The largest number in the world is how. The largest number in the world

Sometimes people who are not involved in mathematics wonder: what is the most big number? On the one hand, the answer is obvious - infinity. Bores will even clarify that “plus infinity” or “+∞” is used by mathematicians. But this answer will not convince the most corrosive, especially since this is not a natural number, but a mathematical abstraction. But having understood the issue well, they can discover a very 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 people's imagination end?

Not to be confused with a trademark of Google Corporation, although they have 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, but it was actively used in mathematics.

It's funny that it was invented by a child - the nephew of the mathematician Edward Kasner. In 1938, my uncle entertained younger relatives reasoning about very large numbers. To the child’s indignation, it turned out that such a wonderful number had no name, and he gave his own 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. But it’s unlikely that anyone will have the patience to count to the end. Therefore, only theoretically.

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

Googolplex

This number is a derivative of googol, but is significantly larger than it. The prefix “plex” means raising ten to a power equal to 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 to be about 1080 degrees. But this did not stop scientists from increasing the number by simply adding the prefix “plex” to it: googolplexplex, googolplexplexplex and so on. And for particularly perverted mathematicians, they invented a variant of magnification without the 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 googoldecaplex and means a tenfold cumulative repetition of the procedure of 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 mentally traumatized.

48th Mersen number

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

Relatively recently, about a year ago, we managed to discover the next, 48th Mersen number. On this moment it is the largest prime number in the world. Let us recall that prime numbers are those that are divisible without a remainder only by one 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 common such numbers are. But the more valuable is the discovery of each next one. For example, the new prime number consists of 17,425,170 digits if represented in the form of the decimal number system familiar to us. The previous one had about 12 million characters.

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

This is what the Graham number looks like in Knuth arrow notation. It’s difficult to say how to decipher this without having a complete higher education in theoretical mathematics. It is also impossible to write it down in our usual decimal form: the observable Universe is simply not able to accommodate it. Building one degree at a time, as is the case with googolplexes, is also not a solution.

Good formula, just unclear

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 included in the Ramsey problem, which is also unclear, but sounds serious. Thirdly, this number is recognized as the largest ever used in mathematics, and not in comic proofs or intellectual games, but to solve a very specific mathematical problem.

Attention! The following information is dangerous for your mental health! By reading it, you accept responsibility for all 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 - it turns out 3*3*3=27. What if we now raise three to this number? The result is 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 realized.

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

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

And 33 is only the first of 64 terms of Graham's number. To get the second one, you need to calculate the result of this mind-blowing formula and substitute the corresponding number of arrows into diagram 3(...)3. And so on, another 63 times.

I wonder if anyone other than him and a dozen other supermathematicians will be able to get to at least the middle of the sequence without going crazy?

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

Why do we need the largest numbers? This is difficult for the average person to understand and comprehend. But with their help, a few specialists are able to introduce new technological toys to ordinary people: phones, computers, tablets. Ordinary people are also unable to understand how they work, but they are happy to use them for their entertainment. And everyone is happy: ordinary people get their toys, “supernerds” have the opportunity to continue playing their mind games.

It is impossible to answer this question correctly, since the number series has no upper limit. So, to any number you just need to add one to get an even larger number. Although the numbers themselves are infinite, they do not have many proper names, since most of them are content with names made up of smaller numbers. So, for example, numbers 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, there must be some largest number. But what is it called and what does it equal? Let's try to figure this out and at the same time find out how large numbers mathematicians came up with.

"Short" and "long" scale

In the Chuquet system, a number between a million and a billion did not have its own name and was simply called “a thousand millions”, similarly called “a thousand billion”, “a thousand trillion”, etc. This was not very convenient, and in 1549 French writer and the scientist Jacques Peletier du Mans (1517–1582) proposed naming such “intermediate” numbers using the same Latin prefixes, but with the ending “-billion”. So, it began to be called “billion”, - “billiard”, - “trillion”, etc.

The Chuquet-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 “billion” or “thousand millions”, but “billion”. Soon this error quickly spread, and a paradoxical situation arose - “billion” became simultaneously synonymous with “billion” () and “million millions” ().

This confusion continued for quite a long time and led to the fact that the United States created its own system for naming large numbers. According to the American system, the names of numbers are constructed in the same way as in the Schuquet system - the Latin prefix and the ending “million”. However, the magnitudes of these numbers are different. If in the Schuquet system names with the ending “illion” received numbers that were powers of a million, then in the American system the ending “-illion” received powers of a thousand. That is, a thousand million () began to be called a “billion”, () - a “trillion”, () - a “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 Chuquet 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".

To avoid confusion, let's summarize:

It is curious that in our country the final transition to a short scale occurred 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 one - in scientific books in astronomy and physics. However, now it is wrong to use a long scale in Russia, although the numbers there are large.

But let's return to the search for the largest number. After decillion, the names of numbers are obtained by combining prefixes. This produces numbers such as undecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion, novemdecillion, etc. However, these names are no longer interesting to us, since we agreed to find the largest number with its own non-compound 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 - “hundred” and mille - “thousand”. The Romans did not have their own names for numbers greater than a thousand. For example, a million () The Romans called it "decies centena milia", that is, "ten times a hundred thousand." According to Chuquet's rule, these three remaining Latin numerals give us such names for numbers as "vigintillion", "centillion" and "millillion".

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 Russia adopted a “long scale” for naming numbers, then the largest number with its own name would be “billion” ().

However, there are names for even larger numbers.

Numbers outside the system

Until the 17th century in Rus' it was used own system names of numbers. Tens of thousands were called "darkness", hundreds of thousands were called "legions", millions were called "leoders", tens of millions were called "ravens", and hundreds of millions were called "decks". This count of up to hundreds of millions was called the “small count,” and in some manuscripts the authors considered “ great score”, in which the same names were used for large numbers, but with a different meaning. So, “darkness” no longer meant ten thousand, but a thousand thousand () , “legion” - the darkness of those () ; "leodr" - legion of legions () , "raven" - leodr leodrov (). For some reason, “deck” in the great Slavic counting was not called “raven of ravens” () , but only ten “ravens”, that is (see table).

Number name	Meaning in "small count"	Meaning in the "great count"	Designation
Dark
Legion
Leodre
Raven (corvid)
Deck
Darkness of topics

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 to Play Chess" he tried to estimate the number possible options chess game. According to it, each game lasts on average of moves and on each move the player makes a choice on average from the 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 famous 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 achieve nirvana.

Nine-year-old Milton Sirotta went down in the history of mathematics not only because he came up with the number googol, but also because at the same time he proposed another number - the “googolplex”, which is equal 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 his proof of the Riemann hypothesis. The first number, which later became known as the “Skuse 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 powers there are in the powers, the more difficult it is to write the numbers and understand their meaning when reading. Moreover, it is possible to come up with such numbers (and, by the way, they have already been invented) when the degrees of degrees simply do not fit on the page. Yes, that's on the page! They won't even fit into a book the size of the entire Universe! In this case, the question arises of how to write such numbers. The problem, fortunately, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who wondered about this problem came up with his own way of writing, which led to the existence of several unrelated methods for writing large numbers - these are the notations of Knuth, Conway, Steinhaus, etc. We now have to deal with some of them.

Other notations

"in a triangle" means "",
"squared" means "in triangles"
"in a circle" means "in squares".

Explaining this method of notation, Steinhaus comes up with the number “mega”, which is 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 the power of , raise the resulting number to the power of , then raise the resulting number to the power of the resulting number, and so on, raise it to the power of times. For example, a calculator in MS Windows cannot calculate due to overflow even in two triangles. This huge number is approximately .

Having determined the “mega” number, Steinhaus invites readers to independently estimate another number - “medzon”, equal in a circle. In another edition of the book, Steinhaus, instead of the medzone, suggests estimating an even larger number - “megiston”, equal in a circle. Following Steinhaus, I 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. Thus, the Canadian mathematician Leo Moser (Leo Moser, 1921–1970) modified the Steinhaus notation, which was limited by the fact that if it were necessary to write numbers much larger than megiston, then difficulties and inconveniences would arise, since it would be necessary to draw many circles one inside another. Moser suggested that after the squares, draw not circles, but pentagons, then hexagons, and so on. He also proposed a formal notation for these polygons so that numbers could be written without drawing complex drawings. Moser notation looks like this:

"triangle" = = ;
"squared" = = "triangles" = ;
"in a pentagon" = = "in squares" = ;
"in -gon" = = "in -gon" = .

Thus, according to Moser’s notation, Steinhaus’s “mega” is written as , “medzone” as , and “megiston” as . In addition, Leo Moser proposed calling a polygon with the number of sides equal to mega - “megagon”. And suggested a number « in megagon", that is. This number became known as the Moser number or simply "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 by the American mathematician Ronald Graham in 1977 when proving one estimate in Ramsey theory, namely when calculating the dimension of certain -dimensional bichromatic hypercubes. Graham's number became famous only after it was described in Martin Gardner's 1989 book, From Penrose Mosaics to Reliable Ciphers.

To explain how large Graham's 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 superpower, which he proposed to write with arrows pointing upward.

Common arithmetic operations - addition, multiplication and exponentiation - naturally can be expanded into a sequence of hyperoperators as follows.

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

For example,

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

For example,

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

For example,

Here and below, the expression is always evaluated from right to left, and Knuth's arrow operators (as well as the operation of exponentiation) by definition have right associativity (order from right to left). According to this definition,

This already leads to quite large numbers, but the notation system 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 “quad arrow” operator:

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

For example:

The notation form is usually used for notation 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 descriptions with a variable number of arrows), or is equivalent to hyperoperators. But some numbers are so huge that even such a notation is insufficient. For example, Graham's number.

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, that is, where , where the superscript of the arrow indicates the total number of arrows. In other words, it is calculated in steps: in the first step we calculate with four arrows between threes, in the second - with arrows between threes, in the third - with arrows between threes, and so on; at the end we calculate with 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 to the corresponding number of objects (for example, the number of stars in the visible part of the Universe is estimated at sextillions - , and the number of atoms that make up Earth has the order of dodecalions), 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, although the notation above is relatively easy to understand. Although this is just the number of towers in this formula for , this number is already a lot more quantity Planck volumes (the smallest possible physical volume) contained in the observable universe (approximately ). After the first member, we are expecting another member of the rapidly growing sequence.

“I see clusters of vague numbers that are hidden there in the darkness, behind the small spot of light that the candle of reason gives. They whisper to each other; conspiring about who knows what. Perhaps they don't like us very much for capturing their little brothers in our minds. Or perhaps they simply lead a single-digit life, out there, beyond our understanding.
Douglas Ray

Sooner or later, everyone is tormented by the question, what is the largest number. There are a million answers to a child's question. What's next? Trillion. And even further? In fact, the answer to the question of what are the largest numbers is simple. Just add one to the largest number, and it will no longer be the largest. This procedure can be continued indefinitely.

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

Now we will find out everything...

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

The American system is built quite simply. All names of large numbers are constructed 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 thousand (lat. mille) and the magnifying suffix -illion (see table). This is how we get the numbers 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 according to 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 former English and spanish colonies. The names of numbers in this system are built like this: like this: the 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 - billion. That is, after a trillion in the English system there is a trillion, and only then a quadrillion, followed by a quadrillion, etc. 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 according to 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 would still be more correct to be called as the Americans call it - billion, since we have adopted the American system. But who in our country does anything according to the rules! ;-) By the way, sometimes the word trillion is used in Russian (you can see this for yourself by running a search in Google or Yandex) and, apparently, it means 1000 trillion, i.e. quadrillion.

In addition to numbers written using Latin prefixes according to the American or English system, so-called non-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 tell you more about them a little later.

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

And now the question arises, what next. What's behind the decillion? In principle, it is, of course, possible, 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 proper names - vigintillion (from Lat.viginti- twenty), centillion (from Lat.centum- one hundred) and 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, the Romans called a million (1,000,000)decies centena milia, that is, "ten hundred thousand." And now, actually, the table:

Thus, according to such a system, numbers are greater than 10 3003 , which would have its own, non-compound name is impossible to obtain! But nevertheless, numbers greater than a million are known - these are the same non-systemic numbers. Let's finally 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. This word, however, is outdated and practically not used, but it is curious that the word “myriads” is widely used, doesn't mean at all a certain number, but an uncountable, uncountable set of something. It is believed that the word myriad (English: myriad) came into European languages ​​from ancient Egypt.

Regarding the origin of this number, there are different opinions. 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, but there were no names for numbers greater than ten thousand. However, in his note “Psammit” (i.e., calculus of sand), Archimedes showed how to systematically construct 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) there 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 (in total a myriad of times more). Archimedes suggested the following names for the numbers:
1 myriad = 10 4 .
1 di-myriad = myriad of myriads = 10 8 .
1 tri-myriad = di-myriad di-myriad = 10 16 .
1 tetra-myriad = three-myriad three-myriad = 10 32 .
etc.

Google(from the English googol) is the number ten to the hundredth power, 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, it was his nine-year-old nephew Milton Sirotta who suggested calling the large number a “googol”. This number became generally known thanks to the search engine named after it. Google. Please note that "Google" is a brand name and googol is a number.

Edward Kasner.

On the Internet you can often find it mentioned that - but this is not so...

In the famous Buddhist treatise Jaina Sutra, dating back to 100 BC, the number appears asankheya(from China asenzi- uncountable), equal to 10 140. It is believed that this number is equal to the number of cosmic cycles required to achieve nirvana.

Googolplex(English) googolplex) - a number also invented by Kasner and his nephew and meaning one with a googol of zeros, that is, 10 10100 . This 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. At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex." A googolplex is much larger 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.

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

But it should be noted that there is a second Skuse number, which in mathematics is denoted as Sk2, which is even greater than the first Skuse number (Sk1). Second Skewes number, was introduced by J. Skuse in the same article to denote a number for which the Riemann hypothesis does not hold. Sk2 equals 1010 10103 , that is 1010 101000 .

As you understand, the more degrees there are, the more difficult it is to understand which number is greater. For example, looking at Skewes numbers, without special calculations, it is almost impossible to understand which of these two numbers is larger. Thus, for super-large 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, that's on the page! They won’t fit even into a book the size of the entire Universe! In this case, the question arises of 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 about this problem came up with his own way of writing, which led to the existence of several, unrelated to each other, methods for writing numbers - these are the notations of Knuth, Conway, Steinhouse, etc.

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

Steinhouse came up with two new superlarge numbers. He named the number - Mega, and the number is Megiston.

Mathematician Leo Moser refined Stenhouse's notation, which was limited by the fact that if it was necessary to write down numbers much larger than a megiston, difficulties and inconveniences arose, since many circles had to be drawn one inside the other. Moser suggested that after the squares, draw not circles, but pentagons, then hexagons, and so on. He also proposed a formal notation for these polygons so that numbers could be written without drawing complex pictures. Moser notation looks like that:

Thus, according to Moser's notation, Steinhouse's mega is written as 2, and megiston as 10. In addition, Leo Moser proposed 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 Moser is not the largest number. The largest number ever used in mathematical proof is the limit known as Graham number(Graham's number), first used in 1977 in the proof of one estimate in Ramsey theory. It is associated with bichromatic hypercubes and cannot be expressed without a special 64-level system of special mathematical symbols, introduced by Knuth in 1976.

Unfortunately, a number written in Knuth's notation cannot be converted into notation in the Moser system. Therefore, we will have to explain this system too. In principle, there is nothing complicated about 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 upward:

IN general view it looks like this:

I think everything is clear, so let’s return to Graham’s number. Graham proposed so-called G-numbers:

The number G63 began to be called Graham number(it is often designated simply as G). This number is the largest known number in the world and is even listed in the Guinness Book of Records. Well, the Graham number is greater than the Moser number.

P.S. In order to bring great benefit to all humanity and become famous throughout the centuries, I decided to come up with and name the largest number myself. This number will be called stasplex and it is equal to the number G100. Remember it, and when your children ask what is the largest number in the world, tell them that this number is called stasplex

So are there numbers greater than Graham's number? There is, of course, for starters there is Graham's number. Concerning significant number...okay, there are some fiendishly complex areas of mathematics (specifically the area known as combinatorics) and computer science in which numbers even larger than Graham's number occur. But we have almost reached the limit of what can be rationally and clearly explained.

The question “What is the largest number in the world?” is, to say the least, incorrect. There are different number systems - decimal, binary and hexadecimal, as well as various categories of numbers - semi-prime and simple, the latter being divided into legal and illegal. In addition, there are Skewes numbers, Steinhouse and other mathematicians who, either as a joke or seriously, invent and present to the public such exotics as “Megiston” or “Moser”.

What is the largest number in the world in decimal system

Of the decimal system, most “non-mathematicians” are familiar with million, billion and trillion. Moreover, if Russians generally associate a million with a dollar bribe that can be carried away in a suitcase, then where to stuff a billion (not to mention a trillion) North American banknotes - most people lack 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 the English decimal system, the most widely used decimal system in the world, the maximum number is considered to be a decillion - 1033.

In 1938, in connection with the development of applied mathematics and the expansion of the micro- and macrocosm, professor at Columbia University (USA), Edward Kasner published in the pages of the journal Scripta Mathematica his nine-year-old nephew’s proposal to use the decimal system as the most the large number "googol" - representing ten to the hundredth power (10100), which on paper is expressed as one followed by one hundred zeros. However, they did not stop there and a few years later proposed introducing a new largest number in the world - the “googolplex”, which represents ten raised to the tenth power and again raised to the hundredth power - (1010)100, expressed by a unit, 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 one. One of the first to record the largest prime number, equal to 2,147,483,647, was the great mathematician Leonhard Euler. As of January 2016, this number is recognized as the expression calculated as 274,207,281 – 1.

Once upon a time in childhood, we learned to count to ten, then to a hundred, then to a thousand. So what's the biggest number you know? A thousand, a million, a billion, a trillion... And then? Petallion, someone will say, and he will be wrong, because he confuses the SI prefix with a completely different concept.

In fact, the question is not as simple as it seems at first glance. Firstly, we are talking about naming the names of powers of a thousand. And here, the first nuance that many know from American films is that they call our billion a billion.

Further, there are two types of scales - long and short. In our country, a short scale is used. In this scale, at each step the mantissa increases by three orders of magnitude, i.e. multiply by a thousand - thousand 10 3, million 10 6, billion/billion 10 9, trillion (10 12). In the long scale, after a billion 10 9 there is a billion 10 12, and subsequently the mantissa increases by six orders of magnitude, and the next number, which is called a trillion, already means 10 18.

But let's return to our native scale. Want to know what comes after a trillion? Please:

10 3 thousand
10 6 million
10 9 billion
10 12 trillion
10 15 quadrillion
10 18 quintillion
10 21 sextillion
10 24 septillion
10 27 octillion
10 30 nonillion
10 33 decillion
10 36 undecillion
10 39 dodecillion
10 42 tredecillion
10 45 quattoordecillion
10 48 quindecillion
10 51 cedecillion
10 54 septdecillion
10 57 duodevigintillion
10 60 undevigintillion
10 63 vigintillion
10 66 anvigintillion
10 69 duovigintillion
10 72 trevigintillion
10 75 quattorvigintillion
10 78 quinvigintillion
10 81 sexvigintillion
10 84 septemvigintillion
10 87 octovigintillion
10 90 novemvigintillion
10 93 trigintillion
10 96 antigintillion

At this number, our short scale cannot stand it, and subsequently the mantis increases progressively.

10 100 googol
10,123 quadragintillion
10,153 quinquagintillion
10,183 sexagintillion
10,213 septuagintillion
10,243 octogintillion
10,273 nonagintillion
10,303 centillion
10,306 centunillion
10,309 centullion
10,312 centtrillion
10,315 centquadrillion
10,402 centretrigintillion
10,603 decentillion
10,903 trcentillion
10 1203 quadringentillion
10 1503 quingentillion
10 1803 sescentillion
10 2103 septingentillion
10 2403 oxtingentillion
10 2703 nongentillion
10 3003 million
10 6003 duo-million
10 9003 three million
10 3000003 mimiliaillion
10 6000003 duomimiliaillion
10 10 100 googolplex
10 3×n+3 zillion

Google(from the English googol) - a number in the decimal number system represented by a unit followed by 100 zeros:
10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
In 1938, American mathematician 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 a hundred zeros, which did not have its own name. One of the nephews, nine-year-old Milton Sirotta, suggested calling this number “googol.” In 1940, Edward Kasner, together with James Newman, wrote the popular science book “Mathematics and Imagination” (“New Names in Mathematics”), where he told mathematics lovers about the googol number.
The term “googol” does not have any serious theoretical or practical meaning. Kasner proposed it to illustrate the difference between an unimaginably large number and infinity, and the term is sometimes used in mathematics teaching for this purpose.

Googolplex(from the English googolplex) - a number represented by a unit with a googol of zeros. Like the googol, the term "googolplex" was coined by American mathematician Edward Kasner and his nephew Milton Sirotta.
The number of googols is greater than the number of all particles in the part of the universe known to us, which ranges from 1079 to 1081. Thus, the number googolplex, consisting of (googol + 1) digits, cannot be written down in the classical “decimal” form, even if all matter in the known parts of the universe turned into paper and ink or computer disk space.

Zillion(English zillion) - a general name for very large numbers.

This term does not have a strict mathematical definition. In 1996, Conway (eng. J. H. Conway) and Guy (eng. R. K. Guy) in their book English. The Book of Numbers defined the nth power zillion as 10 3×n+3 for the short scale number naming system.