What is the largest amount in the world. Not included in the collection of works

Once I read a tragic story, which tells about the Chukchi, whom polar explorers taught to count and write numbers. The magic of numbers impressed him so much that he decided to write down absolutely all the numbers in the world in a row, starting with one, in the notebook donated by the polar explorers. The Chukchi abandons all his affairs, stops communicating even with his own wife, no longer hunts for seals and seals, but writes everything and writes numbers in a notebook .... So a year goes by. In the end, the notebook ends and the Chukchi understands that he was able to write down only a small part of all the numbers. He cries bitterly and, in despair, burns his scribbled notebook in order to start living the simple life of a fisherman again, not thinking any more about the mysterious infinity of numbers ...

We will not repeat the feat of this Chukchi and try to find the largest number, since any number just needs to add one to get an even larger number. Let us ask ourselves, albeit similar, but a different question: which of the numbers that have their own name is the largest?

Obviously, although the numbers themselves are infinite, they do not have so many proper names, since most of them are content with names composed of smaller numbers. So, for example, the numbers 1 and 100 have their own names "one" and "one hundred", and the name of the number 101 is already compound ("one hundred and one"). It is clear that in the finite set of numbers that humanity has awarded with its own name, there must be some largest number. But what is it called and what is it equal to? Let's try to figure it out and find, in the end, this is the largest number!

Number	Latin cardinal number	Russian prefix

"Short" and "Long" scale

The history of the modern system of naming large numbers dates 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 “trillion” for a million cubed. We know about this system thanks to the French mathematician Nicolas Chuquet (c. 1450 - c. 1500): in his treatise "Science of numbers" (Triparty en la science des nombres, 1484), he developed this idea, suggesting further use of Latin cardinal numbers (see table), adding them to the ending "-million". Thus, Schuquet's “bimillion” became a billion, “trillion” into a trillion, and a million to the fourth power became “quadrillion”.

In the Schuke system, the number 10 9, which was between a million and a billion, did not have its own name and was simply called “thousand million”, similarly 10 15 was called “thousand billion”, 10 21 - “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, 10 9 began to be called “billion”, 10 15 - “billiard”, 10 21 - “trillion”, etc.

The Suke-Peletier system gradually became popular and began to be used throughout Europe. However, in the 17th century, an unexpected problem arose. It turned out that some scientists for some reason began to get confused and call the number 10 9 not “billion” or “thousand million”, but “billion”. Soon, this mistake quickly spread, and a paradoxical situation arose - “billion” became simultaneously synonymous with “billion” (10 9) and “million million” (10 18).

This confusion lasted long enough and led to the fact that the United States created its own system of naming large numbers. According to the American system, the names of numbers are constructed in the same way as in the Schuke system - the Latin prefix and the ending "illion". However, the magnitudes of these numbers are different. If in the Shuke system names with the ending “million” received numbers that were degrees of a million, then in the American system the ending “-million” received degrees of a thousand. That is, one thousand million (1000 3 \u003d 10 9) began to be called “billion”, 1000 4 (10 12) - “trillion”, 1000 5 (10 15) - “quadrillion”, etc.

The old system of naming large numbers continued to be used in conservative Great Britain and began to be called "British" throughout the world, despite the fact that it was invented by the French Schuquet and Peletier. However, in the 1970s, Great Britain officially switched to the "American system", which led to the fact that it became somewhat strange to call one system American and the other British. As a result, the American system is now commonly referred to as the "short scale", and the British system, or the Schuke-Peletier system, as the "long scale".

In order not to get confused, let's summarize the intermediate result:

Number name	Short scale value	Long Scale Value

Billion

Billiard
Trillion
Trillion
Quadrillion
Quadrillion
Quintillion
Quintilliard
Sextillion
Sexbillion
Septillion
Septilliard
Octillion
Octilliard
Quintillion
Nonbillion
Decillion
Decilliard

The short naming scale is now used in the United States, United Kingdom, Canada, Ireland, Australia, Brazil and Puerto Rico. Russia, Denmark, Turkey and Bulgaria also use a short scale, except that the number 10 9 is not called “billion”, but “billion”. The long scale is still used 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. 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 scale was used in scientific books on astronomy and physics. However, now it is wrong to use the long scale in Russia, although the numbers there turn out to be large.

But back to finding the largest number. After 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 interesting to us, since we agreed to find the largest number with our own non-composite name.

If we turn to Latin grammar, we 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 "a thousand", the Romans did not have their own names. For example, the Romans called a million (1,000,000) "decies centena milia", that is, "ten times a hundred thousand." By Schücke's rule, these three remaining Latin numerals give us names for numbers such 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 the smaller numbers is "a million" (10 3003). If Russia adopted the "long scale" of naming numbers, then the largest number with its own name would be "milliard" (10 6003).

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-composite name, which are more than a million.

Until the 17th century, Russia used its own system of naming numbers. Tens of thousands were called "darkness", hundreds of thousands - "legions", millions - "leodrs", tens of millions - "crows", and hundreds of millions - "decks". This counting up to hundreds of millions was called the "little count", and in some manuscripts the authors also considered the "great count", which used the same names for large numbers, but with a different meaning. So, "darkness" meant not ten thousand, but a thousand thousand (10 6), "legion" - the darkness of those (10 12); "Leodr" - legion of legions (10 24), "raven" - leodr leodr (10 48). For some reason, the “deck” in the great Slavic account was called not “ravens of ravens” (10 96), but only ten “ravens”, that is, 10 49 (see table).

Number name	Meaning in "small count"	Value in the "grand score"	Designation



Raven (vran)

The number 10 100 also has its own name and was invented by a nine-year-old boy. And it was like this. In 1938, the American mathematician Edward Kasner (1878-1955) walked in the park with his two nephews and discussed large numbers with them. During the conversation, we were talking about a number with one hundred zeros, which did not have its own name. One of the nephews, nine-year-old Milton Sirott, suggested calling the number "googol". In 1940, Edward Kasner, together with James Newman, wrote the popular science book "Mathematics and the Imagination", where he told the lovers of mathematics about the number of googols. Google gained even more prominence in the late 1990s, thanks to the Google search engine named after it.

The name for an even larger number than googol originated in 1950 thanks to the father of computer science, Claude Elwood Shannon (1916-2001). In his article "Programming a Computer for Playing Chess," he tried to estimate the number of possible variants of a chess game. According to him, each game lasts on average 40 moves and on each move the player makes a choice on average out of 30 options, which corresponds to 900 40 (approximately equal to 10 118) options for the game. This work became widely known, and this number became known as the "Shannon number".

In the famous Buddhist treatise Jaina Sutra dating back to 100 BC, the number "asankheya" is found equal to 10 140. It is believed that this number is equal to the number of cosmic cycles required to attain nirvana.

Nine-year-old Milton Sirotta went down in the history of mathematics not only because he came up with the number of googol, but also because at the same time he proposed another number - "googolplex", which is equal to 10 to the power of "googol", that is, one with a googol of zeros.

Two more numbers, larger than the googolplex, were proposed by the South African mathematician Stanley Skewes (1899-1988) in proving the Riemann hypothesis. The first number, which later became known as the "first Skuse number", is e to the extent e to the extent e to the 79th power, that is e e e 79 \u003d 10 10 8.85.10 33. However, the "second Skewes number" is even larger and amounts to 10 10 10 1000.

Obviously, the more degrees there are in degrees, the more difficult it is to write 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 into a book the size of the entire Universe! In this case, the question arises how to write such numbers. The problem is fortunately solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who asked this problem invented 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 now have to deal with some of them.

Other notations

In 1938, the same year that nine-year-old Milton Sirotta invented the numbers googol and googolplex, a book about entertaining mathematics, Mathematical Kaleidoscope, written by Hugo Dionizy Steinhaus (1887-1972), was published in Poland. This book has become very popular, has gone through many editions and has been translated into many languages, including English and Russian. In it, Steinhaus, discussing large numbers, offers a simple way to write them using three geometric shapes - a triangle, a square and a circle:

"N in a triangle "means" n n»,
« n squared "means" n in n triangles ",
« n in a circle "means" n in n squares ".

Explaining this way of writing, Steinhaus comes up with the number "mega", equal to 2 in a circle and shows that it is equal to 256 in a "square" or 256 in 256 triangles. To calculate it, you need to raise 256 to the power of 256, raise the resulting number 3.2.10 616 to the power of 3.2.10 616, then raise the resulting number to the power of the resulting number, and so on, raise the total to the power of 256 times. For example, a calculator in MS Windows cannot calculate because of overflow 256 even in two triangles. This huge number is approximately 10 10 2.10 619.

Having determined the number "mega", Steinhaus invites the readers to independently estimate another number - "mezon", equal to 3 in a circle. In another edition of the book, Steinhaus, instead of the mezzon, proposes to estimate an even larger number - "megiston", equal to 10 in a circle. Following Steinhaus, I would also recommend readers to temporarily break away from this text and try to write these numbers themselves using ordinary degrees in order to feel their gigantic magnitude.

However, there are names for b abouthigher numbers. So, the Canadian mathematician Leo Moser (Leo Moser, 1921-1970) modified the Steinhaus notation, which was limited by the fact that if it was required to write down numbers that are many large megistones, then difficulties and inconveniences would arise, since many circles would have to be drawn one inside another. Moser suggested drawing not circles, but pentagons after the squares, then hexagons, and so on. He also proposed a formal notation for these polygons so that numbers can be written down without drawing complex drawings. Moser's notation looks like this:

« n triangle "\u003d n n = n;
« n squared "\u003d n = « n in n triangles "\u003d n n;
« n in a pentagon "\u003d n = « n in n squares "\u003d n n;
« n in k +1-gon "\u003d n[k+1] \u003d " n in n k-gons "\u003d n[k] n.

Thus, according to Moser's notation, the Steinhaus “mega” is written as 2, the “mezon” as 3, and the “megiston” as 10. In addition, Leo Moser proposed to call a polygon with the number of sides equal to mega - “mega-gon”. And he proposed the number "2 in mega", that is 2. This number became known as Moser's number or simply as "Moser".

But even Moser is not the largest number. So, the largest number ever used in a mathematical proof is the "Graham number". This number was first used by the American mathematician Ronald Graham in 1977 when proving one estimate in Ramsey's theory, namely, when calculating the dimensions of certain n-dimensional bichromatic hypercubes. But Graham's number gained fame only after the story about him in Martin Gardner's book, From Penrose Mosaics to Reliable Ciphers, published in 1989.

To explain how large the Graham number is, we have 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 down with arrows pointing up:

I think everything is clear, so back to Graham's number. Ronald Graham proposed the so-called G-numbers:

Here is the number G 64 and is called the Graham number (it is often denoted simply as G). This number is the largest known number in the world used in mathematical proof, and is even listed in the Guinness Book of Records.

And finally

Having written this article, I cannot help but be tempted to come up with my own number. Let this number be called " stasplex"And will be equal to the number G 100. Remember it, and when your children ask what is the largest number in the world, tell them that this number is called stasplex.

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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 this. 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 number naming systems 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, you need to deal with precisely 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 short-scale naming system for numbers. What are large numbers called in this system? So, 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 turn out to be interesting: in translation from the Latin language, the number “million” can be translated as “thousand”. 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 that were former colonies of 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 added. How can you find out the number of zeros hidden in the 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 of

At this stage, for example, you can consider how the same numbers will be called, but in a different scale.

You can easily see that the same name in different systems means different numbers. For example, a trillion. Therefore, considering a number, you still need to first find out according to which system it is written.

Off-system numbers

It should be said that, in addition to the system numbers, there are also non-system numbers. Perhaps the largest number was lost among them? It's worth looking into this.

Googol. It 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 the scientist Edward Kasner. A very interesting fact: the world search engine "Google" is named after a rather large number at that time - googol. And the name was invented by Kasner's young nephew.
Asankheya. This is a very interesting name, which is translated from Sanskrit as "innumerable". Its numerical value is one with 140 zeros - 10 140. The following fact will be interesting: it 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 is 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, googolctaplex, googoldecaplex, etc.
Graham's number - G. This is the largest number recognized as such in the near 1980 by the Guinness Book of Records. It is significantly larger than googolplex and its derivatives. And scientists did say that the entire Universe is not able to contain the entire decimal notation of Graham's number.
Moser's number, Skuse's number. These numbers are also considered one of the largest and they are most often used when solving various hypotheses and theorems. And since these numbers cannot be written down by all 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, they will be honored to do this. An American scientist from Missouri worked on this project for a long time, his works were crowned with success. On January 25, 2012, he found the new largest number in the world, which is seventeen million digits (which is the 49th Mersenne number). Note: until that time, the largest number was found by a computer in 2008, it consisted of 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 passed 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. He had previously opened the largest numbers. This happened in 2005 and 2006. In 2008, the computer interrupted a series of victories by Curtis Cooper, but in 2012 he regained the palm and the well-deserved title of discoverer.

About the system

How does this all happen, how do scientists find the largest numbers? So, today the computer does most of the work for them. In this case, Cooper used distributed computing. What does it mean? These calculations are carried out by programs installed on computers of Internet users who voluntarily decided to take part in the study. Within the framework of this project, 14 Mersenne numbers were determined, 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 \u003d 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, it is, of course, passion and desire to be a pioneer. However, this also has its own bonuses: for his brainchild, Curtis Cooper received a cash prize of $ 3,000. 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 billion prime numbers. So there is no doubt that a huge number of scientists around the world are working in this direction today.

Simple conclusions

So what's the biggest number today? At the 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 should be said that there can be no end to this search. And it is not surprising if, after a certain time, scientists will submit to us for consideration the next newly discovered largest number in the world. There is no doubt that this will happen as soon as possible.

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 many names of their own, since most of them are content with names made up of smaller numbers. So, for example, numbers and have their own names "one" and "one hundred", and the name of the number is already composite ("one hundred and one"). It is clear that in the finite set of numbers that humanity has awarded with its own name, there 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 have invented.

"Short" and "Long" scale

In the Schücke system, the number between a million and a billion did not have its own name and was simply called “one thousand million”, similarly it was called “one thousand billion”, “one 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" - "trillion", etc.

The Suke-Peletier system gradually became popular and began to be used throughout Europe. However, in the 17th century, an unexpected problem arose. It turned out that some scientists for some reason began to get confused and call the number not a “billion” or “a thousand million”, but “a billion”. Soon, this error spread quickly, and a paradoxical situation arose - “billion” became simultaneously synonymous with “billion” () and “million million” ().

This confusion lasted long enough and led to the fact that the United States created its own system of naming large numbers. According to the American system, the names of numbers are constructed in the same way as in the Schuke system - the Latin prefix and the ending "illion". However, the values \u200b\u200bof these numbers differ. If in the Shuke system names with the ending "million" received numbers that were degrees of a million, then in the American system the ending "-million" received degrees of a thousand. That is, a thousand million () began to be called "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" throughout the world, despite the fact that it was invented by the French Schuquet and Peletier. However, in the 1970s, Great Britain officially switched to the "American system", which led to the fact that it became somewhat strange to call one system American and the other British. As a result, the American system is now commonly referred to as the "short scale" and the British system or the Suquet-Peletier system as the "long scale".

In order not to get confused, let's summarize the intermediate result:

Number name	Short scale value	Long Scale Value
Million
Billion
Billion
Billiard	-
Trillion
Trillion	-
Quadrillion
Quadrillion	-
Quintillion
Quintilliard	-
Sextillion
Sexbillion	-
Septillion
Septilliard	-
Octillion
Octilliard	-
Quintillion
Nonbillion	-
Decillion
Decilliard	-
Vigintillion
Vigintilliard	-
Centillion
Centilliard	-
Million
Milliard	-

The short naming scale is now used in the United States, United Kingdom, Canada, Ireland, Australia, Brazil and Puerto Rico. Russia, Denmark, Turkey and Bulgaria also use a short scale, except that the number is not called “billion” but “billion”. The long scale is still used 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. For example, 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 scale was used in scientific books on astronomy and physics. However, now it is wrong to use the long scale in Russia, although the numbers there turn out to be large.

If we turn to Latin grammar, we 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 "one thousand", the Romans did not have their own names. For example, a million () the Romans called "decies centena milia", that is, "ten times a hundred thousand." By Schuke's rule, these three remaining Latin numerals give us names for numbers like "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 the smaller numbers is "million" (). If the "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

Until the 17th century, Russia used its own system of naming numbers. Tens of thousands were called "darkness", hundreds of thousands - "legions", millions - "leodrs", tens of millions - "crows", and hundreds of millions - "decks". This counting up to hundreds of millions was called the "little count", and in some manuscripts the authors also considered the "great count", which used the same names for large numbers, but with a different meaning. So, "darkness" meant not ten thousand, but a thousand thousand () , "Legion" - the darkness of the () ; "Leodr" - legion of legions () , "Raven" - leodr leodrov (). For some reason, the "deck" in the great Slavic account was not called the "raven of ravens" () , but only ten "ravens", that is (see table).

Number name	Meaning in "small count"	Value in the "grand score"	Designation
Darkness
Legion
Leodre
Raven (vran)
Deck
Darkness of themes

The number also has its own name and was invented by a nine-year-old boy. And it was like this. In 1938, the American mathematician Edward Kasner (1878-1955) walked in the park with his two nephews and discussed large numbers with them. During the conversation, we were talking about a number with one hundred zeros, which did not have its own name. One of the nephews, nine-year-old Milton Sirott, suggested calling the number "googol". In 1940, Edward Kasner, together with James Newman, wrote the popular science book Mathematics and the Imagination, where he told math lovers about the number of googols. Google gained even more prominence in the late 1990s, thanks to the Google search engine named after it.

The name for an even larger number than googol originated in 1950 thanks to the father of computer science, Claude Elwood Shannon (1916–2001). In his article "Programming a Computer for Playing Chess," he tried to estimate the number of possible variants of a chess game. According to him, each game lasts an average of moves and on each move the player makes a choice on average from the options, which corresponds (approximately equal) to the options of the game. This work became widely known, and this number became known as the "Shannon number".

In the famous Buddhist treatise Jaina Sutra, dating back to 100 BC, the number "asankheya" is found equal. It is believed that this number is equal to the number of cosmic cycles required to attain nirvana.

Nine-year-old Milton Sirotta went down in the history of mathematics not only by inventing the number googol, but also by proposing another number at the same time - googolplex, which is equal to the power of googol, that is, one with 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 "the first Skuse number", is equal in degree to degree in degree, that is. However, the "second Skuse number" is even larger and is.

Other notations

In 1938, the same year that nine-year-old Milton Sirotta invented the numbers googol and googolplex, a book about entertaining mathematics, Mathematical Kaleidoscope, written by Hugo Dionizy Steinhaus (1887-1972) was published in Poland. This book has become very popular, has gone through many editions and has been translated into many languages, including English and Russian. In it, Steinhaus, discussing large numbers, offers a simple way to write them using three geometric shapes - a triangle, a square and a circle:

"In a triangle" means "",
"Squared" 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, raise everything to a power of times. For example, a calculator in MS Windows cannot calculate due to overflow even in two triangles. Approximately this huge number is.

Having determined the number "mega", Steinhaus already invites the readers to independently estimate another number - "mezons", equal in the circle. In another edition of the book, Steinhaus, instead of a mezzon, suggests evaluating an even larger number - "megiston", equal in a circle. Following Steinhaus, I will also recommend readers to temporarily break away from this text and try to write these numbers themselves using ordinary degrees in order to feel their gigantic magnitude.

However, there are names for large numbers. For example, the Canadian mathematician Leo Moser (1921-1970) modified the Steinhaus notation, which was limited by the fact that if it was required to write down numbers that are many large megistones, then difficulties and inconveniences would arise, since many circles would have to be drawn one inside another. Moser suggested drawing not circles, but pentagons after the squares, then hexagons, and so on. He also proposed a formal notation for these polygons, so that numbers could be written down without drawing complex drawings. Moser's notation looks like this:

"Triangle" \u003d \u003d;
"Squared" \u003d \u003d "in triangles" \u003d;
"In a pentagon" \u003d \u003d "in squares" \u003d;
"In the -gon" \u003d \u003d "in the -gons" \u003d.

Thus, according to Moser's notation, the Steinhaus "mega" is written as, "mezon" as, and "megiston" as. In addition, Leo Moser suggested calling a polygon with the number of sides equal to a mega - "mega-gon". And suggested the number « in the 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 the "Graham number". This number was first used by the American mathematician Ronald Graham in 1977 when proving one estimate in Ramsey's theory, namely, when calculating the dimensions of certain -dimensional bichromatic hypercubes. But Graham's number gained fame only after the story about him in Martin Gardner's book "From Penrose Mosaics to Reliable Ciphers", published in 1989.

The usual arithmetic operations - addition, multiplication, and exponentiation - can naturally be extended into a sequence of hyperoperators as follows.

Multiplication of natural numbers can be defined through a repeated addition operation ("add copies of a number"):

For instance,

Raising a number to a power can be defined as a repetitive multiplication operation (“multiply copies of a number”), and in Knuth's notation this notation looks like a single arrow pointing up:

For instance,

This single up arrow was used as a degree icon in the Algol programming language.

For instance,

Hereinafter, the expression is always evaluated from right to left, also Knuth's arrow operators (like the exponentiation operation), by definition, have right associativity (order from right to left). 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 the repeated exponentiation of the double arrow operator (also known as pentation):

Then the operator "quadruple arrow":

Etc. General rule operator "-I arrow ", in accordance with the right associativity, continues to the right in a sequential series of operators « arrow ". Symbolically, this can be written as follows,

For instance:

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 preferred (and also for descriptions with a variable number of arrows), or equivalently, to hyperoperators. But some numbers are so huge that even such a record is insufficient. For example, Graham's number.

When using Knuth's Arrow Notation, Graham's 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, that is, where, where the superscript of the arrow shows the total number of arrows. In other words, it is calculated in steps: in the first step we calculate with four arrows between the threes, in the second - with arrows between the threes, in the third - with the arrows between the threes, and so on; at the end we calculate from the arrows between the triplets.

It can be written as, where, where the superscript y means iterating over the functions.

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 sextillons -, and the number of atoms that make up the earth is of the order of dodecalions), then the googol is already "virtual", not to mention about Graham's number. The scale of only the first term is so great that it is almost impossible to grasp it, although the entry 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.

In the names of Arabic numbers, each digit belongs to its own category, and every three digits form a class. Thus, the last digit in a number denotes the number of ones in it and is called, respectively, the ones place. The next, second from the end, number denotes tens (tens place), and the third number from the end indicates the number of hundreds in the number - hundreds place. Further, the discharges are repeated in turn in each class in the same way, denoting already units, tens and hundreds in classes of thousands, millions, and so on. If the number is small and does not contain tens or hundreds, it is customary to take them as zero. Classes group numbers in numbers of three, often in calculators or records between classes, a period or a space is put in order to visually separate them. This is to make it easier to read large numbers. Each class has its own name: the first three digits are the class of units, followed by the class of thousands, then millions, billions (or billions), and so on.

Since we are using the decimal system, the basic unit of measure for quantity is ten, or 10 1. Accordingly, with an increase in the number of digits in a number, the number of tens also increases 10 2, 10 3, 10 4, etc. Knowing the number of tens, you can easily determine the class and place of the number, for example, 10 16 is tens of quadrillion, and 3 × 10 16 is three tens of quadrillion. The decomposition of numbers into decimal components is as follows - each digit is displayed in a separate summand, multiplied by the required coefficient 10 n, where n is the position of the digit from left to right.
For instance: 253 981 \u003d 2 × 10 6 + 5 × 10 5 + 3 × 10 4 + 9 × 10 3 + 8 × 10 2 + 1 × 10 1

Also, the power of 10 is used in writing decimal fractions: 10 (-1) is 0.1 or one tenth. Similarly with the previous paragraph, you can expand the decimal number, n in this case will indicate the position of the digit from the comma from right to left, for example: 0.347629 \u003d 3 × 10 (-1) + 4 × 10 (-2) + 7 × 10 (-3) + 6 × 10 (-4) + 2 × 10 (-5) + 9 × 10 (-6 )

Decimal names. Decimal numbers are read according to the last digit after the decimal point, for example 0.325 - three hundred twenty-five thousandths, where thousandths is the last digit 5.

Table of names of large numbers, digits and classes

1st class unit	1st digit of the unit 2nd rank tens 3rd rank hundreds	1 = 10 0 10 = 10 1 100 = 10 2
2nd class thousand	1st digit units of thousand 2nd rank tens of thousands 3rd rank hundreds of thousands	1 000 = 10 3 10 000 = 10 4 100 000 = 10 5
3rd grade millions	1st digit unit million 2nd rank tens of millions 3rd rank hundreds of millions	1 000 000 = 10 6 10 000 000 = 10 7 100 000 000 = 10 8
4th grade billions	1st digit unit billion 2nd rank tens of billions 3rd rank hundreds of billions	1 000 000 000 = 10 9 10 000 000 000 = 10 10 100 000 000 000 = 10 11
5th grade trillions	1st rank unit trillion 2nd rank tens trillion 3rd rank hundreds trillion	1 000 000 000 000 = 10 12 10 000 000 000 000 = 10 13 100 000 000 000 000 = 10 14
6th grade quadrillion	1st digit unit of quadrillion 2nd grade tens of quadrillion 3rd rank tens of quadrillion	1 000 000 000 000 000 = 10 15 10 000 000 000 000 000 = 10 16 100 000 000 000 000 000 = 10 17
7th grade quintillions	1st digit unit of quintillion 2nd rank tens quintillion 3rd rank hundreds of quintillion	1 000 000 000 000 000 000 = 10 18 10 000 000 000 000 000 000 = 10 19 100 000 000 000 000 000 000 = 10 20
8th grade sextillion	1st rank unit of sextillion 2nd rank tens of sextillions 3rd rank hundreds of sextillions	1 000 000 000 000 000 000 000 = 10 21 10 000 000 000 000 000 000 000 = 10 22 1 00 000 000 000 000 000 000 000 = 10 23
9th grade septillions	1st rank unit of septillion 2nd rank tens septillion 3rd rank hundreds septillion	1 000 000 000 000 000 000 000 000 = 10 24 10 000 000 000 000 000 000 000 000 = 10 25 100 000 000 000 000 000 000 000 000 = 10 26
10th grade octillion	1st digit of the unit of octillion 2nd digit tens of octillion 3rd rank hundreds of octillion	1 000 000 000 000 000 000 000 000 000 = 10 27 10 000 000 000 000 000 000 000 000 000 = 10 28 100 000 000 000 000 000 000 000 000 000 = 10 29

Answering such a difficult question, what is the largest number in the world, first it should be noted that today there are 2 accepted ways of naming numbers - English and American. According to the English system, the suffixes -billion or -million are added to each large number in order, resulting in the number of million, billion, trillion, trillion, and so on. If we proceed from the American system, then according to it, it is necessary to add the suffix-million to each large number, as a result of which the numbers trillion, quadrillion and larger are formed. It should be noted here that the English number system is more widespread in the modern world, and the numbers available in it are quite sufficient for the normal functioning of all systems of our world.

Of course, the answer to the question about the largest number from a logical point of view cannot be unambiguous, because if you only add one to each subsequent digit, then a new larger number is obtained, therefore, this process has no limit. However, oddly enough, the largest number in the world still exists and it is entered in the Guinness Book of Records.

Graham's number is the largest number in the world

It is this number that is recognized in the world as the largest in the Book of Records, while it is very difficult to explain what it is and how large it is. In a general sense, these are triples, multiplied among themselves, as a result of which a number is formed that is 64 orders of magnitude higher than the point of understanding of each person. As a result, we can only give the final 50 digits of Graham's number – 0322234872396701848518 64390591045756272 62464195387.

Googol's number

The history of the emergence of this number is not as complex as the above. So the mathematician from America Edward Kasner, talking with his nephews about large numbers, could not answer the question of how to call numbers with 100 zeros or more. The resourceful nephew suggested his name to such numbers - googol. It should be noted that this number does not have much practical value, however, it is sometimes used in mathematics to express infinity.

Googlex

This number was also invented by mathematician Edward Kasner and his nephew Milton Sirotta. In a general sense, it is the tenth power of a googol. Answering the question of many curious people, how many zeros are in Googleplex, it is worth noting that in the classical version this number cannot be represented, even if you write down all the paper on the planet with classical zeros.

Skewes number

Another contender for the title of the highest number is Skuse's number, proved by John Littlewood in 1914. According to the evidence presented, this number is approximately 8.185 × 10370.

Moser number

This method of naming very large numbers was invented by Hugo Steinhaus, who proposed to denote them by polygons. As a result of three performed mathematical operations, the number 2 is born in a mega-gon (a polygon with mega-sides).

As you can see, a huge number of mathematicians have made efforts to find it - the largest number in the world. To what extent these attempts were crowned with success, of course, is not for us to judge, however, it should be noted that the real applicability of such numbers is doubtful, because they do not even lend themselves to human understanding. In addition, there is always a number that will be greater if you perform a very easy mathematical operation +1.