Countless different numbers surround us every day. Surely many people wondered at least once 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, it is only necessary to add one to the number each time, and it will become more and more - this happens ad infinitum. But if you take apart the numbers that have names, you can find out what the largest number in the world is called.
The emergence of the names of numbers: what methods are used?
Today there are 2 systems according to which numbers are given names - American and English. The first is fairly straightforward, while the second is the most common around the world. American allows you to give names to large numbers like this: first, the ordinal number in Latin is indicated, and then the suffix "illion" is added (the exception here is a million, meaning a thousand). This system is used by the 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 as follows: the numeral in Latin is "plus" with the suffix "illion", and the next (a thousand times larger) number is "plus" "illiard". For example, first comes a trillion, followed by a trillion, followed by a quadrillion, and so on.
So, the same number in different systems can mean different things, for example, the American billion in the English system is called a billion.
Off-system numbers
In addition to numbers that are written according to known systems (above), there are also non-systemic ones. They have their own names, which do not include Latin prefixes.
You can start considering them 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 the innumerable. Even Dahl's dictionary will kindly provide a definition of such a number.
The next after the myriad is googol, denoting 10 to the power of 100. This name was first used in 1938 - by a mathematician from America E. Kasner, who noted that this name was invented by his nephew.
Google (search engine) got its name in honor of googol. Then 1-tsa with a googol of zeros (1010100) is a googolplex - Kasner also invented this name.
Even larger in comparison with the googolplex is the Skuse number (e to the e to the e79 power), proposed by Skuse in the proof of the Rimmann conjecture on primes (1933). There is another Skuse number, but it is applied when the Rimmann hypothesis is not valid. It is rather difficult to say which of them is more, 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 carry out 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 Knut - the reason for this is the connection of the number G with bichromatic hypercubes. The whip invented a superdegree, and in order to make it convenient to make her notes, he suggested using the up arrows. So we learned the name of the largest number in the world. It is worth noting that this G number got on the pages of the famous Book of Records.
June 17th, 2015
“I see clusters of vague numbers that are hiding there, in the darkness, behind a small spot of light that the candle of the mind gives. They whisper to each other; conspiring who knows what. Perhaps they don't like us very much for capturing their little brothers with our minds. Or, perhaps, they simply lead an unambiguous numerical way of life, there, beyond our understanding ''.
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. You just need to add one to the largest number, as it will no longer be the largest. This procedure can be continued indefinitely.
And if you ask the question: what is the largest number that exists, and what is its own name?
Now we will all find out ...
There are two systems for naming numbers - American and English.
The American system is pretty simple. All the names of large numbers are constructed as follows: at the beginning there is a Latin ordinal number, and at the end the suffix-million is added to it. An exception is the name "million" which is the name of the number one thousand (lat. mille) and the increasing suffix-million (see table). This is how the numbers are obtained - trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion and decillion. The American system is used in the USA, Canada, France and Russia. You can find out the number of zeros in a number written in the American system using the simple formula 3 x + 3 (where x is a Latin numeral).
The English naming system is the most common in the world. It is used, for example, in Great Britain and Spain, as well as in most of the former English and Spanish colonies. The names of numbers in this system are built like this: so: the suffix-million is added to the Latin numeral, the next number (1000 times larger) is built according to the principle - the same Latin numeral, but the suffix is -billion. That is, after a trillion in the English system, there is a trillion, and only then a quadrillion, followed by a quadrillion, etc. Thus, a quadrillion in the English and American systems are completely different numbers! You can find out the number of zeros in a number written in the English system and ending with the suffix-million by the formula 6 x + 3 (where x is a Latin numeral) and by the formula 6 x + 6 for numbers ending in -billion.
Only the number billion (10 9) passed from the English system to the Russian language, which would still be more correct to call it as the Americans call it - a billion, since it is the American system that has been adopted in our country. But who in our country does something according to the rules! ;-) By the way, sometimes the word trillion is also used in Russian (you can see for yourself by running a search in Google or Yandex) and it means, apparently, 1000 trillion, i.e. quadrillion.
In addition to numbers written using Latin prefixes according to the American or English system, so-called off-system numbers are also known, i.e. numbers that have their own names without any Latin prefixes. There are several such numbers, but I will talk about them in more detail a little later.
Let's go back to writing using Latin numerals. It would seem that they can write numbers to infinity, but this is not entirely true. Let me explain why. Let's see for a start how the numbers from 1 to 10 33 are called:
And so, now the question arises, what's next. What's behind the decillion? In principle, of course, it is possible, of course, by combining prefixes to generate such monsters as: andecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion and novemdecillion, but these will already be compound names, but we were interested in numbers. Therefore, according to this system, in addition to the above, you can still get only three proper names - vigintillion (from lat.viginti- twenty), centillion (from lat.centum- one hundred) and a million (from lat.mille- thousand). The Romans did not have more than a thousand of their own names for numbers (all numbers over a thousand were composite). For example, the Romans called a million (1,000,000)decies centena milia, that is, "ten hundred thousand". And now, in fact, the table:
Thus, according to a similar system, the numbers are greater than 10 3003 , which would have its own, non-compound name, it is impossible to get! But nevertheless, numbers more than a million million are known - these are the very off-system numbers. Let's finally tell you about them.
The smallest such number is a myriad (it is even in Dahl's dictionary), which means one hundred hundreds, that is, 10,000 does not mean a definite number at all, but an uncountable, uncountable set of something. It is believed that the word myriad came into European languages from ancient Egypt.
There are different opinions about the origin of this number. Some believe that it originated in Egypt, while others believe that it was born only in Ancient Greece. Be that as it may in reality, but the myriad gained fame thanks to the Greeks. Myriad was the name for 10,000, but there were no names for numbers over ten thousand. However, in the note "Psammit" (ie the calculus of sand), Archimedes showed how one can systematically construct and name arbitrarily large numbers. In particular, placing 10,000 (myriad) grains of sand in a poppy seed, he finds that in the Universe (a sphere with a diameter of a myriad of Earth's diameters) no more than 10 63
grains of sand. It is curious that modern calculations of the number of atoms in the visible Universe lead to the number 10 67
(just a myriad of times more). Archimedes suggested the following names for numbers:
1 myriad = 10 4.
1 d-myriad = myriad myriad = 10 8
.
1 three-myriad = di-myriad di-myriad = 10 16
.
1 tetra-myriad = three-myriad three-myriad = 10 32
.
etc.
Googol (from the English googol) is the number ten to the hundredth power, that is, one followed by one hundred zeros. Googol was first written about in 1938 in the article "New Names in Mathematics" in the January issue of Scripta Mathematica by the American mathematician Edward Kasner. According to him, his nine-year-old nephew Milton Sirotta suggested calling a large number "googol". This number became well-known thanks to the search engine named after him. Google... Note that "Google" is a trademark and googol is a number.
Edward Kasner.
On the Internet, you can often find it mentioned that - but it is not ...
In the famous Buddhist treatise Jaina Sutra dating back to 100 BC, the number asankheya (from Ch. asenci- uncountable) equal to 10 140. It is believed that this number is equal to the number of cosmic cycles required to attain nirvana.
Googolplex (eng. googolplex) is a number also invented by Kasner with his nephew and means one with a googol of zeros, that is, 10 10100 ... This is how Kasner himself describes this "discovery":
Words of wisdom are spoken by children at least as often as by scientists. The name "googol" was invented by a child (Dr. Kasner "s nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex." A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out.
Mathematics and the Imagination(1940) by Kasner and James R. Newman.
An even larger number than the googolplex, the Skewes "number, was proposed by Skewes in 1933 (Skewes. J. London Math. Soc. 8, 277-283, 1933.) in proving the Riemann conjecture concerning prime numbers. It means e to the extent e to the extent e to the 79th power, that is, ee e 79 ... Later, Riele (te Riele, H. J. J. "On the Sign of the Difference NS(x) -Li (x). " Math. Comput. 48, 323-328, 1987) reduced Skuse's number to ee 27/4 , which is approximately equal to 8.185 · 10 370. It is clear that since the value of Skuse's number depends on the number e, then it is not an integer, therefore we will not consider it, otherwise we would have to recall other non-natural numbers - pi, e, etc.
But it should be noted that there is a second Skuse number, which in mathematics is denoted as Sk2, which is even greater than the first Skuse number (Sk1). Second Skewes number, was introduced by J. Skuse in the same article to denote a number for which the Riemann hypothesis is not valid. Sk2 is 1010 10103 , that is, 1010 101000 .
As you understand, the more there are in the number of degrees, the more difficult it is to understand which of the numbers is greater. For example, looking at the Skuse numbers, without special calculations, it is almost impossible to understand which of these two numbers is greater. Thus, it becomes inconvenient to use powers for very large numbers. Moreover, you can think of such numbers (and they have already been invented) when the degrees of degrees simply do not fit on the page. Yes, what a page! They will not fit, even in a book the size of the entire Universe! In this case, the question arises how to write them down. The problem, as you understand, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who asked this problem came up with his own way of writing, which led to the existence of several unrelated ways to write numbers - these are the notations of Knuth, Conway, Steinhouse, etc.
Consider the notation of Hugo Steinhaus (H. Steinhaus. Mathematical Snapshots, 3rd edn. 1983), which is pretty simple. Stein House proposed to write large numbers inside geometric shapes - a triangle, a square and a circle:
Steinhaus came up with two new super-large numbers. He named 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 required to write numbers much larger than a megiston, difficulties and inconveniences arose, since many circles had to be drawn inside one another. Moser suggested drawing not circles, but pentagons after the squares, then hexagons, and so on. He also proposed a formal notation for these polygons so that numbers could be written down without drawing complex drawings. Moser's notation looks like this:
Thus, according to Moser's notation, the Steinhouse mega is written as 2, and the megiston as 10. In addition, Leo Moser suggested calling a polygon with the number of sides equal to a mega - megaagon. And he proposed the number "2 in Megagon", that is 2. This number became known as the Moser's number (Moser's number) or simply as moser.
But Moser is not the largest number either. The largest number ever used in mathematical proof is a limiting quantity known as the Graham "s number, first used in 1977 to prove 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 Knuth's notation cannot be translated into the Moser system. Therefore, we will have to explain this system as well. In principle, there is nothing complicated about it either. Donald Knuth (yes, yes, this is the same Knuth who wrote "The Art of Programming" and created the TeX editor) invented the concept of superdegree, which he proposed to write down with arrows pointing up:
In general, it looks like this:
I think everything is clear, so let's go back to Graham's number. Graham proposed the so-called G-numbers:
- 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 overdegree arrows is equal to G62.
The G63 number 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 included in the Guinness Book of Records. And here
The question "What is the largest number in the world?" Is, to say the least, incorrect. There are both different number systems - decimal, binary and hexadecimal, and various categories of numbers - semi-simple and simple, the latter being divided into legal and illegal. In addition, there are the Skewes "number", Steinhouse and other mathematicians who, either jokingly or seriously, invent and publish such exotics as "megiston" or "moser" to the public.
What is the largest decimal number in the world
Of the decimal system, most "non-mathematicians" are well aware of the million, billion and trillion. Moreover, if Russians associate a million with a dollar bribe that can be carried away in a suitcase, then where to shove a billion (not to mention a trillion) North American banknotes - the majority do not have enough imagination. However, in the theory of large numbers, there are concepts such as quadrillion (ten to the fifteenth power - 1015), sextillion (1021) and octillion (1027).
In the English decimal system, the most widely used decimal system in the world, the decimal is considered the maximum number - 1033.
In 1938, in connection with the development of applied mathematics and the expansion of the micro- and macrocosm, a professor at Columbia University (USA), Edward Kasner, published on the pages of the journal "Scripta Mathematica" the proposal of his nine-year-old nephew to use the decimal system of a large number of "googol" ("googol") - representing ten to the hundredth power (10100), which on paper is expressed as one with one hundred zeros. However, they did not stop there and after a few years proposed to introduce into circulation a new largest number in the world - "googolplex", which is ten, raised to the tenth power and once again raised to the hundredth power - (1010) 100, expressed by a unit to which a googol of zeros is assigned to the right. However, for the majority of even professional mathematicians, both "googol" and "googolplex" are of purely speculative interest, and they can hardly be applied to anything in everyday practice.
Exotic numbers
What is the largest number in the world among prime numbers - those that can only be divisible by themselves and by one. One of the first to fix the largest prime number, 2,147,483,647, was the great mathematician Leonard Euler. As of January 2016, this number is recognized as an expression calculated as 274 207 281 - 1.
Have you ever wondered how many zeros there are in one million? This is a pretty straightforward question. What about a billion or a trillion? One with nine zeros (1,000,000,000) - what is the name of the number?
A short list of numbers and their quantitative designation
- Ten (1 zero).
- One hundred (2 zeros).
- Thousand (3 zeros).
- Ten thousand (4 zeros).
- One hundred thousand (5 zeros).
- Million (6 zeros).
- Billion (9 zeros).
- Trillion (12 zeros).
- Quadrillion (15 zeros).
- Quintillon (18 zeros).
- Sextillion (21 zero).
- Septillon (24 zeros).
- Octalion (27 zeros).
- Nonalion (30 zeros).
- Decalion (33 zeros).
Grouping zeros
1,000,000,000 - what is the name of a number that has 9 zeros? This is a billion. For convenience, it is customary to group large numbers into three sets, separated from each other by a space or punctuation marks such as a comma or period.
This is done to make it easier to read and understand the quantitative value. For example, what is the name of the number 1,000,000,000? In this form, it is worthwhile to pretend a little, to count. And if you write 1,000,000,000, then the task is immediately visually easier, so you need to count not zeros, but triples of zeros.
Numbers with very many zeros
The most popular are Million and Billion (1,000,000,000). What is the name of a number with 100 zeros? This is the googol figure, also called Milton Sirotta. This is a wildly huge amount. Do you think this number is large? Then how about a googolplex, a one followed by a googol of zeros? This figure is so large that it is difficult to come up with a meaning for it. In fact, there is no need for such giants, except to count the number of atoms in an infinite universe.
Is 1 billion a lot?
There are two scales of measurement - short and long. Worldwide in the field of science and finance, 1 billion is 1,000 million. This is on a short scale. According to it, this is a number with 9 zeros.
There is also a long scale that is used in some European countries, including France, and was previously used in the UK (until 1971), where a billion was 1 million million, that is, one and 12 zeros. This gradation is also called the long-term scale. The short scale is now dominant in financial and scientific matters.
Some European languages such as Swedish, Danish, Portuguese, Spanish, Italian, Dutch, Norwegian, Polish, German use a billion (or a billion) names in this system. In Russian, a number with 9 zeros is also described for the short scale of a thousand million, and a trillion is a million million. This avoids unnecessary confusion.
Conversational options
In Russian colloquial speech after the events of 1917 - the Great October Revolution - and the period of hyperinflation in the early 1920s. 1 billion rubles was called "Limard". And in the dashing 1990s, a new slang expression “watermelon” appeared for a billion, a million was called “lemon”.
The word "billion" is now used internationally. This is a natural number, which is represented in decimal system as 10 9 (one and 9 zeros). There is also another name - billion, which is not used in Russia and the CIS countries.
Billion = Billion?
Such a word as billion is used to designate a billion only in those states in which the "short scale" is taken as the basis. These are countries such as the Russian Federation, the United Kingdom of Great Britain and Northern Ireland, the United States, Canada, Greece and Turkey. In other countries, the term billion means the number 10 12, that is, one and 12 zeros. In countries with a "short scale", including Russia, this figure corresponds to 1 trillion.
Such confusion appeared in France at a time when the formation of such a science as algebra was taking place. Initially, the billion had 12 zeros. However, everything changed after the appearance of the main textbook on arithmetic (by Tranchan) in 1558), where a billion is already a number with 9 zeros (one thousand million).
For the next several centuries, these two concepts were used on an equal basis with each other. In the middle of the 20th century, namely in 1948, France switched to a long-scale number system. In this regard, the short scale, once borrowed from the French, is still different from the one they use today.
Historically, the United Kingdom has used the long-term billion, but since 1974 the UK's official statistics have used a short-term scale. Since the 1950s, the short-term scale has been increasingly used in the fields of technical writing and journalism, although the long-term scale still persisted.
It is impossible to correctly answer this question, 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 many names of their own, since most of them are content with names made up of smaller numbers. So, for example, numbers and have their own names "one" and "one hundred", and the name of the number is already composite ("one hundred and one"). It is clear that in the finite 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 large the numbers were invented by mathematicians.
"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 large thousand) for a thousand squared, “bimillion” for a million squared and “trillion” for a million cubed. We know about this system thanks to the French mathematician Nicolas Chuquet (c. 1450 - c. 1500): in his treatise "Science of numbers" (Triparty en la science des nombres, 1484), he developed this idea, suggesting further use of Latin cardinal numbers (see table), adding them to the ending "-million". Thus, Schuquet's “bimillion” became a billion, “trillion” into a trillion, and a million to the fourth power became “quadrillion”.
In the Schücke system, the number between a million and a billion had no name of its own and was simply called “one thousand million,” similarly called “one thousand billion,” “one thousand trillion,” and so on. 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" - "trillion", etc.
The Suke-Peletier system gradually became popular and began to be used throughout Europe. However, in the 17th century, an unexpected problem arose. It turned out that some scientists for some reason began to get confused and call the number not “billion” or “thousand million”, but “billion”. Soon this error quickly spread, and a paradoxical situation arose - “billion” became simultaneously synonymous with “billion” () and “million million” ().
This confusion lasted long enough and led to the fact that the United States created its own system of naming large numbers. According to the American system, the names of numbers are constructed in the same way as in the Schuke system - the Latin prefix and the ending “illion”. However, the magnitudes of these numbers are different. If in the Shuke system names with the ending "million" received numbers that were degrees of a million, then in the American system the ending "-million" received degrees of a thousand. That is, one thousand million () began to be called “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" throughout the world, despite the fact that it was invented by the French Schuquet and Peletier. However, in the 1970s, Great Britain officially switched to the "American system", which led to the fact that it became somewhat strange to call one system American and the other British. As a result, the American system is now commonly referred to as the "short scale", and the British system, or the Schuke-Peletier system, as the "long scale."
In order not to get confused, let's summarize the intermediate result:
| Number name | Short scale value | Long Scale Value |
| Million | ||
| Billion | ||
| Billion | ||
| Billiard | - | |
| Trillion | ||
| Trillion | - | |
| Quadrillion | ||
| Quadrillion | - | |
| Quintillion | ||
| Quintilliard | - | |
| Sextillion | ||
| Sexbillion | - | |
| Septillion | ||
| Septilliard | - | |
| Octillion | ||
| Octilliard | - | |
| Quintillion | ||
| Nonbillion | - | |
| Decillion | ||
| Decilliard | - | |
| Vigintillion | ||
| Vigintilliard | - | |
| Centillion | ||
| Centilliard | - | |
| Million | ||
| Milliard | - |
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 a short scale, except that the number is not called “billion”, but “billion”. The long scale, however, continues to be used in most other countries at the present time.
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. For example, Yakov Isidorovich Perelman (1882–1942) in his Entertaining Arithmetic mentions the parallel existence of two scales in the USSR. The short scale, according to Perelman, was used in everyday life and financial calculations, and the long scale was used in scientific books on astronomy and physics. However, now it is wrong to use the long scale in Russia, although the numbers there turn out to be large.
But back to looking for the largest number. After 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 interesting to us, since we agreed to find the largest number with our 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 "a 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 Schücke's rule, these three remaining Latin numerals give us names for numbers like "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 the smaller numbers is “million” (). If the "long scale" of naming numbers were adopted in Russia, then the largest number with its own name would be "milliard" ().
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, recall 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-composite name, which are more than a million.
Until the 17th century, Russia used its own system of naming numbers. Tens of thousands were called "darkness", hundreds of thousands - "legions", millions - "leodra", tens of millions - "crows", and hundreds of millions - "decks". This counting up to hundreds of millions was called the "little count", and in some manuscripts the authors also considered the "great count", in which the same names were used for large numbers, but with a different meaning. So, "darkness" did not mean ten thousand, but a thousand thousand () , "Legion" - the darkness of those () ; "Leodr" - legion of legions () , "Raven" - leodr leodrov (). For some reason, the "deck" in the great Slavic account was not called the "raven of ravens" () , but only ten "ravens", that is (see table).
| Number name | Meaning in "small count" | Value in the "grand score" | Designation |
| Darkness | |||
| Legion | |||
| Leodre | |||
| Raven (vran) | |||
| Deck | |||
| Darkness of themes |  |
The number also has its own name and was invented by a nine-year-old boy. And it was like this. In 1938, the American mathematician Edward Kasner (1878-1955) walked in the park with his two nephews and discussed large numbers with them. During the conversation, they talked about a number with one hundred zeros, which did not have its own name. One of the nephews, nine-year-old Milton Sirott, suggested calling the number "googol". In 1940, Edward Kasner, together with James Newman, wrote the popular science book "Mathematics and the Imagination", where he told math lovers about the number of googols. Google gained even more prominence in the late 1990s, thanks to the Google search engine named after it.
The name for an even larger number than googol originated in 1950 thanks to the father of computer science, Claude Elwood Shannon (1916-2001). In his article "Programming a Computer for Playing Chess," he tried to estimate the number of possible variants of a chess game. According to him, each game lasts an average of moves and on each move the player makes a choice on the average of the options, which corresponds (approximately equal) to the options of the game. 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. It is believed that this number is equal to the number of cosmic cycles required to attain nirvana.
Nine-year-old Milton Sirotta went down in the history of mathematics not only because he came up with the number googol, but also because at the same time he proposed another number - "googolplex", which is equal to the power of "googol", that is, one with a googol of zeros.
Two more numbers, larger than the googolplex, were proposed by the South African mathematician Stanley Skewes (1899-1988) when proving the Riemann hypothesis. The first number, which later came to be called "the first Skuse number", is equal in degree to degree in degree, that is. However, the "second Skewes number" is even larger and is.
Obviously, the more degrees there are in degrees, the more difficult it is to write 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 into a book the size of the entire Universe! In this case, the question arises how to write such numbers. The problem, fortunately, 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 large numbers - these are the notations of Knuth, Conway, Steinhaus, etc. We now have to deal with some of them.
Other notations
In 1938, the same year that nine-year-old Milton Sirotta invented the numbers googol and googolplex, a book about entertaining mathematics, Mathematical Kaleidoscope, written by Hugo Dionizy Steinhaus (1887-1972) was published in Poland. This book has become very popular, has gone through many editions and has been 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 "",
"Squared" 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, raise everything to a power of times. For example, a 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 the readers to independently estimate another number - "mezons", equal in the circle. In another edition of the book, Steinhaus, instead of Medzon, suggests evaluating an even larger number - "megiston", equal in a circle. Following Steinhaus, I will also recommend readers to temporarily break away from this text and try to write these numbers themselves using ordinary degrees in order to feel their gigantic magnitude.
However, there are names for large numbers. For example, the Canadian mathematician Leo Moser (1921-1970) refined the Steinhaus notation, which was limited by the fact that if it was required to write down the numbers many large megistones, then difficulties and inconveniences would arise, since many circles would have to be drawn one inside another. Moser suggested drawing not circles, but pentagons after the squares, then hexagons, and so on. He also proposed a formal notation for these polygons so that numbers could be written down without drawing complex drawings. Moser's notation looks like this:
"Triangle" = =;
"Squared" = = "in triangles" =;
"In a pentagon" = = "in squares" =;
"In the -gon" = = "in the -gons" =.
Thus, according to Moser's notation, the Steinhaus "mega" is written as, "mezon" as, and "megiston" as. In addition, Leo Moser proposed to call a polygon with the number of sides equal to mega - "mega-gon". And suggested the number « in the megagon ", that is. This number became known as the Moser number, or simply as "Moser".
But even the Moser is not the largest number. So, the largest number ever used in a mathematical proof is the "Graham number". This number was first used by the American mathematician Ronald Graham in 1977 when proving one estimate in Ramsey theory, namely, when calculating the dimensions of certain -dimensional bichromatic hypercubes. But Graham's number gained fame only after the story about him in Martin Gardner's book "From Penrose Mosaics to Reliable Ciphers", published in 1989.
To explain how large the Graham number is, we have to explain another way of writing large numbers, introduced by Donald Knuth in 1976. American professor Donald Knuth came up with the concept of superdegree, which he proposed to write down 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 a repeated addition operation ("add copies of a number"):
For example,
Raising a number to a power can be defined as a repetitive multiplication operation (“multiply copies of a number”), and in Knuth's notation it looks like a single arrow pointing up:
For example,
This single up arrow was used as a degree icon in the Algol programming language.
For example,
Hereinafter, the expression is always evaluated from right to left, and Knuth's arrow operators (like the exponentiation operation), by definition, have right associativity (order from right to left). 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 the repeated exponentiation of the double arrow operator (also known as pentation):
Then the operator "quadruple arrow":
Etc. General rule operator "-I am arrow ", in accordance with the right associativity, continues to the right in a sequential series of operators « arrow ". Symbolically, this can be written as follows,
For example:
The notation form is usually used for 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 preferred (and also for descriptions with a variable number of arrows), or equivalently, to hyperoperators. But some numbers are so huge that even such a record is not sufficient. For example, Graham's number.
When using Knuth Arrow notation, Graham's number can be written as
Where the number of arrows in each layer, starting from the top, is determined by the number in the next layer, that is, where, where the superscript of the arrow shows the total number of arrows. In other words, it is calculated in steps: in the first step, we calculate with four arrows between the triplets, in the second - with arrows between the triplets, in the third - with arrows between the triplets, and so on; at the end we calculate from the arrows between the triplets.
It can be written as, where, where the superscript y means iterating over the functions.
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 sextillons -, and the number of atoms that make up the globe is of the order of dodecalions), then the googol is already "virtual", not to mention about Graham's number. The scale of only the first term is so great that it is almost impossible to grasp it, although the entry above is relatively easy to understand. Although this is just the number of towers in this formula for, this number is already much greater 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.