Many are interested in questions about how large numbers are called and what number is the largest in the world. These interesting questions will be dealt with in this article.
Story
The southern and eastern Slavic peoples used alphabetic numbering to write numbers, and only those letters that are in the Greek alphabet. Above the letter, which denoted the number, they put a special “titlo” icon. The numerical values of the letters increased in the same order in which the letters followed in the Greek alphabet (in the Slavic alphabet, the order of the letters was slightly different). In Russia, Slavic numbering was preserved until the end of the 17th century, and under Peter I they switched to “Arabic numbering”, which we still use today.
The names of the numbers also changed. So, until the 15th century, the number “twenty” was designated as “two ten” (two tens), and then it was reduced for faster pronunciation. The number 40 until the 15th century was called “fourty”, then it was replaced by the word “forty”, which originally denoted a bag containing 40 squirrel or sable skins. The name "million" appeared in Italy in 1500. It was formed by adding an augmentative suffix to the number "mille" (thousand). Later, this name came to Russian.
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: septillion (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."
Ways to build names of large numbers
There are 2 main ways to name large numbers:
- American system, which is used in the USA, Russia, France, Canada, Italy, Turkey, Greece, Brazil. The names of large numbers are built quite simply: at the beginning there is a Latin ordinal number, and the suffix “-million” is added to it at the end. The exception is the number "million", which is the name of the number one thousand (mille) and the magnifying suffix "-million". The number of zeros in a number that is written in the American system can be found by the formula: 3x + 3, where x is a Latin ordinal number
- English system most common in the world, it is used in Germany, Spain, Hungary, Poland, Czech Republic, Denmark, Sweden, Finland, Portugal. The names of numbers according to this system are built as follows: the suffix “-million” is added to the Latin numeral, the next number (1000 times larger) is the same Latin numeral, but the suffix “-billion” is added. The number of zeros in a number that is written in the English system and ends with the suffix “-million” can be found by the formula: 6x + 3, where x is a Latin ordinal number. The number of zeros in numbers ending in the suffix “-billion” can be found by the formula: 6x + 6, where x is a Latin ordinal number.
From the English system, only the word billion passed into the Russian language, which is still more correct to call it the way the Americans call it - billion (since the American system for naming numbers is used in Russian).
In addition to numbers that are written in the American or English system using Latin prefixes, non-systemic numbers are known that have their own names without Latin prefixes.
Proper names for large numbers
| Number | Latin numeral | Name | Practical value | |
| 10 1 | 10 | ten | Number of fingers on 2 hands | |
| 10 2 | 100 | hundred | Approximately half the number of all states on Earth | |
| 10 3 | 1000 | one thousand | Approximate number of days in 3 years | |
| 10 6 | 1000 000 | unus (I) | million | 5 times more than the number of drops in a 10-litre. bucket of water |
| 10 9 | 1000 000 000 | duo(II) | billion (billion) | Approximate population of India |
| 10 12 | 1000 000 000 000 | tres(III) | trillion | |
| 10 15 | 1000 000 000 000 000 | quattor(IV) | quadrillion | 1/30 of the length of a parsec in meters |
| 10 18 | quinque (V) | quintillion | 1/18 of the number of grains from the legendary award to the inventor of chess | |
| 10 21 | sex (VI) | sextillion | 1/6 of the mass of the planet Earth in tons | |
| 10 24 | septem(VII) | septillion | Number of molecules in 37.2 liters of air | |
| 10 27 | octo(VIII) | octillion | Half the mass of Jupiter in kilograms | |
| 10 30 | novem(IX) | quintillion | 1/5 of all microorganisms on the planet | |
| 10 33 | decem(X) | decillion | Half the mass of the Sun in grams | |
- Vigintillion (from lat. viginti - twenty) - 10 63
- Centillion (from Latin centum - one hundred) - 10 303
- Milleillion (from Latin mille - thousand) - 10 3003
For numbers greater than a thousand, the Romans did not have their own names (all the names of numbers below were composite).
Compound names for large numbers
In addition to their own names, for numbers greater than 10 33 you can get compound names by combining prefixes.
Compound names for large numbers
| Number | Latin numeral | Name | Practical value |
| 10 36 | undecim (XI) | andecillion | |
| 10 39 | duodecim(XII) | duodecillion | |
| 10 42 | tredecim(XIII) | tredecillion | 1/100 of the number of air molecules on Earth |
| 10 45 | quattuordecim (XIV) | quattordecillion | |
| 10 48 | quindecim (XV) | quindecillion | |
| 10 51 | sedecim (XVI) | sexdecillion | |
| 10 54 | septendecim (XVII) | septemdecillion | |
| 10 57 | octodecillion | So many elementary particles in the sun | |
| 10 60 | novemdecillion | ||
| 10 63 | viginti (XX) | vigintillion | |
| 10 66 | unus et viginti (XXI) | anvigintillion | |
| 10 69 | duo et viginti (XXII) | duovigintillion | |
| 10 72 | tres et viginti (XXIII) | trevigintillion | |
| 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 | triginta (XXX) | trigintillion | |
| 10 96 | antirigintillion |
- 10 123 - quadragintillion
- 10 153 - quinquagintillion
- 10 183 - sexagintillion
- 10 213 - septuagintillion
- 10 243 - octogintillion
- 10 273 - nonagintillion
- 10 303 - centillion
Further names can be obtained by direct or reverse order of Latin numerals (it is not known how to correctly):
- 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
The second spelling is more in line with the construction of numerals in Latin and avoids ambiguities (for example, in the number trecentillion, which in the first spelling is both 10903 and 10312).
- 10 603 - decentillion
- 10 903 - trecentillion
- 10 1203 - quadringentillion
- 10 1503 - quingentillion
- 10 1803 - sescentillion
- 10 2103 - septingentillion
- 10 2403 - octingentillion
- 10 2703 - nongentillion
- 10 3003 - million
- 10 6003 - duomillion
- 10 9003 - tremillion
- 10 15003 - quinquemillion
- 10 308760 -ion
- 10 3000003 - miamimiliaillion
- 10 6000003 - duomyamimiliaillion
myriad– 10,000. The name is obsolete and practically never used. However, the word “myriad” is widely used, which means not a certain number, but an uncountable, uncountable set of something.
googol ( English . googol) — 10 100 . The American mathematician Edward Kasner first wrote about this number in 1938 in the journal Scripta Mathematica in the article “New Names in Mathematics”. According to him, his 9-year-old nephew Milton Sirotta suggested calling the number this way. This number became public knowledge thanks to the Google search engine, named after him.
Asankheyya(from Chinese asentzi - innumerable) - 10 1 4 0. This number is found in the famous Buddhist treatise Jaina Sutra (100 BC). It is believed that this number is equal to the number of cosmic cycles required to gain nirvana.
Googolplex ( English . Googolplex) — 10^10^100. This number was also invented by Edward Kasner and his nephew, it means one with a googol of zeros.
Skewes number (Skewes' number Sk 1) means e to the power of e to the power of e to the power of 79, i.e. e^e^e^79. This number was proposed by Skewes in 1933 (Skewes. J. London Math. Soc. 8, 277-283, 1933.) in proving the Riemann conjecture concerning prime numbers. Later, Riele (te Riele, H. J. J. "On the Sign of the Difference P(x)-Li(x"). Math. Comput. 48, 323-328, 1987) reduced Skuse's number to e^e^27/4, which is approximately equal to 8.185 10^370. However, this number is not an integer, so it is not included in the table of large numbers.
Second Skewes Number (Sk2) equals 10^10^10^10^3, which is 10^10^10^1000. This number was introduced by J. Skuse in the same article to denote the number up to which the Riemann hypothesis is valid.
For super-large numbers, it is inconvenient to use powers, so there are several ways to write numbers - the notations of Knuth, Conway, Steinhouse, etc.
Hugo Steinhaus suggested writing large numbers inside geometric shapes (triangle, square and circle).
The mathematician Leo Moser finalized Steinhaus's notation, suggesting that after the squares, draw not circles, but pentagons, then hexagons, and so on. Moser also proposed a formal notation for these polygons, so that the numbers could be written without drawing complex patterns.
Steinhouse came up with two new super-large numbers: Mega and Megiston. In Moser notation, they are written as follows: Mega – 2, Megiston– 10. Leo Moser suggested also calling a polygon with the number of sides equal to mega – megagon, and also suggested the number "2 in Megagon" - 2. The last number is known as Moser's number or just like Moser.
There are numbers bigger than Moser. The largest number that has been used in a mathematical proof is number Graham(Graham's number). It was first used in 1977 in the proof of one estimate in the Ramsey theory. This number is associated with bichromatic hypercubes and cannot be expressed without a special 64-level system of special mathematical symbols introduced by Knuth in 1976. Donald 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
Graham suggested G-numbers:
The number G 63 is called the Graham number, often simply referred to as G. This number is the largest known number in the world and is listed in the Guinness Book of Records.
“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
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- one 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 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 famous Buddhist treatise Jaina Sutra, dating back to 100 BC, there is a number asankhiya(from 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 more than a googolplex number - Skewes number (Skewes" number) was suggested by Skewes in 1933 (Skewes. J. London Math. soc. 8, 277-283, 1933.) in proving the Riemann conjecture concerning primes. It means e to the extent e to the extent e to the power of 79, i.e. ee e 79 . Later, Riele (te Riele, H. J. J. "On the Sign of the Difference P(x)-Li(x)." Math. Comput. 48, 323-328, 1987) reduced Skuse's number to ee 27/4 , which is approximately equal to 8.185 10 370 . It is clear that since the value of the Skewes number depends on the number e, then it is not an integer, so we will not consider it, otherwise we would have to recall other non-natural numbers - the number pi, the number e, etc.
But it should be noted that there is a second Skewes number, which in mathematics is denoted as Sk2 , which is even larger than the first Skewes number (Sk1 ). Skuse's second number, was introduced by J. Skuse in the same article to denote a number for which the Riemann hypothesis is not valid. Sk2 is 1010 10103 , i.e. 1010 101000 .
As you understand, the more degrees there are, the more difficult it is to understand which of the numbers is greater. For example, looking at the Skewes numbers, without special calculations, it is almost impossible to understand which of these two numbers is larger. Thus, for superlarge numbers, it becomes inconvenient to use powers. Moreover, you can come up with such numbers (and they have already been invented) when the degrees of degrees simply do not fit on the page. Yes, what a page! They won't even fit into a book the size of the entire universe! In this case, the question arises how to write them down. The problem, as you understand, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who asked this problem came up with his own way of writing, which led to the existence of several, unrelated, ways to write numbers - these are the notations of Knuth, Conway, Steinhaus, etc.
Consider the notation of Hugo Stenhaus (H. Steinhaus. Mathematical Snapshots, 3rd edn. 1983), which is quite simple. Steinhouse suggested writing large numbers inside geometric shapes - a triangle, a square and a circle:
Steinhouse came up with two new super-large numbers. He named a 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 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 that:
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 the 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 number(Graham "s number), first used in 1977 in the proof of one estimate in Ramsey theory. It is associated with bichromatic hypercubes and cannot be expressed without a special 64-level system of special mathematical symbols introduced by Knuth in 1976.
Unfortunately, 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:
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 listed in the Guinness Book of Records. And, here, that the Graham number is greater than the Moser number.
P.S. In order to bring great benefit to all mankind and become famous for centuries, I decided to invent 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 bigger than Graham's number? There are, of course, for starters there is a 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 Graham's number. But we have almost reached the limit of what can be rationally and clearly explained.
It is impossible to answer this question correctly, since the number series has no upper limit. So, to any number, it is enough just to add one to get an even larger number. Although the numbers themselves are infinite, they do not have very many proper names, since most of them are content with names made up of smaller numbers. So, for example, the numbers and have their own names "one" and "one hundred", and the name of the number is already compound ("one hundred and one"). It is clear that in the final set of numbers that humanity has awarded with its own name, there must be some largest number. But what is it called and what is it equal to? Let's try to figure it out and at the same time find out how big numbers mathematicians came up with.
"Short" and "long" scale
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 (c. 1450 - c. 1500): in his treatise "The Science of Numbers" (Triparty en la science des nombres, 1484), he developed this idea, proposing to further use the Latin cardinal numbers (see table), adding them to the ending "-million". So, Shuke's "bimillion" turned into a billion, "trimillion" into a trillion, and a million to the fourth power became a "quadrillion".
In Schücke's system, a number that was between a million and a billion did not have its own name and was simply called "a thousand million", similarly it was called "a thousand billion", - "a thousand trillion", etc. It was not very convenient, and in 1549 the French writer and scientist Jacques Peletier du Mans (1517-1582) proposed to name such "intermediate" numbers using the same Latin prefixes, but the ending "-billion". So, it began to be called "billion", - "billiard", - "trilliard", etc.
The Shuquet-Peletier system gradually became popular and was used throughout Europe. However, in the 17th century, an unexpected problem arose. It turned out that for some reason some scientists began to get confused and call the number not “a billion” or “thousand millions”, but “a billion”. Soon this mistake quickly spread, and a paradoxical situation arose - "billion" became at the same time a synonym for "billion" () and "million million" ().
This confusion continued for a long time and led to the fact that in the USA they created their own system for naming large numbers. According to the American system, the names of numbers are built in the same way as in the Schuke system - the Latin prefix and the ending "million". However, these numbers are different. If in the Schuecke system names with the ending "million" received numbers that were powers of a million, then in the American system the ending "-million" received the powers of a thousand. That is, a thousand million () became known as a "billion", () - "trillion", () - "quadrillion", etc.
The old system of naming large numbers continued to be used in conservative Great Britain and began to be called "British" all over the world, despite the fact that it was invented by the French Shuquet and Peletier. However, in the 1970s, the UK officially switched to the "American system", which led to the fact that it became somehow strange to call one system American and another British. As a result, the American system is now commonly referred to as the "short scale" and the British or Chuquet-Peletier system as the "long scale".
In order not to get confused, let's sum up the intermediate result:
| Number name | Value on the "short scale" | Value on the "long scale" |
| Million | ||
| Billion | ||
| Billion | ||
| billiard | - | |
| Trillion | ||
| trillion | - | |
| quadrillion | ||
| quadrillion | - | |
| Quintillion | ||
| quintillion | - | |
| Sextillion | ||
| Sextillion | - | |
| Septillion | ||
| Septilliard | - | |
| Octillion | ||
| Octilliard | - | |
| Quintillion | ||
| Nonilliard | - | |
| Decillion | ||
| Decilliard | - | |
| Vigintillion | ||
| viginbillion | - | |
| Centillion | ||
| Centbillion | - | |
| Milleillion | ||
| Milliilliard | - |
The short naming scale is currently used in the US, UK, Canada, Ireland, Australia, Brazil and Puerto Rico. Russia, Denmark, Turkey, and Bulgaria also use the short scale, except that the number is called "billion" rather than "billion". The long scale continues to be used today in most other countries.
It is curious that in our country the final transition to the short scale took place only in the second half of the 20th century. So, for example, even Yakov Isidorovich Perelman (1882–1942) in his “Entertaining Arithmetic” mentions the parallel existence of two scales in the USSR. The short scale, according to Perelman, was used in everyday life and financial calculations, and the long one was used in scientific books on astronomy and physics. However, now it is wrong to use a long scale in Russia, although the numbers there are large.
But back to finding the largest number. After a decillion, the names of numbers are obtained by combining prefixes. This is how numbers such as undecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion, novemdecillion, etc. are obtained. However, these names are no longer of interest to us, since we agreed to find the largest number with its own non-composite name.
If we turn to Latin grammar, we will find that the Romans had only three non-compound names for numbers more than ten: viginti - "twenty", centum - "one hundred" and mille - "thousand". For numbers greater than "thousand", the Romans did not have their own names. For example, a million () The Romans called it “decies centena milia”, that is, “ten times a hundred thousand”. According to Schuecke's rule, these three remaining Latin numerals give us such names for numbers as "vigintillion", "centillion" and "milleillion".
So, we found out that on the "short scale" the maximum number that has its own name and is not a composite of smaller numbers is "million" (). If a “long scale” of naming numbers were adopted in Russia, then the largest number with its own name would be “millionillion” ().
However, there are names for even larger numbers.
Numbers outside the system
Some numbers have their own name, without any connection with the naming system using Latin prefixes. And there are many such numbers. You can, for example, remember the number e, the number "pi", a dozen, the number of the beast, etc. However, since we are now interested in large numbers, we will consider only those numbers with their own non-compound name that are more than a million.
Until the 17th century, Russia used its own system for naming numbers. Tens of thousands were called "darks", hundreds of thousands were called "legions", millions were called "leodras", tens of millions were called "ravens", and hundreds of millions were called "decks". This account up to hundreds of millions was called the “small account”, and in some manuscripts the authors also considered the “great account”, in which the same names were used for large numbers, but with a different meaning. So, "darkness" meant no longer ten thousand, but a thousand thousand () , "legion" - the darkness of those () ; "leodr" - legion of legions () , "raven" - leodr leodrov (). “Deck” in the great Slavic account for some reason was not called “raven of ravens” () , but only ten "ravens", that is (see table).
| Number name | Meaning in "small count" | Meaning in the "great account" | Designation |
| Dark | |||
| Legion | |||
| Leodr | |||
| Raven (Raven) | |||
| Deck | |||
| Darkness of topics |  |
The number also has its own name and was invented by a nine-year-old boy. And it was like that. In 1938, the American mathematician Edward Kasner (Edward Kasner, 1878–1955) was walking in the park with his two nephews and discussing large numbers with them. During the conversation, we talked about a number with one hundred zeros, which did not have its own name. One of his nephews, nine-year-old Milton Sirott, suggested calling this number "googol". In 1940, Edward Kasner, together with James Newman, wrote the popular science book "Mathematics and Imagination", where he told mathematics lovers about the number of googols. Google became even more widely known in the late 1990s, thanks to the Google search engine named after it.
The name for an even larger number than the googol arose in 1950 thanks to the father of computer science, Claude Shannon (Claude Elwood Shannon, 1916–2001). In his article "Programming a Computer to Play Chess", he tried to estimate the number of possible variants of a chess game. According to it, each game lasts an average of moves, and on each move the player makes an average choice of options, which corresponds to (approximately equal to) the game options. This work became widely known, and this number became known as the "Shannon number".
In the well-known Buddhist treatise Jaina Sutra, dating back to 100 BC, the number "asankheya" is found equal to . It is believed that this number is equal to the number of cosmic cycles required to gain nirvana.
Nine-year-old Milton Sirotta entered the history of mathematics not only by inventing the googol number, but also by suggesting another number at the same time - “googolplex”, which is equal to the power of “googol”, that is, one with the googol of zeros.
Two more numbers larger than the googolplex were proposed by the South African mathematician Stanley Skewes (1899–1988) when proving the Riemann hypothesis. The first number, which later came to be called "Skews's first number", is equal to the power to the power to the power of , that is, . However, the "second Skewes number" is even larger and amounts to .
Obviously, the more degrees in the number of degrees, the more difficult it is to write down numbers and understand their meaning when reading. Moreover, it is possible to come up with such numbers (and they, by the way, have already been invented), when the degrees of degrees simply do not fit on the page. Yes, what a page! They won't even fit in a book the size of the entire universe! In this case, the question arises how to write down such numbers. The problem is, fortunately, resolvable, and mathematicians have developed several principles for writing such numbers. True, 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:
"in a triangle" means "",
"in a square" means "in triangles",
"in a circle" means "in squares".
Explaining this way of writing, Steinhaus comes up with the number "mega", equal in a circle and shows that it is equal in a "square" or in triangles. To calculate it, you need to raise it to a power, raise the resulting number to a power, then raise the resulting number to the power of the resulting number, and so on to raise the power of times. For example, the calculator in MS Windows cannot calculate due to overflow even in two triangles. Approximately this huge number is .
Having determined the number "mega", Steinhaus invites readers to independently evaluate another number - "medzon", equal in a circle. In another edition of the book, Steinhaus, instead of the medzone, proposes to estimate an even larger number - “megiston”, equal in a circle. Following Steinhaus, I will also recommend that readers take a break from this text for a while and try to write these numbers themselves using ordinary powers in order to feel their gigantic magnitude.
However, there are names for large numbers. So, the Canadian mathematician Leo Moser (Leo Moser, 1921–1970) finalized the Steinhaus notation, which was limited by the fact that if it were necessary to write down numbers much larger than a megiston, then difficulties and inconveniences would arise, since 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:
"triangle" = = ;
"in a square" = = "in triangles" =;
"in the pentagon" = = "in the squares" = ;
"in -gon" = = "in -gons" = .
Thus, according to Moser's notation, the Steinhausian "mega" is written as , "medzon" as , and "megiston" as . In addition, Leo Moser proposed to call a polygon with the number of sides equal to mega - "megagon". And offered a number « in a megagon", that is. This number became known as the Moser number, or simply as "moser".
But even "moser" is not the largest number. So, the largest number ever used in 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 -dimensional bichromatic hypercubes. Graham's number gained fame only after the story about it in Martin Gardner's 1989 book "From Penrose Mosaics to Secure Ciphers".
To explain how large the Graham number is, one has to explain another way of writing large numbers, introduced by Donald Knuth in 1976. American professor Donald Knuth came up with the concept of superdegree, which he proposed to write with arrows pointing up.
The usual arithmetic operations - addition, multiplication, and exponentiation - can naturally be extended into a sequence of hyperoperators as follows.
Multiplication of natural numbers can be defined through the repeated operation of addition (“add copies of a number”):
For example,
Raising a number to a power can be defined as a repeated multiplication operation ("multiply copies of a number"), and in Knuth's notation this entry looks like a single arrow pointing up:
For example,
Such a single up arrow was used as a degree icon in the Algol programming language.
For example,
Here and below, the evaluation of the expression always goes from right to left, and Knuth's arrow operators (as well as the exponentiation operation) by definition have right associativity (right-to-left ordering). According to this definition,
This already leads to quite large numbers, but the notation does not end there. The triple arrow operator is used to write repeated exponentiation of the double arrow operator (also known as "pentation"):
Then the "quadruple arrow" operator:
Etc. General rule operator "-I arrow", according to right associativity, continues to the right into a sequential series of operators « arrow". Symbolically, this can be written as follows,
For example:
The notation form is usually used for writing with arrows.
Some numbers are so large that even writing with Knuth's arrows becomes too cumbersome; in this case, the use of the -arrow operator is preferable (and also for a description with a variable number of arrows), or equivalent, to hyperoperators. But some numbers are so huge that even such a notation is not enough. For example, the Graham number.
When using Knuth's Arrow notation, the Graham number can be written as
Where the number of arrows in each layer, starting from the top, is determined by the number in the next layer, i.e. , where , where the superscript of the arrow indicates the total number of arrows. In other words, it is calculated in steps: in the first step we calculate with four arrows between threes, in the second - with arrows between threes, in the third - with arrows between threes, and so on; at the end we calculate from the arrows between the triplets.
This can be written as , where , where the superscript y denotes function iterations.
If other numbers with “names” can be matched with the corresponding number of objects (for example, the number of stars in the visible part of the Universe is estimated in sextillions - , and the number of atoms that make up the globe has the order of dodecallions), then the googol is already “virtual”, not to mention about the Graham number. The scale of the first term alone is so large that it is almost impossible to comprehend it, although the notation above is relatively easy to understand. Although - this is just the number of towers in this formula for , this number is already much larger than the number of Planck volumes (the smallest possible physical volume) that are contained in the observable universe (approximately ). After the first member, another member of the rapidly growing sequence awaits us.
Countless different numbers surround us every day. Surely many people at least once wondered what number is considered the largest. You can simply tell a child that this is a million, but adults are well aware that other numbers follow a million. For example, one has only to add one to the number every time, and it will become more and more - this happens ad infinitum. But if you disassemble the numbers that have names, you can find out what the largest number in the world is called.
The appearance of the names of numbers: what methods are used?
To date, there are 2 systems according to which names are given to numbers - American and English. The first is quite simple, and the second is the most common around the world. The American one allows you to give names to large numbers like this: first, the ordinal number in Latin is indicated, and then the suffix “million” is added (the exception here is a million, meaning a thousand). This system is used by Americans, French, Canadians, and it is also used in our country.
English is widely used in England and Spain. According to it, the numbers are named like this: the numeral in Latin is “plus” with the suffix “million”, and the next (a thousand times greater) number is “plus” “billion”. For example, a trillion comes first, followed by a trillion, a quadrillion follows a quadrillion, and so on.
So, the same number in different systems can mean different things, for example, an American billion in the English system is called a billion.
Off-system numbers
In addition to numbers that are written according to known systems (given above), there are also off-system ones. They have their own names, which do not include Latin prefixes.
You can start their consideration with a number called a myriad. It is defined as one hundred hundreds (10000). But for its intended purpose, this word is not used, but is used as an indication of an innumerable multitude. Even Dahl's dictionary will kindly provide a definition of such a number.
Next after the myriad is the googol, denoting 10 to the power of 100. For the first time this name was used in 1938 by an American mathematician E. Kasner, who noted that his nephew came up with this name.
Google (search engine) got its name in honor of Google. Then 1 with a googol of zeros (1010100) is a googolplex - Kasner also came up with such a name.
Even larger than the googolplex is the Skewes number (e to the power of e to the power of e79), proposed by Skuse when proving the Riemann conjecture on prime numbers (1933). There is another Skewes number, but it is used when the Rimmann hypothesis is unfair. It is rather difficult to say which of them is greater, especially when it comes to large degrees. However, this number, despite its "enormity", cannot be considered the most-most of all those that have their own names.
And the leader among the largest numbers in the world is the Graham number (G64). It was he who was used for the first time to conduct proofs in the field of mathematical science (1977).
When it comes to such a number, you need to know that you cannot do without a special 64-level system created by Knuth - the reason for this is the connection of the number G with bichromatic hypercubes. Knuth invented the superdegree, and in order to make it convenient to record it, he suggested using the up arrows. So we learned what the largest number in the world is called. It is worth noting that this number G got into the pages of the famous Book of Records.
Sometimes people who are not related to mathematics wonder: what is the largest number? On the one hand, the answer is obvious - infinity. The bores will even clarify that "plus infinity" or "+∞" in the notation of mathematicians. But this answer will not convince the most corrosive, especially since this is not a natural number, but a mathematical abstraction. But having well understood the issue, they can open up an interesting problem.
Indeed, there is no size limit in this case, but there is a limit to human imagination. Each number has a name: ten, one hundred, billion, sextillion, and so on. But where does the fantasy of people end?
Not to be confused with a Google Corporation trademark, although they share a common origin. This number is written as 10100, that is, one followed by a tail of one hundred zeros. It is difficult to imagine it, but it was actively used in mathematics.
It's funny what his child came up with - the nephew of the mathematician Edward Kasner. In 1938, my uncle entertained younger relatives with arguments about very large numbers. To the indignation of the child, it turned out that such a wonderful number had no name, and he gave his version. Later, my uncle inserted it into one of his books, and the term stuck.
Theoretically, a googol is a natural number, because it can be used for counting. That's just hardly anyone has the patience to count to its end. Therefore, only theoretically.
As for the name of the company Google, then a common mistake crept in. The first investor and one of the co-founders, when writing the check, was in a hurry, and missed the letter "O", but in order to cash it, the company had to be registered under this spelling.
Googolplex
This number is a derivative of the googol, but significantly larger than it. The prefix "plex" means raising ten to the power of the base number, so guloplex is 10 to the power of 10 to the power of 100, or 101000.
The resulting number exceeds the number of particles in the observable universe, which is estimated at about 1080 degrees. But this did not stop scientists from increasing the number simply by adding the prefix "plex" to it: googolplexplex, googolplexplexplex, and so on. And for especially perverted mathematicians, they invented an option to increase without endless repetition of the prefix "plex" - they simply put Greek numbers in front of it: tetra (four), penta (five) and so on, up to deca (ten). The last option sounds like a googoldekaplex and means a tenfold cumulative repetition of the procedure for raising the number 10 to the power of its base. The main thing is not to imagine the result. You still won’t be able to realize it, but it’s easy to get a trauma to the psyche.
48th Mersen number
Main characters: Cooper, his computer and a new prime number
Relatively recently, about a year ago, it was possible to discover the next, 48th Mersen number. It is currently the largest prime number in the world. Recall that prime numbers are those that are only divisible without a remainder by 1 and themselves. The simplest examples are 3, 5, 7, 11, 13, 17 and so on. The problem is that the further into the wilds, the less often such numbers occur. But the more valuable is the discovery of each next one. For example, a new prime number consists of 17,425,170 digits if it is represented in the form of a decimal number system familiar to us. The previous one had about 12 million characters.
It was discovered by the American mathematician Curtis Cooper, who for the third time delighted the mathematical community with such a record. Just to check his result and prove that this number is really prime, it took 39 days of his personal computer.
This is how Graham's number is written in Knuth's arrow notation. It is difficult to say how to decipher this without having a completed higher education in theoretical mathematics. It is also impossible to write it down in the decimal form we are accustomed to: the observable Universe is simply not able to contain it. Fencing degree for degree, as in the case of googolplexes, is also not an option.
Good formula, but incomprehensible
So why do we need this seemingly useless number? Firstly, for the curious, it was placed in the Guinness Book of Records, and this is already a lot. Secondly, it was used to solve a problem that is part of the Ramsey problem, which is also incomprehensible, but sounds serious. Thirdly, this number is recognized as the largest ever used in mathematics, and not in comic proofs or intellectual games, but for solving a very specific mathematical problem.
Attention! The following information is dangerous for your mental health! By reading it, you accept responsibility for all the consequences!
For those who want to test their mind and meditate on the Graham number, we can try to explain it (but only try).
Imagine 33. It's pretty easy - you get 3*3*3=27. What if we now raise three to this number? It turns out 3 3 to the 3rd power, or 3 27. In decimal notation, this is equal to 7,625,597,484,987. A lot, but for now it can be understood.
In Knuth's arrow notation, this number can be displayed somewhat more simply - 33. But if you add only one arrow, it will turn out to be more difficult: 33, which means 33 to the power of 33 or in power notation. If expanded to decimal notation, we get 7,625,597,484,987 7,625,597,484,987 . Are you still able to follow the thought?
Next step: 33= 33 33 . That is, you need to calculate this wild number from the previous action and raise it to the same power.
And 33 is just the first of the 64 members of Graham's number. To get the second one, you need to calculate the result of this furious formula, and substitute the appropriate number of arrows into the 3(...)3 scheme. And so on, 63 more times.
I wonder if anyone besides him and a dozen other supermathematicians will be able to get at least to the middle of the sequence and not go crazy at the same time?
Did you understand something? We are not. But what a thrill!
Why are the largest numbers needed? It is difficult for the layman to understand and realize this. But a few specialists with their help are able to present new technological toys to the inhabitants: phones, computers, tablets. The townsfolk are also not able to understand how they work, but they are happy to use them for their own entertainment. And everyone is happy: the townsfolk get their toys, "supernerds" - the opportunity to play their mind games for a long time.
