The largest number in the world. The biggest numbers in mathematics

Back in the fourth grade, I was interested in the question: "What are the names of numbers over a billion? And why?" Since then, I have been looking for all the information on this issue for a long time and collecting it bit by bit. But with the advent of Internet access, searches have accelerated significantly. Now I present all the information I have found so that others can also answer the question: "What are the names of large and very large numbers?".

A bit of history

Southern and Eastern Slavic peoples to write numbers, they used alphabetical numbering. Moreover, among the Russians, not all letters played the role of numbers, but only those that are in Greek alphabet... A special "titlo" icon was placed above the letter denoting the number. Wherein numerical values letters increased in the same order in which the letters in the Greek alphabet followed (the order of the letters in the Slavic alphabet was somewhat different).

In Russia, Slavic numbering was preserved until the end of the 17th century. Under Peter I, the so-called "Arabic numbering" prevailed, which we still use today.

There were also changes in the names of the numbers. For example, until the 15th century, the number "twenty" was designated as "two ten" (two tens), but then it was shortened for a faster pronunciation. Until the 15th century, the number "forty" was denoted by the word "fourty", and in the 15th and 16th centuries this word was supplanted by the word "forty", which originally meant a sack containing 40 squirrel or sable skins. There are two options for the origin of the word "thousand": from the old name "fat one hundred" or from a modification of the Latin word centum - "one hundred".

The name "million" first appeared in Italy in 1500 and was formed by adding an augmenting suffix to the number "millet" - a thousand (that is, it meant "a large thousand"), it penetrated into the Russian language later, and before that the same meaning in in Russian it was denoted by the number "leodr". The word "billion" came into use only since the Franco-Prussian war (1871), when the French had to pay Germany an indemnity of 5,000,000,000 francs. Like “million,” the word “billion” comes from the root “thousand” with the addition of an Italian augmentation suffix. In Germany and America for some time the word "billion" meant the number 100,000,000; this explains that the word billionaire was used in America before any of the wealthy had $ 1,000,000,000. In the old (XVIII century) "Arithmetic" of Magnitsky, a table of the names of numbers is given, brought to "quadrillion" (10 ^ 24, according to the system after 6 digits). Perelman Ya.I. in the book "Entertaining arithmetic" the names of large numbers of that time are given, somewhat different from those of today: septillion (10 ^ 42), octalion (10 ^ 48), nonalion (10 ^ 54), decallion (10 ^ 60), endecalion (10 ^ 66), dodecalion (10 ^ 72) and it is written that "there are no further names".

Naming Principles and List of Large Numbers
All the names of large numbers are constructed in a rather simple way: 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 (mille) and the augmentation suffix-million. There are two main types of names for large numbers in the world:
3x + 3 system (where x is a Latin ordinal number) - this system is used in Russia, France, USA, Canada, Italy, Turkey, Brazil, Greece
and the 6x system (where x is a Latin ordinal number) - this system is the most common in the world (for example: Spain, Germany, Hungary, Portugal, Poland, Czech Republic, Sweden, Denmark, Finland). In it, the missing intermediate 6x + 3 end with the suffix -billion (from it we borrowed a billion, which is also called a billion).

The general list of numbers used in Russia is presented below:

Number	Name	Latin numeral	Increasing prefix SI	Reducing prefix SI	Practical value
10 1	ten		deca	deci-	Number of fingers on 2 hands
10 2	hundred		hecto-	centi-	About half the number of all states on Earth
10 3	thousand		kilo	Milli-	Approximate number of days in 3 years
10 6	million	unus (I)	mega-	micro-	5 times the number of drops in a 10 liter bucket of water
10 9	billion (billion)	duo (II)	giga-	nano-	Approximate population of India
10 12	trillion	tres (III)	tera-	pico	1/13 of the gross domestic product of Russia in rubles for 2003
10 15	quadrillion	quattor (IV)	peta-	femto-	1/30 parsec length in meters
10 18	quintillion	quinque (V)	ex-	atto-	1/18 of the number of grains from the legendary chess inventor award
10 21	sextillion	sex (VI)	zetta-	chain	1/6 the mass of planet Earth in tons
10 24	septillion	septem (VII)	yotta-	yokto-	The number of molecules in 37.2 liters of air
10 27	octillion	octo (VIII)	no-	sieve-	Half the mass of Jupiter in kilograms
10 30	quintillion	novem (IX)	de-	thread-	1/5 of all microorganisms on the planet
10 33	decillion	decem (X)	una-	roaring	Half the mass of the Sun in grams

The pronunciation of the numbers below is often different.

Number	Name	Latin numeral	Practical value
10 36	andecillion	undecim (XI)
10 39	duodecillion	duodecim (XII)
10 42	tredecillion	tredecim (XIII)	1/100 of the number of air molecules on Earth
10 45	quattordecillion	quattuordecim (XIV)
10 48	quindecillion	quindecim (XV)
10 51	sexdecillion	sedecim (XVI)
10 54	septemdecillion	septendecim (XVII)
10 57	octodecillion		So many elementary particles in the sun
10 60	novemdecillion
10 63	vigintillion	viginti (XX)
10 66	anvigintillion	unus et viginti (XXI)
10 69	duovigintillion	duo et viginti (XXII)
10 72	trevigintillion	tres et viginti (XXIII)
10 75	quattorvigintillion
10 78	quinvigintillion
10 81	sexvigintillion		So many elementary particles in the universe
10 84	septemwigintillion
10 87	octovigintillion
10 90	novemvigintillion
10 93	trigintillion	triginta (XXX)
10 96	antrigintillion

10 100 - googol (the number was invented by the 9-year-old nephew of the American mathematician Edward Kasner)

10 123 - quadragintillion (quadraginta, XL)

10 153 - quinquaginta, L

10,183 - sexaginta (LX)

10 213 - septuagintillion (septuaginta, LXX)

10 243 - octogintillion (octoginta, LXXX)

10 273 - nonagintillion (nonaginta, XC)

10,303 - centillion (Centum, C)

Further names can be obtained either by direct or reverse order of Latin numerals (as it is correct, it is not known):

10 306 - antcentillion or centunillion

10 309 - duocentillion or centduollion

10 312 - trecentillion or centtrillion

10 315 - quattorcentillion or centquadrillion

10 402 - tretrigintacentillion or centtretrigintillion

I believe that the second spelling option will be the most correct, since it is more consistent with the construction of numbers in Latin and avoids ambiguities (for example, in the number trecentillion, which, according to the first spelling, is both 10 903 and 10 312).
Numbers further:
Some literary references:

Perelman Ya.I. "Entertaining arithmetic". - M .: Triada-Litera, 1994, pp. 134-140

Vygodsky M. Ya. "Handbook of Elementary Mathematics". - S-Pb., 1994, pp. 64-65

"Encyclopedia of Knowledge". - comp. IN AND. Korotkevich. - St. Petersburg: Owl, 2006, p. 257

"Interesting about physics and mathematics." - Library Kvant. no. 50. - M .: Nauka, 1988, p. 50

As a child, I was tormented by the question of what is the largest number, and I tormented almost everyone with this stupid question. Having learned the number one million, I asked if there was a number more than a million. Billion? And more than a billion? Trillion? More than a trillion? Finally, there was someone clever who explained to me that the question is stupid, since it is enough to just add one to the largest number, and it turns out that it was never the largest, since there are even more numbers.

And now, many years later, I decided to ask another question, namely: what is the largest number that has its own name? Fortunately, now there is an Internet and they can be puzzled by patient search engines that will not call my questions idiotic ;-). Actually, this is what I did, and this is what I found out as a result.

Number	Latin name	Russian prefix
1	unus	an-
2	duo	duo-
3	tres	three-
4	quattuor	quadri-
5	quinque	quinti-
6	sex	sex-
7	septem	septi-
8	octo	oct-
9	novem	non-
10	decem	deci-

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

The American system is pretty simple. All the names of large numbers are constructed as follows: at the beginning there is a Latin ordinal number, and at the end the suffix-million is added to it. An exception is the name "million" which is the name of the number one thousand (lat. mille) and the increasing suffix-million (see table). This is how 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: so: 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 is -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 in 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 by the formula 6 x + 3 (where x is a Latin numeral) and by the formula 6 x + 6 for numbers ending in -billion.

From English system only the number billion passed into the Russian language (10 9), which would still be more correct to call it as the Americans call it - a billion, since we have adopted exactly 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 according to the American or English system, 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. Let me explain why. Let's see for a start how the numbers from 1 to 10 33 are called:

Name	Number
Unit	10 0
Ten	10 1
Hundred	10 2
Thousand	10 3
Million	10 6
Billion	10 9
Trillion	10 12
Quadrillion	10 15
Quintillion	10 18
Sextillion	10 21
Septillion	10 24
Octillion	10 27
Quintillion	10 30
Decillion	10 33

And so, now the question arises, what's next. What's behind the decillion? In principle, of course, 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, but we were interested in numbers. Therefore, according to this system, in addition to the above, you can still get only three proper names - vigintillion (from lat. viginti- twenty), centillion (from lat. centum- one hundred) and a million (from lat. mille- thousand). The Romans did not have more than a thousand of their own 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, in fact, the table:

Thus, according to such a system, the number is greater than 10 3003, which would have its own, non-compound name, it is impossible to get! But nevertheless, numbers more than a million million are known - these are the very off-system numbers. Let's finally tell you about them.

Name	Number
Myriad	10 4
Googol	10 100
Asankheya	10 140
Googolplex	10 10 100
Second Skewes number	10 10 10 1000
Mega	2 (in Moser notation)
Megiston	10 (in Moser notation)
Moser	2 (in Moser notation)
Graham's number	G 63 (in Graham notation)
Stasplex	G 100 (in Graham notation)

The smallest such number is myriad(it is even in Dahl's dictionary), which means a hundred hundred, that is, 10,000. This word, however, 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 things. It is believed that the word myriad came to European languages from ancient Egypt.

Googol(from the English googol) is the number ten to the hundredth power, that is, one with one hundred zeros. Googol was first written about in 1938 in the article "New Names in Mathematics" in the January issue of 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... Please note that "Google" is trademark and googol is a number.

In the famous Buddhist treatise of the Jaina Sutra, dating back to 100 BC, there is a number asankheya(from whale. asenci- uncountable) equal to 10 140. It is believed that this number is equal to the number of cosmic cycles required to attain nirvana.

Googolplex(eng. googolplex) is a number also invented by Kasner with his nephew and means one with a googol of zeros, that is, 10 10 100. 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 the googolplex, the 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 the extent e to the extent e to the 79th power, that is, e e e 79. Later, Riele (te Riele, H. J. J. "On the Sign of the Difference NS(x) -Li (x). " Math. Comput. 48 , 323-328, 1987) reduced the Skewes number to e e 27/4, which is approximately 8.185 10 370. It is clear that since the value of Skuse's 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 - pi, e, Avogadro's number, etc.

But it should be noted that there is a second Skuse number, which in mathematics is denoted as Sk 2, which is even greater than the first Skuse number (Sk 1). Second Skewes number, was introduced by J. Skuse in the same article to denote the number up to which the Riemann hypothesis is valid. Sk 2 is equal to 10 10 10 10 3, that is, 10 10 10 1000.

As you understand, the more there are in the number of degrees, the more difficult it is to understand which of the numbers is greater. For example, looking at the Skuse numbers, without special calculations, it is almost impossible to understand which of these two numbers is greater. Thus, it becomes inconvenient to use powers for very large numbers. Moreover, you can think of 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 will not fit, even in 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 wondered about 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.

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

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

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

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

But Moser is not the largest number either. The largest number ever used in mathematical proof is limit value known as Graham's number(Graham "s number), first used in 1977 to prove one estimate in Ramsey's 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 Knuth's notation cannot be translated into the Moser system. Therefore, we will have to explain this system as well. 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) invented the concept of superdegree, which he proposed to write down with arrows pointing up:

V general view it looks like this:

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

The number G 63 became known as Graham number(it is often denoted simply as G). This number is the largest known number in the world and is even included in the Guinness Book of Records. Ah, here's that Graham's number is greater than Moser's.

P.S. In order to bring great benefit to all mankind and become famous for 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 G 100. Memorize it, and when your children ask what is the largest number in the world, tell them that this number is called stasplex.

Update (4.09.2003): Thanks everyone for the comments. It turned out that I made several mistakes while writing the text. I'll try to fix it now.

I made several mistakes at once by simply mentioning Avogadro's number. First, several people pointed out to me that in fact 6,022 10 23 is the most that neither is natural number... And secondly, there is an opinion, and it seems to me correct, that Avogadro's number is not at all a number in the proper, mathematical sense of the word, since it depends on the system of units. Now it is expressed in "mole -1", but if you express it, for example, in moles or something else, it will be expressed in a completely different number, but this will not stop being Avogadro's number at all.
10,000 - darkness
100,000 - legion
1,000,000 - leodr
10,000,000 - a raven or a lie
100,000,000 - deck
Interestingly, the ancient Slavs also loved large numbers and knew how to count up to a billion. Moreover, they called such an account "small account". In some manuscripts, the authors considered and " great score", reaching the number 10 50. About numbers greater than 10 50 it was said:" And the human mind cannot understand more than this. " meant no longer 10,000, but a million, legion - darkness of those (a million million); leodr - legion of legions (10 to 24 degrees), then it was said - ten leodr, one hundred leodr, ..., and, finally, one hundred thousand leodr legion leodr (10 in 47); leodr leodr (10 in 48) was called a raven and, finally, a deck (10 in 49).
The topic of national names for numbers can be expanded if we recall the forgotten Japanese system of naming numbers, which is very different from the English and American systems (I will not draw hieroglyphs, if someone is interested, they are):
10 0 - ichi
10 1 - jyuu
10 2 - hyaku
10 3 - sen
10 4 - man
10 8 - oku
10 12 - chou
10 16 - kei
10 20 - gai
10 24 - jyo
10 28 - jyou
10 32 - kou
10 36 - kan
10 40 - sei
10 44 - sai
10 48 - goku
10 52 - gougasya
10 56 - asougi
10 60 - nayuta
10 64 - fukashigi
10 68 - muryoutaisuu
Concerning the numbers of Hugo Steinhaus (in Russia, for some reason, his name was translated as Hugo Steinhaus). botev assures that the idea of writing super-large numbers in the form of numbers in circles belongs not to Steinhaus, but to Daniil Kharms, who published this idea for nothing in the article "Raising the Number". I also want to thank Evgeny Sklyarevsky, the author of the most interesting site for entertaining math on the Russian-language Internet - Watermelon, for information that Steinhaus came up with not only the mega and megiston numbers, but also suggested another number mezzon, equal (in its notation) to "3 in a circle".
Now about the number myriad or myrioi. As for 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 reality, but the myriad gained fame thanks to the Greeks. Myriad was the name for 10,000, but there were no names for numbers over ten thousand. However, in the note "Psammit" (ie the calculus of sand), Archimedes showed how one can 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 sphere with a diameter of a myriad of Earth's diameters) no more than 1063 grains of sand would fit (in our notation). It is curious that modern calculations of the number of atoms in the visible Universe lead to the number 10 67 (just a myriad of times more). Archimedes suggested the following names for numbers:
1 myriad = 10 4.
1 d-myriad = myriad of myriads = 10 8.
1 three-myriad = di-myriad of di-myriads = 10 16.
1 tetra-myriad = three-myriad three-myriad = 10 32.
etc.

If there are any comments -

There are numbers that are so incredibly, incredibly large that even to write them down would require the entire universe. But here's what really drives you crazy ... some of these inconceivably large numbers are extremely important to understanding the world.

When I say “the largest number in the universe,” I really mean the largest significant number, the largest possible number that is useful in some way. There are many contenders for this title, but I immediately warn you: there is indeed a risk that trying to understand all of this will blow your mind. And besides, with too much math, you have little fun.

Googol and googolplex

Edward Kasner

We could start with two, very likely the largest numbers you have ever heard of, and these are indeed the two largest numbers that have generally accepted definitions in English language... (There is a fairly accurate nomenclature used to denote numbers as large as you would like, but these two numbers are not currently found in dictionaries.) Google, since it became world famous (albeit with errors, note. in fact it is googol) in Google form, was born in 1920 as a way to get kids interested in big numbers.

To this end, Edward Kasner (pictured) took his two nephews, Milton and Edwin Sirotte, for a stroll through the New Jersey Palisades. He invited them to put forward any ideas, and then nine-year-old Milton suggested "googol". Where he got this word from is unknown, but Kasner decided that or a number in which there are one hundred zeros behind the unit will henceforth be called googol.

But young Milton did not stop there, he proposed an even larger number, a googolplex. This is a number, according to Milton, in which there is 1 in the first place, followed by as many zeros as you could write before you get tired. While this idea is fascinating, Kasner decided that a more formal definition was needed. As he explained in his 1940 book Mathematics and the Imagination, Milton's definition leaves open the risky possibility that the occasional jester could become a mathematician superior to Albert Einstein simply because he has more endurance.

So Kasner decided that the googolplex would be equal, or 1, and then the googol of zeros. Otherwise, and in notation similar to those with which we will deal for other numbers, we will say that a googolplex is. To show how mesmerizing this is, Carl Sagan once remarked that it is physically impossible to write down all the zeros of a googolplex, because there simply isn't enough room in the universe. If you fill the entire volume of the observable Universe with fine dust particles about 1.5 microns in size, then the number of different ways of arranging these particles will be approximately equal to one googolplex.

Linguistically speaking, googol and googolplex are probably the two largest significant numbers (in English at least), but, as we will now establish, there are infinitely many ways to define “significance”.

Real world

If we are talking about the largest significant number, there is a reasonable argument that this really means that we need to find the largest number with a real value in the world. We can start with the current human population, which is currently about 6,920 million. World GDP in 2010 was estimated at about $ 61.96 billion, but both numbers are insignificant compared to the roughly 100 trillion cells that make up the human body. Of course, none of these numbers can compare with full number particles in the Universe, which, as a rule, is considered equal to approximately, and this number is so large that our language does not have a corresponding word.

We can play a little with the systems of measures, making the numbers bigger and bigger. So, the mass of the Sun in tons will be less than in pounds. An excellent way to do this is to use the Planck system of units, which are the smallest possible units for which the laws of physics remain valid. For example, the age of the universe in Planck's time is about. If we go back to the first unit of Planck time after the Big Bang, we will see what the density of the universe was then. We are getting more and more, but we haven't even gotten to googol yet.

The largest number with any real world application - or, in this case, a real world application - is probably one of the most recent estimates of the number of universes in the multiverse. This number is so large that the human brain will literally be unable to perceive all these different universes, since the brain is only capable of approximately configurations. In fact, this number is probably the largest number with any practical meaning unless you take into account the idea of the multiverse as a whole. However, there are still much larger numbers hiding there. But in order to find them, we must go to the realm of pure mathematics, and no better start than prime numbers.

Mersenne primes

Part of the difficulty is coming up with good definition what a "significant" number is. One way is to think in terms of prime and composite numbers. A prime number, as you probably remember from school mathematics, Is any natural number (note not equal to one), which is divisible only by and itself. So, and are prime numbers, and and are composite numbers. This means that any composite number can ultimately be represented by its prime divisors. In a sense, a number is more important than, say, because there is no way to express it in terms of the product of smaller numbers.

Obviously, we can go a little further. for example, it is really simple, which means that in a hypothetical world where our knowledge of numbers is limited to a number, a mathematician can still express a number. But already the next number is prime, which means that the only way to express it - to know directly about its existence. This means that the largest known prime numbers play an important role, but, say, googol - which is ultimately just a collection of numbers and multiplied among themselves - actually does not. And since primes are mostly random, there is no known way to predict that an incredibly large number will actually be prime. To this day, discovering new primes is difficult.

Mathematicians Ancient Greece had a concept of primes at least as early as 500 BC, and 2000 years later, people still knew which numbers were prime only up to about 750. Thinkers of Euclid's time saw the possibility of simplification, but until the Renaissance, mathematics could not really use it in practice. These numbers are known as the Mersenne numbers and are named after the 17th century French scientist Marina Mersenne. The idea is quite simple: the Mersenne number is any number of the form. So, for example, and this number is prime, the same is true for.

It is much faster and easier to identify Mersenne primes than any other kind of prime, and computers have been working hard to find them for the past six decades. Until 1952, the largest known prime number was a number - a number with digits. In the same year, a computer calculated that the number is prime, and this number consists of numbers, which makes it much larger than a googol.

Computers have been on the hunt ever since, and Mersenne's nth number is currently the largest prime number known to mankind. Discovered in 2008, it is - a number with almost a million digits. It is the largest known number that cannot be expressed in terms of any smaller numbers, and if you want to help find an even larger Mersenne number, you (and your computer) can always join the search at http: //www.mersenne. org /.

Skuse's number

Stanley Skewes

Let's go back to prime numbers. As I said, they behave fundamentally wrong, which means that there is no way to predict what the next prime will be. Mathematicians were forced to turn to some rather fantastic measurements in order to come up with some way to predict future primes, even in some obscure way. The most successful of these attempts is probably the prime counting function that he came up with in late XVIII century legendary mathematician Karl Friedrich Gauss.

I'll save you the more complicated math - one way or another, we still have a lot to come - but the essence of the function is this: for any integer, you can estimate how many less primes there are. For example, if, the function predicts that there should be primes, if - primes, less, and if, then there are fewer numbers that are prime.

The arrangement of the primes is indeed irregular and it is just an approximation of the actual number of primes. In fact, we know that there are primes, less, primes less, and primes. This is an excellent grade, to be sure, but it is always only an assessment ... and, more specifically, an upper grade.

In all known cases before, the prime count function slightly exaggerates the actual count of less primes. Mathematicians once thought that it would always be this way, ad infinitum, that this certainly applies to some unimaginably huge numbers, but in 1914 John Edenzor Littlewood proved that for some unknown, unimaginably huge number, this function would begin to produce fewer primes, and then it will switch between upper bound and lower bound an infinite number of times.

The hunt was on the starting point of the races, and here Stanley Skewes appeared (see photo). In 1933, he proved that the upper bound when a function that approximates the number of prime numbers first gives a smaller value is a number. It is difficult to truly understand, even in the most abstract sense, what this number actually represents, and from that point of view, it was the largest number ever used in serious mathematical proof. Since then, mathematicians have been able to reduce the upper bound to a relatively small number, but the original number has remained known as the Skuse number.

So how big is the number that makes even the mighty googolplex dwarf? In The Penguin Dictionary of Curious and Interesting Numbers, David Wells describes one way that Hardy mathematician was able to comprehend the size of Skuse's number:

“Hardy thought it was“ the largest number ever to serve any specific purpose in mathematics, ”and suggested that if you played chess with all the particles in the universe as pieces, one move would be to swap two particles. and the game would end when the same position would be repeated a third time, then the number of all possible games would be approximately equal to Skuse's number. ''

One last thing before moving on: we talked about the lesser of the two Skuse numbers. There is another Skuse number, which the mathematician found in 1955. The first number is obtained on the basis that the so-called Riemann hypothesis is true - this is a particularly difficult hypothesis of mathematics, which remains unproven, very useful when it comes about prime numbers. However, if the Riemann hypothesis is false, Skuse found that the start point of the jumps increases to.

The magnitude problem

Before we get to the number that even Skuse's number looks tiny next to, we need to talk a little about scale, because otherwise we have no way of estimating where we are going to go. Let's take a number first - it's a tiny number so small that people can actually have an intuitive understanding of what it means. There are very few numbers that fit this description, since numbers greater than six cease to be separate numbers and become “several”, “many”, etc.

Now let's take, i.e. ... Although we really cannot intuitively, as it was for a number, it is very easy to understand what it is, to imagine what it is. So far so good. But what happens if we go to? It is equal to, or. We are very far from being able to imagine this value, like any other, very large - we lose the ability to comprehend individual parts somewhere around a million. (True, insane a large number of It would take time to really count to a million of whatever, but the point is, we can still perceive that number.)

However, while we cannot imagine, we are at least able to understand in general terms what 7.6 billion is, perhaps comparing it to something like US GDP. We have moved from intuition to representation and to simple understanding, but at least we still have some gap in understanding what a number is. This is about to change as we move one step up the ladder.

To do this, we need to go to a notation introduced by Donald Knuth, known as arrow notation. In these designations, it can be written as. When we then go to, the number we get is equal to. This is equal to where there is a total of threes. We have now vastly and truly surpassed all the other numbers that have already been spoken of. After all, even the largest of them had only three or four terms in the row of indicators. For example, even Skuse's super-number is “only” - even if adjusted for the fact that both the base and the indicators are much larger than, it is still absolutely nothing compared to the size of the number tower with a billion members.

Obviously, there is no way to comprehend such huge numbers ... and yet, the process by which they are created can still be understood. We could not understand the real number that is given by a tower of powers, in which there are billions of triples, but we can basically imagine such a tower with many members, and a really decent supercomputer can store such towers in memory even if it cannot calculate their actual values. ...

This is becoming more and more abstract, but it will only get worse. You might think that it is a tower of powers whose exponent length is (moreover, in the previous version of this post I made exactly this mistake), but it's simple. In other words, imagine that you have the ability to calculate exact value a power tower of triplets, which consists of elements, and then you took this value and created a new tower with as many in it ... that it gives.

Repeat this process with each successive number ( note. starting from the right) until you do it once, and then you finally get it. This is a number that is simply incredibly large, but at least the steps to get it seem to be understandable, if everything is done very slowly. We can no longer understand the number or imagine the procedure by which it is obtained, but at least we can understand the basic algorithm, only in a long enough time.

Now let's prepare the mind to really blow it up.

Graham's number (Graham)

Ronald Graham

This is how you get the Graham number, which ranks in the Guinness Book of World Records as the largest number ever used in mathematical proof. It is completely impossible to imagine how great it is, and it is just as difficult to explain exactly what it is. Basically, Graham's number appears when dealing with hypercubes, which are theoretical geometric shapes with more than three dimensions. The mathematician Ronald Graham (see photo) wanted to find out at what the smallest number measurements, certain properties of the hypercube will remain stable. (Sorry for such a vague explanation, but I'm sure we all need to complete at least two degrees in mathematics to make it more accurate.)

In any case, the Graham number is an upper bound for this minimum number of dimensions. So how big is this upper bound? Let's go back to a number so large that we can only vaguely understand the algorithm for obtaining it. Now, instead of just jumping up one more level to, we will count the number in which there are arrows between the first and last three. Now we are far beyond even the slightest understanding of what this number is, or even what needs to be done to calculate it.

Now we repeat this process once ( note. at each next step we write the number of arrows, equal to the number obtained in the previous step).

This, ladies and gentlemen, is Graham's number, which is about an order of magnitude higher than the point of human understanding. This number, which is so much larger than any number you can imagine - is much more than any infinity you could ever hope to imagine - it just defies even the most abstract description.

But here's the weird thing. Since Graham's number is basically just triples multiplied among themselves, we know some of its properties without actually calculating it. We cannot represent Graham's number using any notation we know, even if we used the entire universe to write it down, but I can tell you the last twelve digits of Graham's number right now:. And that's not all: we know at least the last digits of Graham's number.

Of course, it is worth remembering that this number is only the upper bound in the original Graham problem. It is possible that the actual number of measurements required to complete desired property much, much less. In fact, since the 1980s, it was believed, according to most experts in this field, that in fact the number of dimensions is only six - a number so small that we can understand it intuitively. Since then, the lower bound has been increased to, but there is still a very good chance that the solution to Graham's problem does not lie next to a number as large as Graham's number.

To infinity

So there are numbers greater than Graham's number? There is, of course, the Graham number for starters. Concerning significant number... well, there are some devilishly complex areas of mathematics (in particular, 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 I can hopefully ever be able to reasonably explain. For those reckless enough enough to go even further, further reading is offered at your own risk.

Well, now an amazing quote attributed to Douglas Ray ( note. to be honest, it sounds pretty funny):

“I see clusters of vague numbers that are hiding there, in the darkness, behind a small spot of light that the candle of the mind gives. They whisper to each other; conspiring who knows what. Perhaps they don't like us very much for capturing their little brothers with our minds. Or, perhaps, they simply lead an unambiguous numerical way of life, there, beyond our understanding ''.

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

Of course, the answer to the question about the largest number from a logical point of view cannot be unambiguous, because you only need 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 entered in the Guinness Book of Records.

Graham's number - the largest number in the world

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

Googol's number

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

Googlex

This number was also invented by mathematician Edward Kasner and his nephew Milton Sirotta. In a general sense, it is the tenth power of a googol. Answering the question of many curious people, how many zeros are in Googleplex, it is worth noting that in classic version there is no way to imagine this number, even if you write down all the paper on the planet with classical zeros.

Skuse's number

Another contender for the title of the highest number is the Skuse number, proved by John Littlewood 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 proposed to denote them by polygons. As a result of three performed mathematical operations, the number 2 is born in a mega-gon (a polygon with mega sides).

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

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. You just need to add one to the largest number, as it will no longer be the largest. This procedure can be continued indefinitely.

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

Now we will all find out ...

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

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: so: 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 is -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 is 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 by the formula 6 x + 3 (where x is a Latin numeral) and by the formula 6 x + 6 for numbers ending in -billion.

Only the number billion (10 9) passed from the English system to the Russian language, which would still be more correct to call it as the Americans call it - a billion, since it is the American system that has been adopted in our country. 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.

And so, now the question arises, what's next. What's behind the decillion? In principle, of course, 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, but we were interested in numbers. Therefore, according to this system, in addition to the above, you can still get only three proper names - vigintillion (from lat.viginti- twenty), centillion (from lat.centum- one hundred) and a million (from lat.mille- thousand). The Romans did not have more than a thousand of their own 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, in fact, the table:

Thus, according to a similar system, the numbers are greater than 10 3003 , which would have its own, non-compound name, it is impossible to get! But nevertheless, numbers more than a million million are known - these are the very off-system numbers. Let's finally tell you about them.

The smallest such number is a myriad (it is even in Dahl's dictionary), which means one hundred hundreds, that is, 10,000 does not mean a definite number at all, but an uncountable, uncountable set of something. It is believed that the word myriad came into 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 reality, but the myriad gained fame thanks to the Greeks. Myriad was the name for 10,000, but there were no names for numbers over ten thousand. However, in the note "Psammit" (ie the calculus of sand), Archimedes showed how one can 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 sphere with a diameter of a myriad of Earth's diameters) 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 (just a myriad of times more). Archimedes suggested the following names for numbers:
1 myriad = 10 4.
1 d-myriad = myriad myriad = 10 8 .
1 three-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. Googol was first written about in 1938 in the article "New Names in Mathematics" in the January issue of 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 it mentioned that - but it is not ...

In the famous Buddhist treatise of the Jaina Sutra, dating back to 100 BC, there is a number asankheya(from whale. asenci- uncountable) equal to 10 140. It is believed that this number is equal to the number of cosmic cycles required to attain nirvana.

Googolplex(eng. googolplex) is a number also invented by Kasner with his nephew and means 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.

Even more than a googolplex number - Skewes number (Skewes "number) was proposed by Skewes in 1933 (Skewes. J. London Math. Soc. 8, 277-283, 1933.) in proving the Riemann conjecture concerning prime numbers. It means e to the extent e to the extent e to the 79th power, that is, ee e 79 ... Later, Riele (te Riele, H. J. J. "On the Sign of the Difference NS(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 Skuse's number depends on the number e, then it is not an integer, therefore we will not consider it, otherwise we would have to remember other non-natural numbers - pi, 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 is not valid. Sk2 is 1010 10103 , that is, 1010 101000 .

As you understand, the more there are in the number of degrees, the more difficult it is to understand which of the numbers is greater. For example, looking at the Skuse numbers, without special calculations, it is almost impossible to understand which of these two numbers is greater. Thus, it becomes inconvenient to use powers for very large numbers. Moreover, you can think of 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 will not fit, even in 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.

Consider the notation of Hugo Steinhaus (H. Steinhaus. Mathematical Snapshots, 3rd edn. 1983), which is pretty simple. Stein House proposed to write large numbers inside geometric shapes - a triangle, a square and a circle:

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

But Moser is not the largest number either. The largest number ever used in mathematical proof is a limiting value known as Graham's number(Graham "s number), first used in 1977 to prove one estimate in Ramsey's 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.

In general, it looks like this:

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

The number G63 became known as Graham number(it is often denoted simply as G). This number is the largest known number in the world and is even included in the Guinness Book of Records. Ah, here's that Graham's number is greater than Moser's.

P.S. In order to bring great benefit to all mankind and become famous for 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. Memorize it, and when your children ask what is the largest number in the world, tell them that this number is called stasplex

So there are numbers greater than Graham's number? There is, of course, there is Graham's number for starters.... As for the significant number ... well, there are some devilishly complex areas of mathematics (in particular, 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 reasonably and intelligibly explained.

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