What is the largest number in the whole world. The largest numbers in mathematics

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 most big 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. What are they called in this system? big numbers? 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. It will be interesting the following fact: translated 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?

1. If the number ends in "-million", you will need the formula 6 * x + 3 ("x" is a Latin numeral).
2. If the number ends in "-billion", you will need the formula 6 * x + 6 (where "x", again, is a Latin numeral).

Examples

At 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 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.

1. 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: worldwide search system"Google" is named after a rather large number at that time - googol. And the name came up with Kasner's young nephew.
2. Asankhiya. This is 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.
3. 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.
4. 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.
5. 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 defined, 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.

The question "What is the largest number in the world?" is, to say the least, incorrect. There are both different systems of calculus - decimal, binary and hexadecimal, as well as various categories of numbers - semi-simple and prime, the latter being divided into legal and illegal. In addition, there are the numbers of Skewes (Skewes "number), Steinhaus and other mathematicians who either jokingly or seriously invent and spread to the public such exotics as "megiston" or "moser".

What is the largest decimal number in the world

From the decimal system, most "non-mathematicians" are well aware of the million, billion and trillion. Moreover, if a million among Russians is mainly associated with a dollar bribe that can be carried away in a suitcase, then where to shove a billion (not to mention a trillion) North American banknotes - most do not have enough imagination. However, in the theory of large numbers, there are such concepts as quadrillion (ten to the fifteenth power - 1015), sextillion (1021) and octillion (1027).

In English, the most widely used decimal system in the world, the maximum number is decillion - 1033.

In 1938, in connection with the development of applied mathematics and the expansion of the micro- and macrocosms, Professor of Columbia University (USA), Edward Kasner (Edward Kasner) published on the pages of the journal "Scripta Mathematica" the proposal of his nine-year-old nephew to use the decimal system as the most a large number "googol" ("googol") - representing ten to the hundredth power (10100), which on paper is expressed as a unit with one hundred zeros. However, they did not stop there and a few years later proposed to put into circulation the new largest number in the world - "googolplex" (googolplex), which is ten raised to the tenth power and again raised to the hundredth power - (1010) 100, expressed by one, to which a googol of zeros is assigned to the right. However, for the majority of even professional mathematicians, both "googol" and "googolplex" are of purely speculative interest, and it is unlikely that they can be applied to anything in everyday practice.

exotic numbers

What is the largest number in the world among prime numbers - those that can only be divided by themselves and by one. One of the first to record the largest prime number, 2,147,483,647, was great mathematician Leonard Euler. As of January 2016, this number is an expression calculated as 274 207 281 - 1.

John Sommer

Put zeros after any number or multiply with tens raised to an arbitrarily large power. It won't seem like much. It will seem like a lot. But naked recordings, after all, are not too impressive. The heaping zeros in the humanities cause not so much surprise as a slight yawn. In any case, to any largest number in the world that you can imagine, you can always add one more ... And the number will come out even more.

And yet, are there words in Russian or any other language for designating very large numbers? Those that are more than a million, billion, trillion, billion? And in general, a billion is how much?

It turns out that there are two systems for naming numbers. But not Arabic, Egyptian, or any other ancient civilizations, but American and English.

In the American system numbers are called like this: the Latin numeral is taken + - million (suffix). Thus, the numbers are obtained:

Trillion - 1,000,000,000,000 (12 zeros)

Quadrillion - 1,000,000,000,000,000 (15 zeros)

Quintillion - 1 and 18 zeros

Sextillion - 1 and 21 zero

Septillion - 1 and 24 zero

octillion - 1 followed by 27 zeros

Nonillion - 1 and 30 zeros

Decillion - 1 and 33 zero

The formula is simple: 3 x + 3 (x is a Latin numeral)

In theory, there should also be numbers anilion (unus in Latin- one) and duolion (duo - two), but, in my opinion, such names are not used at all.

English naming system more widespread.

Here, too, the Latin numeral is taken and the suffix -million is added to it. However, the name of the next number, which is 1,000 times greater than the previous one, is formed using the same Latin number and the suffix - billion. I mean:

Trillion - 1 and 21 zero (in the American system - sextillion!)

Trillion - 1 and 24 zeros (in the American system - septillion)

Quadrillion - 1 and 27 zeros

Quadribillion - 1 followed by 30 zeros

Quintillion - 1 and 33 zero

Quinilliard - 1 followed by 36 zeros

Sextillion - 1 followed by 39 zeros

Sextillion - 1 and 42 zero

The formulas for counting the number of zeros are:

For numbers ending in - illion - 6 x+3

For numbers ending in - billion - 6 x+6

As you can see, confusion is possible. But let's not be afraid!

In Russia, the American system for naming numbers has been adopted. From the English system, we borrowed the name of the number "billion" - 1,000,000,000 \u003d 10 9

And where is the "cherished" billion? - Why, a billion is a billion! American style. And although we use the American system, we took the "billion" from the English one.

Using the Latin names of numbers and the American system, let's call the numbers:

- vigintillion- 1 and 63 zeros

- centillion- 1 and 303 zeros

- Million- one and 3003 zeros! Oh-hoo...

But this, it turns out, is not all. There are also off-system numbers.

And the first one is probably myriad- one hundred hundreds = 10,000

googol(it is in honor of him that the famous search engine is named) - one and one hundred zeros

In one of the Buddhist treatises, a number is named asankhiya- one and one hundred and forty zeros!

Number name googolplex(like Google) was invented by the English mathematician Edward Kasner and his nine-year-old nephew - unit c - dear mother! - googol zeros!!!

But that's not all...

The mathematician Skewes named the Skewes number after himself. It means e to the extent e to the extent e to the power of 79, i.e. e e e 79

And then a big problem arose. You can think of names for numbers. But how to write them down? The number of degrees of degrees of degrees is already such that it simply does not fit on the page! :)

And then some mathematicians began to write numbers in geometric shapes. And the first, they say, such a method of recording was invented by the outstanding writer and thinker Daniil Ivanovich Kharms.

And yet, what is the BIGGEST NUMBER IN THE WORLD? - It is called STASPLEX and is equal to G 100,

where G is the Graham number, the largest number ever used in mathematical proofs.

This number - stasplex - was invented by a wonderful person, our compatriot Stas Kozlovsky, to LJ to which I address you :) - ctac

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. The American one allows you to give names to large numbers like this: 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 various systems can mean different things, for example, an American billion in English system 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 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 on 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 we are talking about 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 got into the pages of the famous Book of Records.

It is impossible to answer this question correctly because 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 USA 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 greater 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 “millionillion” ().

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 “ great score”, which used the same names 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 this 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, 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 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 break away 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. So, 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 mathematical proof, is the "Graham number". This number was first used 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.

The usual arithmetic operations - addition, multiplication, and exponentiation - can naturally be extended 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 notation 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, also 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 at 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 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 - 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.