Answering such a difficult question, what is it, the largest number in the world, it should first 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 in turn to each large number, resulting in the numbers million, billion, trillion, trilliard, 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 large are formed. It should also be noted here that the English system of calculus is more common in 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 one has only to 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 listed 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, resulting in a number 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 the Graham number – 0322234872396701848518 64390591045756272 62464195387.

Googol number

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

Googleplex

This number was also invented by mathematician Edward Kasner and his nephew Milton Sirotta. In a general sense, it is a number to the tenth power of a googol. Answering the question of many inquisitive natures, how many zeros are in the Googleplex, it is worth noting that in the classical version this number is not possible to represent, even if all the paper available on the planet is covered with classical zeros.

Skewes number

Another contender for the title of the largest number is the Skewes number, proved by John Littwood in 1914. According to the evidence given, this number is approximately 8.185 10370.

Moser number

This method of naming very large numbers was invented by Hugo Steinhaus, who suggested that they be denoted by polygons. As a result of three mathematical operations performed, the number 2 is born in a megagon (a polygon with mega sides).

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

Naming systems for large numbers

There are two systems for naming numbers - American and European (English).

In the American system, all the names of large numbers are built like this: at the beginning there is a Latin ordinal number, and at the end the suffix "million" is added to it. The exception is the name "million", which is the name of the number one thousand (Latin mille) and the magnifying suffix "million". This is how numbers are obtained - trillion, quadrillion, quintillion, sextillion, etc. The American system is used in the USA, Canada, France and Russia. The number of zeros in a number written in the American system is determined by the formula 3 x + 3 (where x is a Latin numeral).

The European (English) naming system is the most common in the world. It is used, for example, in Great Britain and Spain, as well as in most of the former English and Spanish colonies. The names of numbers in this system are constructed as follows: the suffix "million" is added to the Latin numeral, the name of the next number (1,000 times larger) is formed from the same Latin numeral, but with the suffix "billion". That is, after a trillion in this system comes a trillion, and only then a quadrillion, followed by a quadrillion, etc. The number of zeros in a number written in the European system and ending in the suffix "million" is determined by the formula 6 x + 3 (where x - Latin numeral) and by the formula 6 x + 6 for numbers ending in "billion". In some countries using the American system, for example, in Russia, Turkey, Italy, the word "billion" is used instead of the word "billion".

Both systems come from France. French physicist and mathematician Nicolas Chuquet coined the words "billion" (byllion) and "trillion" (tryllion) and used them to represent the numbers 1012 and 1018 respectively, which formed the basis of the European system.

But some French mathematicians in the 17th century used the words "billion" and "trillion" for the numbers 109 and 1012, respectively. This naming system took hold in France and America, and became known as the American one, while the original Choquet system continued to be used in Great Britain and Germany. France in 1948 returned to the Choquet (ie European) system.

In recent years, the American system has been supplanting the European one, partly in the UK and so far hardly noticeable in other European countries. Basically, this is due to the fact that Americans in financial transactions insist that 1,000,000,000 dollars should be called a billion dollars. In 1974, the government of Prime Minister Harold Wilson announced that the word billion would be 10 9 instead of 10 12 in UK official records and statistics.

12 - Dozen(from French douzaine or Italian dozzina, which in turn came from Latin duodecim.)
A measure of the piece count of homogeneous objects. Widely used before the introduction of the metric system. For example, a dozen handkerchiefs, a dozen forks. 12 dozen make a gross. For the first time in Russian, the word "dozen" is mentioned since 1720. It was originally used by sailors.

13 - Baker's dozen

The number is considered unlucky. Many western hotels do not have rooms with the number 13, but office buildings have 13th floors. There are no seats with this number in Italian opera houses. Almost on all ships, after the 12th cabin, the 14th immediately follows.

144 - Gross- "big dozen" (from German Gro? - big)

A counting unit equal to 12 dozen. It was usually used when counting small haberdashery and stationery items - pencils, buttons, writing pens, etc. A dozen grosses is a mass.

1728 - Weight

Mass (obsolete) - a measure of the account, equal to a dozen grosses, i.e. 144 * 12 = 1728 pieces. Widely used before the introduction of the metric system.

666 or 616 - Number of the beast

A special number mentioned in the Bible (Revelation 13:18, 14:2). It is assumed that in connection with the assignment of a numerical value to the letters of the ancient alphabets, this number can mean any name or concept, the sum of the numerical values ​​​​of the letters of which is 666. Such words can be: "Lateinos" (means in Greek everything Latin; proposed by Jerome ), "Nero Caesar", "Bonaparte" and even "Martin Luther". In some manuscripts, the number of the beast is read as 616.

Myriad - the word is outdated and practically not used, but the word "myriad" - (astronomer.) is widely used, which means an uncountable, uncountable set of something.

Myriad was the largest number for which the ancient Greeks had a name. However, in the work "Psammit" ("Calculation of grains of sand"), Archimedes showed how one can systematically build and name arbitrarily large numbers. All numbers from 1 to myriad (10,000) Archimedes called the first numbers, he called the myriad of myriads (10 8) the unit of numbers of the second (dimyriad), the myriad of myriads of second numbers (10 16) he called the unit of numbers of the third (trimiriad), etc. .

10 000 - dark
100 000 - legion
1 000 000 - leodre
10 000 000 - raven or raven
100 000 000 - deck

The ancient Slavs also loved large numbers, they knew how to count up to a billion. Moreover, they called such an account a “small account”. In some manuscripts, the authors also considered the "great count", which reached the number 10 50 . About numbers greater than 10 50 it was said: "And more than this to bear the human mind to understand." The names used in the "small account" were transferred to the "great account", but with a different meaning. So, darkness meant no longer 10,000, but a million, legion - the darkness of those (million millions); leodrus - legion of legions - 10 24, then it was said - ten leodres, one hundred leodres, ..., and, finally, one hundred thousand legions of leodres - 10 47; leodr leodrov -10 48 was called a raven and, finally, a deck of -10 49 .

10 140 - Asankhei I (from Chinese asentzi - innumerable)

Mentioned in the famous Buddhist treatise Jaina Sutra, dating back to 100 BC. It is believed that this number is equal to the number of cosmic cycles required to gain nirvana.

googol(from English. googol) - 10 100 , that is, one followed by one hundred zeros.

The "googol" was first written about in 1938 in the article "New Names in Mathematics" in the January issue of the journal Scripta Mathematica by the American mathematician Edward Kasner. According to him, his nine-year-old nephew Milton Sirotta suggested calling a large number "googol". This number became well-known thanks to the search engine named after him. Google. Note that " Google" - This trademark, A googol - number.

Googolplex(English googolplex) 10 10 100 - 10 to the power of googol.

The number was also invented by Kasner and his nephew and means one with a googol of zeros, that is, 10 to the power of a googol. Here is how Kasner himself describes this "discovery":

Words of wisdom are spoken by children at least as often as by scientists. The name "googol" was invented by a child (Dr. Kasner\"s nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. than a googol, but is still finite, as the inventor of the name was quick to point out.

Mathematics and the Imagination (1940) by Kasner and James R. Newman.

Skewes number(Skewes` number)- Sk 1 e e e 79 - means e to the power of e to the power of e to the power of 79.

It was proposed by J. Skewes in 1933 (Skewes. J. London Math. Soc. 8, 277-283, 1933.) in proving the Riemann conjecture concerning primes. Later, Riele (te Riele, H. J. J. "On the Sign of the Difference P(x)-Li(x"). Math. Comput. 48, 323-328, 1987) reduced Skuse's number to e e 27/4, which is approximately equal to 8.185 10 370 .

It was introduced by J. Skuse in the same article to denote the number up to which the Riemann hypothesis is not valid. Sk 2 is equal to 10 10 10 10 3 .

As you understand, the more degrees there are, the more difficult it is to understand which of the numbers is larger. For example, looking at the Skewes numbers, without special calculations, it is almost impossible to understand which of these two numbers is larger. Thus, for superlarge numbers, it becomes inconvenient to use powers. Moreover, you can come up with such numbers (and they have already been invented) when the degrees of degrees simply do not fit on the page. Yes, what a page! They won't even fit into a book the size of the entire universe!

In this case, the question arises how to write them down. The problem, as you understand, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who asked this problem came up with his own way of writing, which led to the existence of several, unrelated, ways to write numbers - these are the notations of Knuth, Conway, Steinhouse, etc.

Hugo Stenhouse notation(H. Steinhaus. Mathematical Snapshots, 3rd edn. 1983) is quite simple. Steinhaus (German: Steihaus) suggested writing large numbers inside geometric shapes - a triangle, a square and a circle.

Steinhouse came up with super-large numbers and called the number 2 in a circle - Mega, 3 in a circle - Medzone, and the number 10 in a circle - Megiston.

Mathematician Leo Moser finalized Stenhouse's notation, which was limited by the fact that if it was required to write numbers much larger than megiston, difficulties and inconveniences arose, since many circles had to be drawn one inside the other. 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:

• "n triangle" = nn = n.
• "n squared" = n = "n in n triangles" = nn.
• "n in a pentagon" = n = "n in n squares" = nn.
• n = "n in n k-gons" = n[k]n.

In Moser's notation, the Steinhaus mega is written as 2, and the megiston as 10. Leo Moser suggested calling a polygon with the number of sides equal to mega - megagon. And he also proposed the number "2 in Megagon", that is, 2. This number became known as Moser number(Moser`s number) or simply as a moser. But the Moser number is not the largest number.

The largest number ever used in a mathematical proof is the limiting value known as Graham number(Graham`s number), first used in 1977 in the proof of one estimate in the Ramsey theory. It is associated with bichromatic hypercubes and cannot be expressed without a special 64-level system of special mathematical symbols introduced by D. Knuth in 1976.

Sometimes people who are not related to mathematics wonder: what is the largest number? On the one hand, the answer is obvious - infinity. The bores will even clarify that "plus infinity" or "+∞" in the notation of mathematicians. But this answer will not convince the most corrosive, especially since this is not a natural number, but a mathematical abstraction. But having well understood the issue, they can open up an interesting problem.

Indeed, there is no size limit in this case, but there is a limit to human imagination. Each number has a name: ten, one hundred, billion, sextillion, and so on. But where does the fantasy of people end?

Not to be confused with a Google Corporation trademark, although they share a common origin. This number is written as 10100, that is, one followed by a tail of one hundred zeros. It is difficult to imagine it, but it was actively used in mathematics.

It's funny what his child came up with - the nephew of the mathematician Edward Kasner. In 1938, my uncle entertained younger relatives with arguments about very large numbers. To the indignation of the child, it turned out that such a wonderful number had no name, and he gave his version. Later, my uncle inserted it into one of his books, and the term stuck.

Theoretically, a googol is a natural number, because it can be used for counting. That's just hardly anyone has the patience to count to its end. Therefore, only theoretically.

As for the name of the company Google, then a common mistake crept in. The first investor and one of the co-founders was in a hurry when he wrote the check, and missed the letter “O”, but in order to cash it, the company had to be registered under this spelling.

Googolplex

This number is a derivative of the googol, but significantly larger than it. The prefix "plex" means raising ten to the power of the base number, so guloplex is 10 to the power of 10 to the power of 100, or 101000.

The resulting number exceeds the number of particles in the observable universe, which is estimated at about 1080 degrees. But this did not stop scientists from increasing the number simply by adding the prefix "plex" to it: googolplexplex, googolplexplexplex, and so on. And for especially perverted mathematicians, they invented an option to increase without endless repetition of the prefix "plex" - they simply put Greek numbers in front of it: tetra (four), penta (five) and so on, up to deca (ten). The last option sounds like a googoldekaplex and means a tenfold cumulative repetition of the procedure for raising the number 10 to the power of its base. The main thing is not to imagine the result. You still won’t be able to realize it, but it’s easy to get a trauma to the psyche.

48th Mersen number

Main characters: Cooper, his computer and a new prime number

Relatively recently, about a year ago, it was possible to discover the next, 48th Mersen number. It is currently the largest prime number in the world. Recall that prime numbers are those that are only divisible without a remainder by 1 and themselves. The simplest examples are 3, 5, 7, 11, 13, 17 and so on. The problem is that the further into the wilds, the less often such numbers occur. But the more valuable is the discovery of each next one. For example, a new prime number consists of 17,425,170 digits if it is represented in the form of a decimal number system familiar to us. The previous one had about 12 million characters.

It was discovered by the American mathematician Curtis Cooper, who for the third time delighted the mathematical community with such a record. Just to check his result and prove that this number is really prime, it took 39 days of his personal computer.

This is how Graham's number is written in Knuth's arrow notation. It is difficult to say how to decipher this without having a completed higher education in theoretical mathematics. It is also impossible to write it down in the decimal form we are accustomed to: the observable Universe is simply not able to contain it. Fencing degree for degree, as in the case of googolplexes, is also not an option.

Good formula, but incomprehensible

So why do we need this seemingly useless number? Firstly, for the curious, it was placed in the Guinness Book of Records, and this is already a lot. Secondly, it was used to solve a problem that is part of the Ramsey problem, which is also incomprehensible, but sounds serious. Thirdly, this number is recognized as the largest ever used in mathematics, and not in comic proofs or intellectual games, but for solving a very specific mathematical problem.

Attention! The following information is dangerous for your mental health! By reading it, you accept responsibility for all the consequences!

For those who want to test their mind and meditate on the Graham number, we can try to explain it (but only try).

Imagine 33. It's pretty easy - you get 3*3*3=27. What if we now raise three to this number? It turns out 3 3 to the 3rd power, or 3 27. In decimal notation, this is equal to 7,625,597,484,987. A lot, but for now it can be understood.

In Knuth's arrow notation, this number can be displayed somewhat more simply - 33. But if you add only one arrow, it will turn out to be more difficult: 33, which means 33 to the power of 33 or in power notation. If expanded to decimal notation, we get 7,625,597,484,987 7,625,597,484,987 . Are you still able to follow the thought?

Next step: 33= 33 33 . That is, you need to calculate this wild number from the previous action and raise it to the same power.

And 33 is just the first of the 64 members of Graham's number. To get the second one, you need to calculate the result of this furious formula, and substitute the appropriate number of arrows into the 3(...)3 scheme. And so on, 63 more times.

I wonder if anyone besides him and a dozen other supermathematicians will be able to get at least to the middle of the sequence and not go crazy at the same time?

Did you understand something? We are not. But what a thrill!

Why are the largest numbers needed? It is difficult for the layman to understand and realize this. But a few specialists with their help are able to present new technological toys to the inhabitants: phones, computers, tablets. The townsfolk are also not able to understand how they work, but they are happy to use them for their own entertainment. And everyone is happy: the townsfolk get their toys, "supernerds" - the opportunity to play their mind games for a long time.

In the names of Arabic numbers, each digit belongs to its category, and every three digits form a class. Thus, the last digit in a number indicates the number of units in it and is called, accordingly, the place of units. The next, second from the end, digit indicates tens (the tens digit), and the third digit from the end indicates the number of hundreds in the number - the hundreds digit. Further, the digits are repeated in the same way in turn in each class, denoting units, tens and hundreds in the classes of thousands, millions, and so on. If the number is small and does not contain a tens or hundreds digit, it is customary to take them as zero. Classes group numbers in numbers of three, often in computing devices or records a period or space is placed between classes to visually separate them. This is done 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 use the decimal system, the basic unit of quantity is the ten, or 10 1 . Accordingly, with an increase in the number of digits in a number, the number of tens of 10 2, 10 3, 10 4, etc. also increases. Knowing the number of tens, you can easily determine the class and category of the number, for example, 10 16 is tens of quadrillions, and 3 × 10 16 is three tens of quadrillions. The decomposition of numbers into decimal components occurs as follows - each digit is displayed in a separate term, multiplied by the required coefficient 10 n, where n is the position of the digit in the count from left to right.
For example: 253 981=2×10 6 +5×10 5 +3×10 4 +9×10 3 +8×10 2 +1×10 1

Also, the power of 10 is also used in writing decimals: 10 (-1) is 0.1 or one tenth. Similarly with the previous paragraph, a decimal number can also be decomposed, in which case n will indicate the position of the digit from the comma from right to left, for example: 0.347629= 3x10 (-1) +4x10 (-2) +7x10 (-3) +6x10 (-4) +2x10 (-5) +9x10 (-6) )

Names of decimal numbers. Decimal numbers are read by the last digit after the decimal point, for example 0.325 - three hundred and twenty-five thousandths, where thousandths are the digit of the last digit 5.

Table of names of large numbers, digits and classes

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

We continue ours. Today we have numbers...

Sooner or later, everyone is tormented by the question, what is the largest number. A child's question can be answered in a million. What's next? Trillion. And even further? In fact, the answer to the question of what are the largest numbers is simple. It is simply worth adding one to the largest number, as it will no longer be the largest. This procedure can be continued indefinitely.

But if you ask yourself: what is the largest number that exists, and what is its own name?

Now we all know...

There are two systems for naming numbers - American and English.

The American system is built quite simply. All names of large numbers are built like this: at the beginning there is a Latin ordinal number, and at the end the suffix -million is added to it. The exception is the name "million" which is the name of the number one thousand (lat. mille) and the magnifying suffix -million (see table). So the numbers are obtained - trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion and decillion. The American system is used in the USA, Canada, France and Russia. You can find out the number of zeros in a number written in the American system using the simple formula 3 x + 3 (where x is a Latin numeral).

The English naming system is the most common in the world. It is used, for example, in Great Britain and Spain, as well as in most of the former English and Spanish colonies. The names of numbers in this system are built like this: like this: a suffix -million is added to the Latin numeral, the next number (1000 times larger) is built according to the principle - the same Latin numeral, but the suffix is ​​-billion. That is, after a trillion in the English system comes a trillion, and only then a quadrillion, followed by a quadrillion, and so on. Thus, a quadrillion according to the English and American systems are completely different numbers! You can find out the number of zeros in a number written in the English system and ending with the suffix -million using the formula 6 x + 3 (where x is a Latin numeral) and using the formula 6 x + 6 for numbers ending in -billion.

Only the number billion (10 9 ) passed from the English system into the Russian language, which, nevertheless, would be more correct to call it the way the Americans call it - a billion, since we have adopted the American system. But who in our country does something according to the rules! ;-) By the way, sometimes the word trillion is also used in Russian (you can see for yourself by running a search in Google or Yandex) and it means, apparently, 1000 trillion, i.e. quadrillion.

In addition to numbers written using Latin prefixes in the American or English system, the so-called off-system numbers are also known, i.e. numbers that have their own names without any Latin prefixes. There are several such numbers, but I will talk about them in more detail a little later.

Let's go back to writing using Latin numerals. It would seem that they can write numbers to infinity, but this is not entirely true. Now I will explain why. Let's first see how the numbers from 1 to 10 33 are called:

And so, now the question arises, what next. What is a decillion? In principle, it is possible, of course, by combining prefixes to generate such monsters as: andecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion and novemdecillion, but these will already be compound names, and we were interested in our own names numbers. Therefore, according to this system, in addition to those indicated above, you can still get only three - vigintillion (from lat.viginti- twenty), centillion (from lat.percent- one hundred) and a million (from lat.mille- thousand). The Romans did not have more than a thousand proper names for numbers (all numbers over a thousand were composite). For example, a million (1,000,000) Romans calledcentena miliai.e. ten hundred thousand. And now, actually, the table:

Thus, according to a similar system, numbers are greater than 10 3003 , which would have its own, non-compound name, it is impossible to get! But nevertheless, numbers greater than a million are known - these are the very non-systemic numbers. Finally, let's talk about them.

The smallest such number is a myriad (it is even in Dahl's dictionary), which means a hundred hundreds, that is, 10,000. True, this word is outdated and is practically not used, but it is curious that the word "myriad" is widely used, which does not mean a certain number at all, but an uncountable, uncountable set of something. It is believed that the word myriad (English myriad) came to European languages ​​from ancient Egypt.

There are different opinions about the origin of this number. Some believe that it originated in Egypt, while others believe that it was born only in ancient Greece. Be that as it may, in fact, the myriad gained fame precisely thanks to the Greeks. Myriad was the name for 10,000, and there were no names for numbers over ten thousand. However, in the note "Psammit" (i.e., the calculus of sand), Archimedes showed how one can systematically build and name arbitrarily large numbers. In particular, placing 10,000 (myriad) grains of sand in a poppy seed, he finds that in the Universe (a ball with a diameter of a myriad of Earth diameters) would fit (in our notation) no more than 10 63 grains of sand. It is curious that modern calculations of the number of atoms in the visible universe lead to the number 10 67 (only a myriad of times more). The names of the numbers Archimedes suggested are as follows:
1 myriad = 10 4 .
1 di-myriad = myriad myriad = 10 8 .
1 tri-myriad = di-myriad di-myriad = 10 16 .
1 tetra-myriad = three-myriad three-myriad = 10 32 .
etc.

Googol (from the English googol) is the number ten to the hundredth power, that is, one with one hundred zeros. The "googol" was first written about in 1938 in the article "New Names in Mathematics" in the January issue of the journal Scripta Mathematica by the American mathematician Edward Kasner. According to him, his nine-year-old nephew Milton Sirotta suggested calling a large number "googol". This number became well-known thanks to the search engine named after him. Google. Note that "Google" is a trademark and googol is a number.

Edward Kasner.

On the Internet, you can often find mention that - but this is not so ...

In the well-known Buddhist treatise Jaina Sutra, dating back to 100 BC, the number Asankheya (from the Chinese. asentzi- incalculable), equal to 10 140. It is believed that this number is equal to the number of cosmic cycles required to gain nirvana.

Googolplex (English) googolplex) - a number also invented by Kasner with his nephew and meaning one with a googol of zeros, that is, 10 10100 . Here is how Kasner himself describes this "discovery":

Words of wisdom are spoken by children at least as often as by scientists. The name "googol" was invented by a child (Dr. Kasner"s nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. a googol, but is still finite, as the inventor of the name was quick to point out.

Mathematics and the Imagination(1940) by Kasner and James R. Newman.

Even larger than the googolplex number, Skewes' number was proposed by Skewes in 1933 (Skewes. J. London Math. soc. 8, 277-283, 1933.) in proving the Riemann conjecture concerning primes. It means e to the extent e to the extent e to the power of 79, i.e. ee e 79 . Later, Riele (te Riele, H. J. J. "On the Sign of the Difference P(x)-Li(x)." Math. Comput. 48, 323-328, 1987) reduced Skuse's number to ee 27/4 , which is approximately equal to 8.185 10 370 . It is clear that since the value of the Skewes number depends on the number e, then it is not an integer, so we will not consider it, otherwise we would have to recall other non-natural numbers - the number pi, the number e, etc.

But it should be noted that there is a second Skewes number, which in mathematics is denoted as Sk2 , which is even larger than the first Skewes number (Sk1 ). Skuse's second number, was introduced by J. Skuse in the same article to denote a number for which the Riemann hypothesis is not valid. Sk2 is 1010 10103 , i.e. 1010 101000 .

As you understand, the more degrees there are, the more difficult it is to understand which of the numbers is greater. For example, looking at the Skewes numbers, without special calculations, it is almost impossible to understand which of these two numbers is larger. Thus, for superlarge numbers, it becomes inconvenient to use powers. Moreover, you can come up with such numbers (and they have already been invented) when the degrees of degrees simply do not fit on the page. Yes, what a page! They won't even fit into a book the size of the entire universe! In this case, the question arises how to write them down. The problem, as you understand, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who asked this problem came up with his own way of writing, which led to the existence of several, unrelated, ways to write numbers - these are the notations of Knuth, Conway, Steinhaus, etc.

Consider the notation of Hugo Stenhaus (H. Steinhaus. Mathematical Snapshots, 3rd edn. 1983), which is quite simple. Steinhouse suggested writing large numbers inside geometric shapes - a triangle, a square and a circle:

Steinhouse came up with two new super-large numbers. He called the number - Mega, and the number - Megiston.

The mathematician Leo Moser refined Stenhouse's notation, which was limited by the fact that if it was necessary to write numbers much larger than a megiston, difficulties and inconveniences arose, since many circles had to be drawn one inside the other. Moser suggested 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:

Thus, according to Moser's notation, Steinhouse's mega is written as 2, and megiston as 10. In addition, Leo Moser suggested calling a polygon with the number of sides equal to mega - megagon. And he proposed the number "2 in Megagon", that is, 2. This number became known as Moser's number or simply as moser.

But the moser is not the largest number. The largest number ever used in a mathematical proof is the limiting value known as Graham's number, first used in 1977 in the proof of one estimate in Ramsey theory. It is associated with bichromatic hypercubes and cannot be expressed without the special 64-level system of special mathematical symbols introduced by Knuth in 1976.

Unfortunately, the number written in the Knuth notation cannot be translated into the Moser notation. Therefore, this system will also have to be explained. In principle, there is nothing complicated in it either. Donald Knuth (yes, yes, this is the same Knuth who wrote The Art of Programming and created the TeX editor) came up with the concept of superpower, which he proposed to write with arrows pointing up:

In general, it looks like this:

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

1. G1 = 3..3, where the number of superdegree arrows is 33.


2. G2 = ..3, where the number of superdegree arrows is equal to G1 .


3. G3 = ..3, where the number of superdegree arrows is equal to G2 .



4. G63 = ..3, where the number of superpower arrows is G62 .

The number G63 became known as the Graham number (it is often denoted simply as G). This number is the largest known number in the world and is even listed in the Guinness Book of Records. And here