What is the latest number in the world. What is the largest number? What are they, giant numbers
Once I read a tragic story about a Chukchi who was taught to count and write numbers by polar explorers. The magic of numbers impressed him so much that he decided to write down absolutely all the numbers in the world in a row, starting from one, in the notebook donated by the polar explorers. The Chukchi abandons all his affairs, stops communicating even with his own wife, no longer hunts seals and seals, but writes and writes numbers in a notebook .... So a year goes by. In the end, the notebook ends and the Chukchi realizes that he was able to write down only a small part of all the numbers. He weeps bitterly and in despair burns his scribbled notebook in order to start living the simple life of a fisherman again, no longer thinking about the mysterious infinity of numbers...
We will not repeat the feat of this Chukchi and try to find the largest number, since it is enough for any number to just add one to get an even larger number. Let's ask ourselves a similar but different question: which of the numbers that have their own name is the largest?
Obviously, although the numbers themselves are infinite, they do not have very many proper names, since most of them are content with names made up of smaller numbers. So, for example, the numbers 1 and 100 have their own names "one" and "one hundred", and the name of the number 101 is already compound ("one hundred and one"). It is clear that in the finite set of numbers that humanity has awarded own name must be some largest number. But what is it called and what is it equal to? Let's try to figure it out and find, in the end, this is the largest number!
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"Short" and "long" scale
The history of the modern naming system for large numbers dates back to the middle of the 15th century, when in Italy they began to use the words "million" (literally - a big thousand) for a thousand squared, "bimillion" for a million squared and "trimillion" for a million cubed. We know about this system thanks to the French mathematician Nicolas Chuquet (Nicolas Chuquet, c. 1450 - c. 1500): in his treatise "The Science of Numbers" (Triparty en la science des nombres, 1484), he developed this idea, proposing to further use the Latin cardinal numbers (see table), adding them to the ending "-million". So, Shuke's "bimillion" turned into a billion, "trimillion" into a trillion, and a million to the fourth power became a "quadrillion".
In Schücke's system, the number 10 9 , which was between a million and a billion, did not have its own name and was simply called "a thousand million", similarly, 10 15 was called "a thousand billion", 10 21 - "a thousand trillion", etc. It was not very convenient, and in 1549 the French writer and scientist Jacques Peletier du Mans (1517-1582) proposed to name such "intermediate" numbers using the same Latin prefixes, but the ending "-billion". So, 10 9 became known as "billion", 10 15 - "billiard", 10 21 - "trillion", etc.
The Shuquet-Peletier system gradually became popular and was used throughout Europe. However, in the 17th century, an unexpected problem arose. It turned out that for some reason some scientists began to get confused and call the number 10 9 not “a billion” or “a thousand million”, but “a billion”. Soon this error quickly spread, and a paradoxical situation arose - "billion" became simultaneously a synonym for "billion" (10 9) and "million million" (10 18).
This confusion continued for a long time and led to the fact that in the USA they created their own system for naming large numbers. According to the American system, the names of numbers are built in the same way as in the Schücke system - the Latin prefix and the ending "million". However, these numbers are different. If in the Schuecke system names with the ending "million" received numbers that were powers of a million, then in the American system the ending "-million" received the powers of a thousand. That is, a thousand million (1000 3 \u003d 10 9) began to be called a "billion", 1000 4 (10 12) - "trillion", 1000 5 (10 15) - "quadrillion", etc.
The old system of naming large numbers continued to be used in conservative Great Britain and began to be called "British" all over the world, despite the fact that it was invented by the French Shuquet and Peletier. However, in the 1970s, the UK officially switched to the "American system", which led to the fact that it became somehow strange to call one system American and another British. As a result, the American system is now commonly referred to as the "short scale" and the British or Chuquet-Peletier system as the "long scale".
In order not to get confused, let's sum up the intermediate result:
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The short naming scale is now used in the United States, United Kingdom, Canada, Ireland, Australia, Brazil and Puerto Rico. Russia, Denmark, Turkey, and Bulgaria also use the short scale, except that the number 109 is not called "billion" but "billion". The long scale continues to be used today in most other countries.
It is curious that in our country the final transition to the short scale took place only in the second half of the 20th century. So, for example, even Yakov Isidorovich Perelman (1882-1942) in his "Entertaining Arithmetic" mentions the parallel existence of two scales in the USSR. The short scale, according to Perelman, was used in everyday life and financial calculations, and the long one was used in scientific books on astronomy and physics. However, now it is wrong to use a long scale in Russia, although the numbers there are large.
But back to finding the largest number. After a decillion, the names of numbers are obtained by combining prefixes. This is how numbers such as undecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion, novemdecillion, etc. are obtained. However, these names are no longer of interest to us, since we agreed to find the largest number with its own non-composite name.
If we turn to Latin grammar, we will find that the Romans had only three non-compound names for numbers greater than ten: viginti - "twenty", centum - "one hundred" and mille - "thousand". For numbers greater than "thousand", the Romans did not have their own names. For example, the Romans called a million (1,000,000) "decies centena milia", that is, "ten times a hundred thousand". According to Schuecke's rule, these three remaining Latin numerals give us such names for numbers as "vigintillion", "centillion" and "milleillion".
So, we found out that on the "short scale" the maximum number that has its own name and is not a composite of smaller numbers is "million" (10 3003). If a “long scale” of naming numbers were adopted in Russia, then the largest number with its own name would be “million” (10 6003).
However, there are names for even larger numbers.
Numbers outside the system
Some numbers have their own name, without any connection with the naming system using Latin prefixes. And there are many such numbers. You can, for example, remember the number e, the number "pi", a dozen, the number of the beast, etc. However, since we are now interested in large numbers, we will consider only those numbers with their own non-compound name that are more than a million.
Until the 17th century, Rus' used its own system for naming numbers. Tens of thousands were called "darks," hundreds of thousands were called "legions," millions were called "leodres," tens of millions were called "ravens," and hundreds of millions were called "decks." This account up to hundreds of millions was called the “small account”, and in some manuscripts the authors also considered the “great account”, in which the same names were used for large numbers, but with a different meaning. So, "darkness" meant not ten thousand, but a thousand thousand (10 6), "legion" - the darkness of those (10 12); "leodr" - legion of legions (10 24), "raven" - leodr of leodres (10 48). For some reason, the “deck” in the great Slavic count was not called the “raven of ravens” (10 96), but only ten “ravens”, that is, 10 49 (see table).
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The number 10100 also has its own name and was invented by a nine-year-old boy. And it was like that. In 1938, the American mathematician Edward Kasner (Edward Kasner, 1878-1955) was walking in the park with his two nephews and discussing large numbers with them. During the conversation, we talked about a number with one hundred zeros, which did not have its own name. One of his nephews, nine-year-old Milton Sirott, suggested calling this number "googol". In 1940, Edward Kasner, together with James Newman, wrote the non-fiction book Mathematics and the Imagination, where he taught mathematics lovers about the googol number. Google became even more widely known in the late 1990s, thanks to the Google search engine named after it.
The name for an even larger number than googol arose in 1950 thanks to the father of computer science, Claude Shannon (Claude Elwood Shannon, 1916-2001). In his article "Programming a Computer to Play Chess," he tried to estimate the number options chess game. According to him, each game lasts an average of 40 moves, and on each move the player chooses an average of 30 options, which corresponds to 900 40 (approximately equal to 10 118) 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 10 140. It is believed that this number is equal to the number of cosmic cycles required to gain nirvana.
Nine-year-old Milton Sirotta entered the history of mathematics not only by inventing the googol number, but also by suggesting another number at the same time - “googolplex”, which is equal to 10 to the power of “googol”, that is, one with a googol of zeros.
Two more numbers larger than the googolplex were proposed by the South African mathematician Stanley Skewes (1899-1988) when proving the Riemann hypothesis. The first number, which later came to be called "Skeuse's first number", is equal to e to the extent e to the extent e to the power of 79, that is e e e 79 = 10 10 8.85.10 33 . However, the "second Skewes number" is even larger and is 10 10 10 1000 .
Obviously, the more degrees in the number of degrees, the more difficult it is to write down numbers and understand their meaning when reading. Moreover, it is possible to come up with such numbers (and they, by the way, have already been invented), when the degrees of degrees simply do not fit on the page. Yes, what a page! They won't even fit in a book the size of the entire universe! In this case, the question arises how to write down such numbers. The problem is, fortunately, resolvable, and mathematicians have developed several principles for writing such numbers. True, each mathematician who asked this problem came up with his own way of writing, which led to the existence of several unrelated ways to write large numbers - these are the notations of Knuth, Conway, Steinhaus, etc. We will now have to deal with some of them.
Other notations
In 1938, the same year that nine-year-old Milton Sirotta came up with the googol and googolplex numbers, Hugo Dionizy Steinhaus, 1887-1972, a book about entertaining mathematics, The Mathematical Kaleidoscope, was published in Poland. This book became very popular, went through many editions and was translated into many languages, including English and Russian. In it, Steinhaus, discussing large numbers, offers a simple way to write them using three geometric shapes - a triangle, a square and a circle:
"n in a triangle" means " n n»,
« n square" means " n V n triangles",
« n in a circle" means " n V n squares."
Explaining this way of writing, Steinhaus comes up with the number "mega" equal to 2 in a circle and shows that it is equal to 256 in a "square" or 256 in 256 triangles. To calculate it, you need to raise 256 to the power of 256, raise the resulting number 3.2.10 616 to the power of 3.2.10 616, then raise the resulting number to the power of the resulting number, and so on to raise to the power of 256 times. For example, the calculator in MS Windows cannot calculate due to overflow 256 even in two triangles. Approximately this huge number is 10 10 2.10 619 .
Having determined the number "mega", Steinhaus invites readers to independently evaluate another number - "medzon", equal to 3 in a circle. In another edition of the book, Steinhaus instead of the medzone proposes to estimate an even larger number - “megiston”, equal to 10 in a circle. Following Steinhaus, I will also recommend that readers break away from this text for a while and try to write these numbers themselves using ordinary powers in order to feel their gigantic magnitude.
However, there are names for O higher numbers. So, the Canadian mathematician Leo Moser (Leo Moser, 1921-1970) finalized the Steinhaus notation, which was limited by the fact that if it were necessary to write down numbers much larger than a megiston, then difficulties and inconveniences would arise, since one would have to draw many circles one inside another. Moser suggested drawing not circles after squares, but pentagons, then hexagons, and so on. He also proposed a formal notation for these polygons, so that numbers could be written without drawing complex patterns. Moser notation looks like this:
« n triangle" = n n = n;
« n in a square" = n = « n V n triangles" = nn;
« n in a pentagon" = n = « n V n squares" = nn;
« n V k+ 1-gon" = n[k+1] = " n V n k-gons" = n[k]n.
Thus, according to Moser's notation, the Steinhausian "mega" is written as 2, "medzon" as 3, and "megiston" as 10. In addition, Leo Moser suggested calling a polygon with a number of sides equal to mega - "megagon". And he proposed the number "2 in megagon", that is, 2. This number became known as the Moser number or simply as "moser".
But even "moser" is not the largest number. So, the largest number ever used in a mathematical proof is "Graham's 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 dimensions of certain n-dimensional bichromatic hypercubes. Graham's number gained fame only after the story about it in Martin Gardner's 1989 book "From Penrose Mosaics to Secure Ciphers".
To explain how large the Graham number is, one has to explain another way of writing large numbers, introduced by Donald Knuth in 1976. American professor Donald Knuth came up with the concept of superdegree, which he proposed to write with arrows pointing up:
I think that everything is clear, so let's get back to Graham's number. Ronald Graham proposed the so-called G-numbers:
Here is the number G 64 and is called the Graham number (it is often denoted simply as G). This number is the largest known number in the world used in a mathematical proof, and is even listed in the Guinness Book of Records.
And finally
Having written this article, I can not resist the temptation and come up with my own number. Let this number be called stasplex» and will be 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.
Partner news
Back in the fourth grade, I was interested in the question: "What are the numbers more than a billion called? 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 access to the Internet, the search has accelerated significantly. Now I present all the information I found so that others can answer the question: "What are large and very large numbers called?".
A bit of history
The southern and eastern Slavic peoples used alphabetical numbering to record numbers. Moreover, among the Russians, not all letters played the role of numbers, but only those that are in the Greek alphabet. Above the letter, denoting a number, a special "titlo" icon was placed. At the same time, the numerical values of the letters increased in the same order as the letters in the Greek alphabet followed (the order of the letters of the Slavic alphabet was somewhat different).
In Russia, Slavic numbering survived 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 reduced for faster pronunciation. Until the 15th century, the number "forty" was denoted by the word "fourty", and in the 15-16th centuries this word was supplanted by the word "forty", which originally meant a bag in which 40 squirrel or sable skins were placed. There are two options about the origin of the word "thousand": from the old name "fat 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 augmentative suffix to the number "mille" - a thousand (i.e. it meant "big thousand"), it penetrated into the Russian language later, and before that the same meaning in Russian was denoted by the number "leodr". The word "billion" came into use only from the time of 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 magnifying suffix. In Germany and America, for some time, the word "billion" meant the number 100,000,000; this explains why the word billionaire was used in America before any of the rich had $1,000,000,000. In the old (XVIII century) "Arithmetic" of Magnitsky, there is a table of names of numbers, brought to the "quadrillion" (10 ^ 24, according to the system through 6 digits). Perelman Ya.I. in the book "Entertaining Arithmetic" the names of large numbers of that time are given, somewhat different from today: septillon (10 ^ 42), octalion (10 ^ 48), nonalion (10 ^ 54), decalion (10 ^ 60), endecalion (10 ^ 66), dodecalion (10 ^ 72) and it is written that "there are no further names".
Principles of naming and the 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 thousand (mille) and the magnifying 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 ends 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 | SI magnifier | SI diminutive prefix | Practical value |
| 10 1 | ten | deca- | deci- | Number of fingers on 2 hands | |
| 10 2 | one hundred | hecto- | centi- | Approximately 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 of the length of a parsec in meters |
| 10 18 | quintillion | quinque (V) | exa- | atto- | 1/18 of the number of grains from the legendary award to the inventor of chess |
| 10 21 | sextillion | sex (VI) | zetta- | zepto- | 1/6 of the mass of the planet Earth in tons |
| 10 24 | septillion | septem(VII) | yotta- | yocto- | 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) | dea- | tredo- | 1/5 of all microorganisms on the planet |
| 10 33 | decillion | decem(X) | una- | revo- | Half the mass of the Sun in grams |
The pronunciation of the numbers that follow 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 | septemvigintillion | ||
| 10 87 | octovigintillion | ||
| 10 90 | novemvigintillion | ||
| 10 93 | trigintillion | triginta (XXX) | |
| 10 96 | antirigintillion |
- ...
- 10 100 - googol (the number was invented by the 9-year-old nephew of the American mathematician Edward Kasner)
- 10 123 - quadragintillion (quadragaginta, XL)
- 10 153 - quinquagintillion (quinquaginta, L)
- 10 183 - sexagintillion (sexaginta, LX)
- 10 213 - septuagintillion (septuaginta, LXX)
- 10 243 - octogintillion (octoginta, LXXX)
- 10 273 - nonagintillion (nonaginta, XC)
- 10 303 - centillion (Centum, C)
- 10 306 - ancentillion or centunillion
- 10 309 - duocentillion or centduollion
- 10 312 - trecentillion or centtrillion
- 10 315 - quattorcentillion or centquadrillion
- 10 402 - tretrigintacentillion or centtretrigintillion
Numbers next:
Some literary references:
- Perelman Ya.I. "Entertaining arithmetic". - M.: Triada-Litera, 1994, pp. 134-140
- Vygodsky M.Ya. "Handbook of Elementary Mathematics". - St. Petersburg, 1994, pp. 64-65
- "Encyclopedia of Knowledge". - comp. IN AND. Korotkevich. - St. Petersburg: Owl, 2006, p. 257
- "Entertaining about physics and mathematics." - Kvant Library. issue 50. - M.: Nauka, 1988, p. 50
“I see clumps of vague numbers lurking out there in the dark, behind the little spot of light that the mind candle gives. They whisper to each other; talking about who knows what. Perhaps they do not like us very much for capturing their little brothers with our minds. Or maybe they just lead an unambiguous numerical way of life, out there, beyond our understanding.''
Douglas Ray
We continue ours. Today we have numbers...
Sooner or later, everyone is tormented by the question, what is the largest number. A child's question can be answered in a million. What's next? Trillion. And even further? In fact, the answer to the question of what are the largest numbers is simple. It is simply worth adding one to the largest number, as it will no longer be the largest. This procedure can be continued indefinitely.
But if you ask yourself: what is the largest number that exists, and what is its own name?
Now we all know...
There are two systems for naming numbers - American and English.
The American system is built quite simply. All names of large numbers are built like this: at the beginning there is a Latin ordinal number, and at the end the suffix -million is added to it. The exception is the name "million" which is the name of the number one thousand (lat. mille) and the magnifying suffix -million (see table). So the numbers are obtained - trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion and decillion. The American system is used in the USA, Canada, France and Russia. You can find out the number of zeros in a number written in the American system using the simple formula 3 x + 3 (where x is a Latin numeral).
The English naming system is the most common in the world. It is used, for example, in Great Britain and Spain, as well as in most of the former English and Spanish colonies. The names of numbers in this system are built like this: like this: a suffix -million is added to the Latin numeral, the next number (1000 times larger) is built according to the principle - the same Latin numeral, but the suffix is -billion. That is, after a trillion in the English system comes a trillion, and only then a quadrillion, followed by a quadrillion, and so on. Thus, a quadrillion according to the English and American systems are completely different numbers! You can find out the number of zeros in a number written in the English system and ending with the suffix -million using the formula 6 x + 3 (where x is a Latin numeral) and using the formula 6 x + 6 for numbers ending in -billion.
Only the number billion (10 9 ) passed from the English system into the Russian language, which, nevertheless, would be more correct to call it the way the Americans call it - a billion, since we have adopted the American system. But who in our country does something according to the rules! ;-) By the way, sometimes the word trillion is also used in Russian (you can see for yourself by running a search in Google or Yandex) and it means, apparently, 1000 trillion, i.e. quadrillion.
In addition to numbers written using Latin prefixes in the American or English system, the so-called off-system numbers are also known, i.e. numbers that have their own names without any Latin prefixes. There are several such numbers, but I will talk about them in more detail a little later.
Let's go back to writing using Latin numerals. It would seem that they can write numbers to infinity, but this is not entirely true. Now I will explain why. Let's first see how the numbers from 1 to 10 33 are called:
And so, now the question arises, what next. What is a decillion? In principle, it is possible, of course, by combining prefixes to generate such monsters as: andecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion and novemdecillion, but these will already be compound names, and we were interested in our own names numbers. Therefore, according to this system, in addition to those indicated above, you can still get only three - vigintillion (from lat.viginti- twenty), centillion (from lat.percent- one hundred) and a million (from lat.mille- thousand). The Romans did not have more than a thousand proper names for numbers (all numbers over a thousand were composite). For example, a million (1,000,000) Romans calledcentena miliai.e. ten hundred thousand. And now, actually, the table:
Thus, according to a similar system, numbers are greater than 10 3003 , which would have its own, non-compound name, it is impossible to get! But nevertheless, numbers greater than a million are known - these are the very non-systemic numbers. Finally, let's talk about them.
The smallest such number is a myriad (it is even in Dahl's dictionary), which means a hundred hundreds, that is, 10,000. True, this word is outdated and is practically not used, but it is curious that the word "myriad" is widely used, which does not mean a certain number at all, but an uncountable, uncountable set of something. It is believed that the word myriad (English myriad) came to European languages from ancient Egypt.
There are different opinions about the origin of this number. Some believe that it originated in Egypt, while others believe that it was born only in ancient Greece. Be that as it may, in fact, the myriad gained fame precisely thanks to the Greeks. Myriad was the name for 10,000, and there were no names for numbers over ten thousand. However, in the note "Psammit" (i.e., the calculus of sand), Archimedes showed how one can systematically build and name arbitrarily large numbers. In particular, placing 10,000 (myriad) grains of sand in a poppy seed, he finds that in the Universe (a ball with a diameter of a myriad of Earth diameters) would fit (in our notation) no more than 10 63
grains of sand. It is curious that modern calculations of the number of atoms in the visible universe lead to the number 10 67
(only a myriad of times more). The names of the numbers Archimedes suggested are as follows:
1 myriad = 10 4 .
1 di-myriad = myriad myriad = 10 8
.
1 tri-myriad = di-myriad di-myriad = 10 16
.
1 tetra-myriad = three-myriad three-myriad = 10 32
.
etc.
Googol (from the English googol) is the number ten to the hundredth power, that is, one with one hundred zeros. The "googol" was first written about in 1938 in the article "New Names in Mathematics" in the January issue of the journal Scripta Mathematica by the American mathematician Edward Kasner. According to him, his nine-year-old nephew Milton Sirotta suggested calling a large number "googol". This number became well-known thanks to the search engine named after him. Google. Note that "Google" is a trademark and googol is a number.
Edward Kasner.
On the Internet, you can often find mention that - but this is not so ...
In the well-known Buddhist treatise Jaina Sutra, dating back to 100 BC, the number Asankheya (from the Chinese. asentzi- incalculable), equal to 10 140. It is believed that this number is equal to the number of cosmic cycles required to gain nirvana.
Googolplex (English) googolplex) - a number also invented by Kasner with his nephew and meaning one with a googol of zeros, that is, 10 10100 . Here is how Kasner himself describes this "discovery":
Words of wisdom are spoken by children at least as often as by scientists. The name "googol" was invented by a child (Dr. Kasner"s nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. a googol, but is still finite, as the inventor of the name was quick to point out.
Mathematics and the Imagination(1940) by Kasner and James R. Newman.
Even larger than the googolplex number, Skewes' number was proposed by Skewes in 1933 (Skewes. J. London Math. soc. 8, 277-283, 1933.) in proving the Riemann conjecture concerning primes. It means e to the extent e to the extent e to the power of 79, i.e. ee e 79 . Later, Riele (te Riele, H. J. J. "On the Sign of the Difference P(x)-Li(x)." Math. Comput. 48, 323-328, 1987) reduced Skuse's number to ee 27/4 , which is approximately equal to 8.185 10 370 . It is clear that since the value of the Skewes number depends on the number e, then it is not an integer, so we will not consider it, otherwise we would have to recall other non-natural numbers - the number pi, the number e, etc.
But it should be noted that there is a second Skewes number, which in mathematics is denoted as Sk2 , which is even larger than the first Skewes number (Sk1 ). Skuse's second number, was introduced by J. Skuse in the same article to denote a number for which the Riemann hypothesis is not valid. Sk2 is 1010 10103 , i.e. 1010 101000 .
As you understand, the more degrees there are, the more difficult it is to understand which of the numbers is greater. For example, looking at the Skewes numbers, without special calculations, it is almost impossible to understand which of these two numbers is larger. Thus, for superlarge numbers, it becomes inconvenient to use powers. Moreover, you can come up with such numbers (and they have already been invented) when the degrees of degrees simply do not fit on the page. Yes, what a page! They won't even fit into a book the size of the entire universe! In this case, the question arises how to write them down. The problem, as you understand, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who asked this problem came up with his own way of writing, which led to the existence of several, unrelated, ways to write numbers - these are the notations of Knuth, Conway, Steinhaus, etc.
Consider the notation of Hugo Stenhaus (H. Steinhaus. Mathematical Snapshots, 3rd edn. 1983), which is quite simple. Steinhouse suggested writing large numbers inside geometric shapes- triangle, square and circle:
Steinhouse came up with two new super-large numbers. He called the number - Mega, and the number - Megiston.
The mathematician Leo Moser refined Stenhouse's notation, which was limited by the fact that if it was necessary to write numbers much larger than a megiston, difficulties and inconveniences arose, since many circles had to be drawn one inside the other. Moser suggested drawing not circles after squares, but pentagons, then hexagons, and so on. He also proposed a formal notation for these polygons, so that numbers could be written without drawing complex patterns. Moser notation looks like this:
Thus, according to Moser's notation, Steinhouse's mega is written as 2, and megiston as 10. In addition, Leo Moser suggested calling a polygon with the number of sides equal to mega - megagon. And he proposed the number "2 in Megagon", that is, 2. This number became known as Moser's number or simply as moser.
But the moser is not the largest number. The largest number ever used in a mathematical proof is the limiting value known as Graham's number, first used in 1977 in the proof of one estimate in Ramsey theory. It is associated with bichromatic hypercubes and cannot be expressed without the special 64-level system of special mathematical symbols introduced by Knuth in 1976.
Unfortunately, the number written in the Knuth notation cannot be translated into the Moser notation. Therefore, this system will also have to be explained. In principle, there is nothing complicated in it either. Donald Knuth (yes, yes, this is the same Knuth who wrote The Art of Programming and created the TeX editor) came up with the concept of superpower, which he proposed to write with arrows pointing up:
In general, it looks like this:
I think that everything is clear, so let's get back to Graham's number. Graham proposed the so-called G-numbers:
- G1 = 3..3, where the number of superdegree arrows is 33.
- G2 = ..3, where the number of superdegree arrows is equal to G1 .
- G3 = ..3, where the number of superdegree arrows is equal to G2 .
- G63 = ..3, where the number of superpower arrows is G62 .
The number G63 became known as the Graham number (it is often denoted simply as G). This number is the largest known number in the world and is even listed in the Guinness Book of Records. And here
In the names of Arabic numbers, each digit belongs to its category, and every three digits form a class. Thus, the last digit in a number indicates the number of units in it and is called, accordingly, the place of units. The next, second from the end, digit indicates tens (the tens digit), and the third digit from the end indicates the number of hundreds in the number - the hundreds digit. Further, the digits are repeated in the same way in turn in each class, denoting units, tens and hundreds in the classes of thousands, millions, and so on. If the number is small and does not contain a tens or hundreds digit, it is customary to take them as zero. Classes group numbers in numbers of three, often in computing devices or records a period or space is placed between classes to visually separate them. This is done to make it easier to read large numbers. Each class has its own name: the first three digits are the class of units, followed by the class of thousands, then millions, billions (or billions), and so on.
Since we use the decimal system, the basic unit of quantity is the ten, or 10 1 . Accordingly, with an increase in the number of digits in a number, the number of tens of 10 2, 10 3, 10 4, etc. also increases. Knowing the number of tens, you can easily determine the class and category of the number, for example, 10 16 is tens of quadrillions, and 3 × 10 16 is three tens of quadrillions. The decomposition of numbers into decimal components occurs as follows - each digit is displayed in a separate term, multiplied by the required coefficient 10 n, where n is the position of the digit in the count from left to right.
For example: 253 981=2×10 6 +5×10 5 +3×10 4 +9×10 3 +8×10 2 +1×10 1
Also, the power of 10 is also used in writing decimals: 10 (-1) is 0.1 or one tenth. Similarly with the previous paragraph, a decimal number can also be decomposed, in which case n will indicate the position of the digit from the comma from right to left, for example: 0.347629= 3x10 (-1) +4x10 (-2) +7x10 (-3) +6x10 (-4) +2x10 (-5) +9x10 (-6) )
Names of decimal numbers. Decimal numbers are read by the last digit after the decimal point, for example 0.325 - three hundred and twenty-five thousandths, where thousandths are the digit of the last digit 5.
Table of names of large numbers, digits and classes
| 1st class unit | 1st unit digit 2nd place ten 3rd rank hundreds |
1 = 10 0 10 = 10 1 100 = 10 2 |
| 2nd class thousand | 1st digit units of thousands 2nd digit tens of thousands 3rd rank hundreds of thousands |
1 000 = 10 3 10 000 = 10 4 100 000 = 10 5 |
| 3rd grade millions | 1st digit units million 2nd digit tens of millions 3rd digit hundreds of millions |
1 000 000 = 10 6 10 000 000 = 10 7 100 000 000 = 10 8 |
| 4th grade billions | 1st digit units billion 2nd digit tens of billions 3rd digit hundreds of billions |
1 000 000 000 = 10 9 10 000 000 000 = 10 10 100 000 000 000 = 10 11 |
| 5th grade trillions | 1st digit trillion units 2nd digit tens of trillions 3rd digit hundred trillion |
1 000 000 000 000 = 10 12 10 000 000 000 000 = 10 13 100 000 000 000 000 = 10 14 |
| 6th grade quadrillions | 1st digit quadrillion units 2nd digit tens of quadrillions 3rd digit tens of quadrillions |
1 000 000 000 000 000 = 10 15 10 000 000 000 000 000 = 10 16 100 000 000 000 000 000 = 10 17 |
| 7th grade quintillions | 1st digit units of quintillions 2nd digit tens of quintillions 3rd rank hundred quintillion |
1 000 000 000 000 000 000 = 10 18 10 000 000 000 000 000 000 = 10 19 100 000 000 000 000 000 000 = 10 20 |
| 8th grade sextillions | 1st digit sextillion units 2nd digit tens of sextillions 3rd rank hundred sextillions |
1 000 000 000 000 000 000 000 = 10 21 10 000 000 000 000 000 000 000 = 10 22 1 00 000 000 000 000 000 000 000 = 10 23 |
| 9th grade septillion | 1st digit units of septillion 2nd digit tens of septillions 3rd rank hundred septillion |
1 000 000 000 000 000 000 000 000 = 10 24 10 000 000 000 000 000 000 000 000 = 10 25 100 000 000 000 000 000 000 000 000 = 10 26 |
| 10th grade octillion | 1st digit octillion units 2nd digit ten octillion 3rd rank hundred octillion |
1 000 000 000 000 000 000 000 000 000 = 10 27 10 000 000 000 000 000 000 000 000 000 = 10 28 100 000 000 000 000 000 000 000 000 000 = 10 29 |
There are numbers that are so incredibly, incredibly large that it would take the entire universe to even write them down. But here's what's really maddening... some of these incomprehensibly 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 maximum possible number that is useful in some way. There are many contenders for this title, but I warn you right away: there is indeed a risk that trying to understand all this will blow your mind. And besides, with too much math, you get little fun.
Googol and googolplex
Edward Kasner
We could start with two, very likely the biggest numbers you've ever heard of, and these are indeed the two biggest numbers that have commonly accepted definitions in English language. (There is a fairly precise nomenclature used for 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 the form of Google, was born in 1920 as a way to get children interested in big numbers.
To this end, Edward Kasner (pictured) took his two nephews, Milton and Edwin Sirott, on a New Jersey Palisades tour. He invited them to come up with any ideas, and then the nine-year-old Milton suggested “googol”. Where he got this word from is unknown, but Kasner decided that 
or a number in which one hundred zeros follow the one will henceforth be called a googol.
But young Milton didn't stop there, he came up with an even bigger number, the googolplex. It's a number, according to Milton, that has a 1 first and then as many zeros as you can write before you get tired. While the idea is fascinating, Kasner felt a more formal definition was needed. As he explained in his 1940 book Mathematics and the Imagination, Milton's definition leaves open the perilous 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 , or 1, followed by a googol of zeros. Otherwise, and in a notation similar to that with which we will deal with other numbers, we will say that the googolplex is . To show how mesmerizing this is, Carl Sagan once remarked that it was physically impossible to write down all the zeros of a googolplex because there simply wasn't enough room in the universe. If the entire volume of the observable universe is filled with fine dust particles approximately 1.5 microns in size, then the number various ways the location of these particles will be approximately equal to one googolplex.
Linguistically speaking, googol and googolplex are probably the two largest significant numbers (at least in English), but, as we will now establish, there are infinitely many ways to define “significance”.
Real world
If we talk about the largest significant number, there is a reasonable argument that this really means that you need to find the largest number with a value that actually exists in the world. We can start with the current human population, which is currently around 6920 million. World GDP in 2010 was estimated to be around $61,960 billion, but both of these numbers are small compared to the roughly 100 trillion cells that make up the human body. Of course, none of these numbers can compare with the total number of particles in the universe, which is usually considered to be about , and this number is so large that our language does not have a word for it.
We can play around with measurement systems a bit, making the numbers bigger and bigger. Thus, the mass of the Sun in tons will be less than in pounds. A great way to do this is to use the Planck units, which are the smallest possible measures for which the laws of physics still hold. For example, the age of the universe in Planck time is about . If we go back to the first Planck time unit after the Big Bang, we will see that the density of the Universe was then . We're getting more and more, but we haven't even reached a googol yet.
The largest number with any real world application—or, in this case, real world application—is probably , one of the latest estimates of the number of universes in the multiverse. This number is so large that human brain will literally be unable to perceive all these different universes, since the brain is only capable of roughly configurations. In fact, this number is probably the largest number with any practical meaning, if you do not take into account the idea of the multiverse as a whole. However, there are still much larger numbers lurking there. But in order to find them, we must go into the realm of pure mathematics, and there is no better place to start than prime numbers.
Mersenne primes
Part of the difficulty is coming up with a good definition of what a “meaningful” number is. One way is to think in terms of primes and composites. A prime number, as you probably remember from school mathematics, is any natural number (not equal to one) that is divisible only by itself. So, and are prime numbers, and and are composite numbers. This means that any composite number can eventually be represented by its prime divisors. In a sense, the 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, is actually just , which means that in a hypothetical world where our knowledge of numbers is limited to , a mathematician can still express . But the next number is already prime, which means that the only way to express it is to directly know about its existence. This means that the largest known prime numbers play an important role, but, say, a googol - which is ultimately just a collection of numbers and , multiplied together - actually does not. And since prime numbers are mostly random, there is no known way to predict that an incredibly large number will actually be prime. To this day, discovering new prime numbers is a difficult task.
The mathematicians of ancient Greece had a concept of prime numbers at least as early as 500 BC, and 2000 years later people still only knew what prime numbers were up to about 750. Euclid's thinkers saw the possibility of simplification, but until the Renaissance mathematicians couldn't really use it in practice. These numbers are known as Mersenne numbers and are named after the 17th century French scientist Marina Mersenne. The idea is quite simple: a Mersenne number is any number of the form . So, for example, and this number is prime, the same is true for .
Mersenne primes are much faster and easier to determine than any other kind of prime, and computers have been hard at work finding them for the past six decades. Until 1952, the largest known prime number was a number—a number with digits. In the same year, it was calculated on a computer that the number is prime, and this number consists of digits, which makes it already much larger than a googol.
Computers have been on the hunt ever since, and the th Mersenne number is currently the largest prime number known to mankind. Discovered in 2008, it is a number with almost millions of digits. This 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/.
Skewes number
Stanley Skuse
Let's go back to prime numbers. As I said before, they behave fundamentally wrong, which means that there is no way to predict what the next prime number will be. Mathematicians have been forced to turn to some rather fantastic measurements in order to come up with some way to predict future primes, even in some nebulous way. The most successful of these attempts is probably the function that counts prime numbers, which he came up with in late XVIII century legendary mathematician Carl Friedrich Gauss.
I'll spare you the more complicated math - anyway, we still have a lot to come - but the essence of the function is this: for any integer, it is possible to estimate how many primes there are less than . For example, if , the function predicts that there should be prime numbers, if - prime numbers less than , and if , then there are smaller numbers that are prime.
The arrangement of primes is indeed irregular, and is only an approximation of the actual number of primes. In fact, we know that there are primes less than , primes less than , and primes less than . It's a great estimate, to be sure, but it's always just an estimate... and more specifically, an estimate from above.
In all known cases up to , the function that finds the number of primes slightly exaggerates the actual number of primes less than . Mathematicians once thought that this would always be the case, ad infinitum, and that this certainly applies to some unimaginably huge numbers, but in 1914 John Edensor Littlewood proved that for some unknown, unimaginably huge number, this function will begin to produce fewer primes, and then it will switch between overestimation and underestimation an infinite number of times.
The hunt was for the starting point of the races, and that's where Stanley Skuse appeared (see photo). In 1933, he proved that the upper limit, when a function that approximates the number of primes for the first time gives a smaller value, is the number. It is difficult to truly understand, even in the most abstract sense, what this number really is, and from this point of view it was the largest number ever used in a 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 Skewes 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 in which the mathematician Hardy was able to make sense of the size of the Skewes number:
"Hardy thought it was 'the largest number ever to serve any particular purpose in mathematics' and suggested that if chess was played with all the particles of the universe as pieces, one move would consist of swapping two particles, and the game would stop when the same position was repeated a third time, then the number of all possible games would be equal to about the number of Skuse''.
One last thing before moving on: we talked about the smaller of the two Skewes numbers. There is another Skewes number, which the mathematician found in 1955. The first number is derived on the grounds that the so-called Riemann Hypothesis is true - a particularly difficult hypothesis in mathematics that remains unproven, very useful when it comes to prime numbers. However, if the Riemann Hypothesis is false, Skewes found that the jump start point increases to .
The problem of magnitude
Before we get to a number that makes even Skuse's number look tiny, we need to talk a little about scale because otherwise we have no way of estimating where we're going. 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 can't really intuitively, as we did for the number , figure out what , imagine what it is, it's very easy. So far everything is going well. But what happens if we go to ? This is equal to , or . We are very far from being able to imagine this value, like any other very large one - we are losing the ability to comprehend individual parts somewhere around a million. (Admittedly, it would take an insanely long time to actually count to a million of anything, but the point is that we are still able to perceive that number.)
However, although we cannot imagine, we are at least able to understand in general terms, which is 7600 billion, perhaps comparing it to something like US GDP. We have gone from intuition to representation to mere understanding, but at least we still have some gap in our understanding of what a number is. This is about to change as we move one more rung up the ladder.
To do this, we need to switch to the notation introduced by Donald Knuth, known as arrow notation. These notations can be written as . When we then go to , the number we get will be . This is equal to where the total of triplets is. We have now vastly and truly surpassed all the other numbers already mentioned. After all, even the largest of them had only three or four members in the index series. For example, even Skuse's super number is "only" - even with the fact that both the base and the exponents are much larger than , it is still absolutely nothing compared to the size of the number tower with billions of 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 given by the tower of powers, which is a billion triples, but we can basically imagine such a tower with many members, and a really decent supercomputer will be able to store such towers in memory, even if it cannot calculate their real values .
It's getting more and more abstract, but it's only going to get worse. You might think that a tower of powers whose exponent length is (moreover, in a previous version of this post I made exactly that mistake), but it's just . In other words, imagine that you have the ability to calculate the exact value of a power tower of triples, which consists of elements, and then you take this value and create a new tower with so many in it ... that gives .
Repeat this process with each successive number ( note starting from the right) until you do this once, and then finally you get . This is a number that is simply incredibly large, but at least the steps to get it seem to be clear if everything is done very slowly. We can no longer understand numbers or imagine the procedure by which they are obtained, but at least we can understand the basic algorithm, only in a sufficiently long time.
Now let's prepare the mind to actually blow it up.
Graham's (Graham's) number
Ronald Graham
This is how you get Graham's number, which ranks in the Guinness Book of World Records as the largest number ever used in a mathematical proof. It is absolutely impossible to imagine how big it is, and it is just as difficult to explain exactly what it is. Basically, Graham's number comes into play when dealing with hypercubes, which are theoretical geometric shapes with more than three dimensions. The mathematician Ronald Graham (see photo) wanted to find out what was the smallest number of dimensions that would keep certain properties of a hypercube stable. (Sorry for this vague explanation, but I'm sure we all need at least two math degrees to make it more accurate.)
In any case, the Graham number is an upper estimate of this minimum number of dimensions. So how big is this upper bound? Let's get back to a number so large that we can understand the algorithm for obtaining it rather vaguely. Now, instead of just jumping up one more level to , we'll count the number that has arrows between the first and last triples. Now we are far beyond even the slightest understanding of what this number is or even of what needs to be done to calculate it.
Now repeat this process times ( note at each next step, we write the number of arrows equal to the number obtained at the previous step).
This, ladies and gentlemen, is Graham's number, which is about an order of magnitude above the point of human understanding. It is a number that is so much larger than any number you can imagine - it is much larger than any infinity you could ever hope to imagine - it simply defies even the most abstract description.
But here's the weird thing. Since Graham's number is basically just triplets multiplied together, we know some of its properties without actually calculating it. We can't represent Graham's number in any notation we're familiar with, even if we used the entire universe to write it down, but I can give 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's worth remembering that this number is only an upper bound in Graham's original problem. It is possible that the actual number of measurements required to fulfill the desired property is much, much less. In fact, since the 1980s, it has been believed by most experts in the field that there are actually only six dimensions - a number so small that we can understand it on an intuitive level. The lower bound has since been increased to , but there is still a very good chance that the solution to Graham's problem does not lie near a number as large as Graham's.
To infinity
So there are numbers bigger than Graham's number? There are, of course, for starters there is the Graham number. As for the significant number... well, there are some fiendishly difficult areas of mathematics (in particular, the area known as combinatorics) and computer science, in which there are numbers even larger than the Graham number. But we have almost reached the limit of what I can hope can ever reasonably explain. For those who are reckless enough to go even further, additional reading is offered at your own risk.
Well, now an amazing quote that is attributed to Douglas Ray ( note To be honest, it sounds pretty funny:
“I see clumps of vague numbers lurking out there in the dark, behind the little spot of light that the mind candle gives. They whisper to each other; talking about who knows what. Perhaps they do not like us very much for capturing their little brothers with our minds. Or maybe they just lead an unambiguous numerical way of life, out there, beyond our understanding.''