - "one hundred" and mille - "thousand". For numbers greater than "thousand", the Romans did not have their own names. For example, a million () The Romans called it “decies centena milia”, that is, “ten times a hundred thousand”. According to Schuecke's rule, these three remaining Latin numerals give us such names for numbers as "vigintillion", "centillion" and "milleillion".
So, we found out that on the "short scale" the maximum number that has its own name and is not a composite of smaller numbers is "million" (). If a “long scale” of naming numbers were adopted in Russia, then the largest number with its own name would be “million billion” ().
However, there are names for even larger numbers.
Numbers outside the system
Some numbers have their own name, without any connection with the naming system using Latin prefixes. And there are many such numbers. You can, for example, remember the number e, the number "pi", a dozen, the number of the beast, etc. However, since we are now interested in large numbers, we will consider only those numbers with their own non-compound name that are more than a million.
Until the 17th century, Russia used its own system for naming numbers. Tens of thousands were called "darks", hundreds of thousands were called "legions", millions were called "leodras", tens of millions were called "ravens", and hundreds of millions were called "decks". This account up to hundreds of millions was called the “small account”, and in some manuscripts the authors also considered the “great account”, in which the same names were used for large numbers, but with a different meaning. So, "darkness" meant no longer ten thousand, but a thousand thousand () , "legion" - the darkness of those () ; "leodr" - legion of legions () , "raven" - leodr leodrov (). “Deck” in the great Slavic account for some reason was not called “raven of ravens” () , but only ten "ravens", that is (see table).
| Number name | Meaning in "small count" | Meaning in the "great account" | Designation |
| Dark | |||
| Legion | |||
| Leodr | |||
| Raven (Raven) | |||
| Deck | |||
| Darkness of topics |  |
The number also has its own name and was invented by a nine-year-old boy. And it was like that. In 1938, the American mathematician Edward Kasner (Edward Kasner, 1878–1955) was walking in the park with his two nephews and discussing large numbers with them. During the conversation, we talked about a number with one hundred zeros, which did not have its own name. One of his nephews, nine-year-old Milton Sirott, suggested calling this number "googol". In 1940, Edward Kasner, together with James Newman, wrote the popular science book "Mathematics and Imagination", where he told mathematics lovers about the number of googols. Google became even more widely known in the late 1990s, thanks to the Google search engine named after it.
The name for an even larger number than the googol arose in 1950 thanks to the father of computer science, Claude Shannon (Claude Elwood Shannon, 1916–2001). In his article "Programming a Computer to Play Chess," he tried to estimate the number options chess game. According to it, each game lasts an average of moves, and on each move the player makes an average choice of options, which corresponds to (approximately equal to) the game options. This work became widely known and given number became known as the Shannon number.
In the well-known Buddhist treatise Jaina Sutra, dating back to 100 BC, the number "asankheya" is found equal to . It is believed that this number is equal to the number of cosmic cycles required to gain nirvana.
Nine-year-old Milton Sirotta entered the history of mathematics not only by inventing the googol number, but also by suggesting another number at the same time - “googolplex”, which is equal to the power of “googol”, that is, one with the googol of zeros.
Two more numbers larger than the googolplex were proposed by the South African mathematician Stanley Skewes (1899–1988) when proving the Riemann Hypothesis. The first number, which later came to be called "Skews's first number", is equal to the power to the power to the power of , that is, . However, the "second Skewes number" is even larger and amounts to .
Obviously, the more degrees in the number of degrees, the more difficult it is to write down numbers and understand their meaning when reading. Moreover, it is possible to come up with such numbers (and they, by the way, 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 in a book the size of the entire universe! In this case, the question arises how to write down such numbers. The problem is, fortunately, resolvable, and mathematicians have developed several principles for writing such numbers. True, each mathematician who asked this problem came up with his own way of writing, which led to the existence of several unrelated ways to write large numbers - these are the notations of Knuth, Conway, Steinhaus, etc. We will now have to deal with some of them.
Other notations
In 1938, the same year that nine-year-old Milton Sirotta came up with the googol and googolplex numbers, Hugo Dionizy Steinhaus (1887–1972), a book about entertaining mathematics, The Mathematical Kaleidoscope, was published in Poland. This book became very popular, went through many editions and was translated into many languages, including English and Russian. In it, Steinhaus, discussing large numbers, offers a simple way to write them using three geometric figures- triangle, square and circle:
"in a triangle" means "",
"in a square" means "in triangles",
"in a circle" means "in squares".
Explaining this way of writing, Steinhaus comes up with the number "mega", equal in a circle and shows that it is equal in a "square" or in triangles. To calculate it, you need to raise it to a power, raise the resulting number to a power, then raise the resulting number to the power of the resulting number, and so on to raise the power of times. For example, the calculator in MS Windows cannot calculate due to overflow even in two triangles. Approximately this huge number is .
Having determined the number "mega", Steinhaus invites readers to independently evaluate another number - "medzon", equal in a circle. In another edition of the book, Steinhaus, instead of the medzone, proposes to estimate an even larger number - “megiston”, equal in a circle. Following Steinhaus, I will also recommend that readers take a break from this text for a while and try to write these numbers themselves using ordinary powers in order to feel their gigantic magnitude.
However, there are names for large numbers. Thus, the Canadian mathematician Leo Moser (Leo Moser, 1921–1970) finalized the Steinhaus notation, which was limited by the fact that if it were necessary to write down numbers much larger than a megiston, then difficulties and inconveniences would arise, since many circles would have to be drawn one inside another. Moser suggested drawing not circles after squares, 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:
"triangle" = = ;
"in a square" = = "in triangles" =;
"in the pentagon" = = "in the squares" = ;
"in -gon" = = "in -gons" = .
Thus, according to Moser's notation, the Steinhausian "mega" is written as , "medzon" as , and "megiston" as . In addition, Leo Moser proposed to call a polygon with the number of sides equal to mega - "megagon". And offered a number « in a megagon", that is. This number became known as the Moser number, or simply as "moser".
But even "moser" is not the largest number. So, the largest number ever used in a mathematical proof is "Graham's number". This number was first used by the American mathematician Ronald Graham in 1977 when proving one estimate in Ramsey theory, namely when calculating the dimensions of certain -dimensional bichromatic hypercubes. Graham's number gained fame only after the story about it in Martin Gardner's 1989 book "From Penrose Mosaics to Secure Ciphers".
To explain how large the Graham number is, one has to explain another way of writing large numbers, introduced by Donald Knuth in 1976. American professor Donald Knuth came up with the concept of superdegree, which he proposed to write with arrows pointing up.
Common arithmetic operations - addition, multiplication and exponentiation - naturally can be expanded into a sequence of hyperoperators as follows.
Multiplication natural numbers can be defined through a repetitive addition operation (“add copies of a number”):
For example,
Raising a number to a power can be defined as a repeated multiplication operation ("multiply copies of a number"), and in Knuth's notation this entry looks like a single arrow pointing up:
For example,
Such a single up arrow was used as a degree icon in the Algol programming language.
For example,
Here and below, the evaluation of the expression always goes from right to left, and Knuth's arrow operators (as well as the exponentiation operation) by definition have right associativity (right-to-left ordering). According to this definition,
This already leads to quite large numbers, but the notation does not end there. The triple arrow operator is used to write repeated exponentiation of the double arrow operator (also known as "pentation"):
Then the "quadruple arrow" operator:
Etc. General rule operator "-I arrow", according to right associativity, continues to the right into a sequential series of operators « arrow". Symbolically, this can be written as follows,
For example:
The notation form is usually used for writing with arrows.
Some numbers are so large that even writing with Knuth's arrows becomes too cumbersome; in this case, the use of the -arrow operator is preferable (and also for a description with a variable number of arrows), or equivalent, to hyperoperators. But some numbers are so huge that even such a notation is not enough. For example, the Graham number.
When using Knuth's Arrow notation, the Graham number can be written as
Where the number of arrows in each layer, starting from the top, is determined by the number in the next layer, i.e. , where , where the superscript of the arrow indicates the total number of arrows. In other words, it is calculated in steps: in the first step we calculate with four arrows between threes, in the second - with arrows between threes, in the third - with arrows between threes, and so on; at the end we calculate from the arrows between the triplets.
This can be written as , where , where the superscript y denotes function iterations.
If other numbers with "names" can be matched with the corresponding number of objects (for example, the number of stars in the visible part of the Universe is estimated in sextillions - , and the number of atoms that make up Earth has the order of dodecallions), then the googol is already "virtual", not to mention the Graham number. The scale of the first term alone is so large that it is almost impossible to comprehend it, although the notation above is relatively easy to understand. Although - this is just the number of towers in this formula for , this number is already much larger than the number of Planck volumes (the smallest possible physical volume) that are contained in the observable universe (approximately ). After the first member, another member of the rapidly growing sequence awaits us.