The name of the largest number in the world. Big numbers have big names
I once read a tragic story about a Chukchi who was taught by polar explorers to count and write down numbers. The magic of numbers amazed him so much that he decided to write down absolutely all the numbers in the world in a row, starting with one, in a notebook donated by polar explorers. The Chukchi abandons all his affairs, stops communicating even with his own wife, no longer hunts ringed seals and seals, but keeps writing and writing numbers in a notebook…. This is how a year goes by. In the end, the notebook runs out 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 again begin to live the simple life of a fisherman, no longer thinking about the mysterious infinity of numbers...
Let's not repeat the feat of this Chukchi and try to find the largest number, since any number only needs to add one to get an even larger number. Let us ask ourselves a similar but different question: which of the numbers that have their own name is the largest?
It is obvious that although the numbers themselves are infinite, they do not have so 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 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 does it equal? Let's try to figure this out and find, in the end, this is the largest number!
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"Short" and "long" scale
The history of the modern system of naming large numbers dates back to the middle of the 15th century, when in Italy they began to use the words “million” (literally - large 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, “bimillion” for Schuke turned into a billion, “trimillion” became a trillion, and a million to the fourth power became “quadrillion”.
In the Schuquet system, the number 10 9, located between a million and a billion, did not have its own name and was simply called “a thousand millions”, similarly 10 15 was called “a thousand billions”, 10 21 - “a thousand trillion”, etc. This was not very convenient, and in 1549 the French writer and scientist Jacques Peletier du Mans (1517-1582) proposed naming such “intermediate” numbers using the same Latin prefixes, but with the ending “-billion”. Thus, 10 9 began to be called “billion”, 10 15 - “billiard”, 10 21 - “trillion”, etc.
The Chuquet-Peletier system gradually became popular and was used throughout Europe. However, in the 17th century an unexpected problem arose. It turned out that for some reason some scientists began to get confused and call the number 10 9 not “billion” or “thousand millions”, but “billion”. Soon this error quickly spread, and a paradoxical situation arose - “billion” became simultaneously synonymous with “billion” (10 9) and “million millions” (10 18).
This confusion continued for quite a long time and led to the fact that the United States created its own system for naming large numbers. According to the American system, the names of numbers are constructed in the same way as in the Chuquet system - the Latin prefix and the ending “million”. However, the magnitudes of these numbers are different. If in the Schuquet system names with the ending “illion” received numbers that were powers of a million, then in the American system the ending “-illion” received powers of a thousand. That is, a thousand million (1000 3 = 10 9) began to be called a “billion”, 1000 4 (10 12) - a “trillion”, 1000 5 (10 15) - a “quadrillion”, etc.
The old system of naming large numbers continued to be used in conservative Great Britain and began to be called “British” throughout the world, despite the fact that it was invented by the French Chuquet and Peletier. However, in the 1970s, the UK officially switched to the “American system”, which led to the fact that it became somehow strange to call one system American and another British. As a result, the American system is now commonly referred to as the "short scale" and the British or Chuquet-Peletier system as the "long scale".
To avoid confusion, let's summarize:
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The short naming scale is now used in the US, UK, Canada, Ireland, Australia, Brazil and Puerto Rico. Russia, Denmark, Turkey and Bulgaria also use a short scale, except that the number 10 9 is called "billion" rather than "billion". The long scale continues to be used in most other countries.
It is curious that in our country the final transition to a short scale occurred only in the second half of the 20th century. For example, Yakov Isidorovich Perelman (1882-1942) in his “Entertaining Arithmetic” mentions the parallel existence of two scales in the USSR. The short scale, according to Perelman, was used in everyday life and financial calculations, and the long scale 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 let's return to the search for the largest number. After decillion, the names of numbers are obtained by combining prefixes. This produces numbers such as undecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion, novemdecillion, etc. However, these names are no longer interesting to us, since we agreed to find the largest number with its own non-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 - “hundred” and mille - “thousand”. The Romans did not have their own names for numbers greater than a thousand. For example, the Romans called a million (1,000,000) “decies centena milia,” that is, “ten times a hundred thousand.” According to Chuquet's rule, these three remaining Latin numerals give us such names for numbers as "vigintillion", "centillion" and "millillion".
So, we found out that on the “short scale” the maximum number that has its own name and is not a composite of smaller numbers is “million” (10 3003). If Russia adopted a “long scale” for naming numbers, then the largest number with its own name would be “billion” (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, number “pi”, dozen, 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 that are greater than a million.
Until the 17th century, Rus' used its own system for naming numbers. Tens of thousands were called "darkness", hundreds of thousands were called "legions", millions were called "leoders", tens of millions were called "ravens", and hundreds of millions were called "decks". This count up to hundreds of millions was called the “small 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” no longer meant ten thousand, but a thousand thousand (10 6), “legion” - the darkness of those (10 12); “leodr” - legion of legions (10 24), “raven” - leodr of leodrov (10 48). For some reason, “deck” in the great Slavic counting was not called “raven of ravens” (10 96), but only ten “ravens”, that is, 10 49 (see table).
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The number 10,100 also has its own name and was invented by a nine-year-old boy. And it was like this. In 1938, American mathematician Edward Kasner (1878-1955) was walking in the park with his two nephews and discussing large numbers with them. During the conversation, we talked about a number with a hundred zeros, which did not have its own name. One of the nephews, nine-year-old Milton Sirott, suggested calling this number “googol.” In 1940, Edward Kasner, together with James Newman, wrote the popular science book Mathematics and the Imagination, where he told mathematics lovers about the googol number. Googol 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 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 on average 40 moves and on each move the player makes a choice from 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 achieve nirvana.
Nine-year-old Milton Sirotta went down in the history of mathematics not only because he came up with the number googol, but also because at the same time he proposed another number - the “googolplex”, which is equal to 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 became known as the "Skuse number", is equal to e to a degree e to a degree 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 powers there are in the powers, the more difficult it is to write the numbers and understand their meaning when reading. Moreover, it is possible to come up with such numbers (and, by the way, they have already been invented) when the degrees of degrees simply do not fit on the page. Yes, that's on the page! They won't even fit into a book the size of the entire Universe! In this case, the question arises of how to write such numbers. The problem, fortunately, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who asked about this problem came up with his own way of writing, which led to the existence of several unrelated methods for writing large numbers - these are the notations of Knuth, Conway, Steinhaus, etc. We now have to deal with some of them.
Other notations
In 1938, the same year that nine-year-old Milton Sirotta invented the numbers googol and googolplex, a book about entertaining mathematics, A Mathematical Kaleidoscope, written by Hugo Dionizy Steinhaus (1887-1972), 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 figures - a triangle, a square and a circle:
"n in a triangle" means " n n»,
« n squared" means " n V n triangles",
« n in a circle" means " n V n squares."
Explaining this method of notation, 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, raise it to the power 256 times. For example, a calculator in MS Windows cannot calculate due to overflow of 256 even in two triangles. Approximately this huge number is 10 10 2.10 619.
Having determined the “mega” number, Steinhaus invites readers to independently estimate another number - “medzon”, equal to 3 in a circle. In another edition of the book, Steinhaus, instead of medzone, suggests estimating an even larger number - “megiston”, equal to 10 in a circle. Following Steinhaus, I also recommend that readers break away from this text for a while and try to write these numbers themselves using ordinary powers in order to feel their gigantic magnitude.
However, there are names for b O larger numbers. Thus, the Canadian mathematician Leo Moser (Leo Moser, 1921-1970) modified the Steinhaus notation, which was limited by the fact that if it were necessary to write numbers much larger than megiston, then difficulties and inconveniences would arise, since it would be necessary to draw many circles one inside another. Moser suggested that after the squares, draw not circles, but pentagons, then hexagons, and so on. He also proposed a formal notation for these polygons so that numbers could be written without drawing complex pictures. Moser notation looks like this:
« n triangle" = n n = n;
« n squared" = 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, Steinhaus’s “mega” is written as 2, “medzone” as 3, and “megiston” as 10. In addition, Leo Moser proposed calling a polygon with the number of sides equal to mega - “megagon”. And he proposed the number “2 in megagon”, that is, 2. This number became known as the Moser number or simply as “Moser”.
But even “Moser” is not the largest number. So, the largest number ever used in mathematical proof is the "Graham number". This number was first used by the American mathematician Ronald Graham in 1977 when proving one estimate in Ramsey theory, namely when calculating the dimension of certain n-dimensional bichromatic hypercubes. Graham's number became famous only after it was described in Martin Gardner's 1989 book, From Penrose Mosaics to Reliable Ciphers.
To explain how large Graham's number is, we have to explain another way of writing large numbers, introduced by Donald Knuth in 1976. American professor Donald Knuth came up with the concept of superpower, which he proposed to write with arrows pointing upward:
I think everything is clear, so let’s return to Graham’s number. Ronald Graham proposed the so-called G-numbers:
The number G 64 is called the Graham number (it is often designated 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’t help but resist the temptation to come up with my own number. Let this number be called " stasplex"and will be equal to the number G 100. Remember it, and when your children ask what the largest number in the world is, tell them that this number is called stasplex.
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Have you ever thought how many zeros there are in one million? This is a pretty simple question. What about a billion or a trillion? One followed by nine zeros (1000000000) - what is the name of the number?
A short list of numbers and their quantitative designation
- Ten (1 zero).
- One hundred (2 zeros).
- One thousand (3 zeros).
- Ten thousand (4 zeros).
- One hundred thousand (5 zeros).
- Million (6 zeros).
- Billion (9 zeros).
- Trillion (12 zeros).
- Quadrillion (15 zeros).
- Quintilion (18 zeros).
- Sextillion (21 zeros).
- Septillion (24 zeros).
- Octalion (27 zeros).
- Nonalion (30 zeros).
- Decalion (33 zeros).
Grouping of zeros
1000000000 - what is the name of a number that has 9 zeros? This is a billion. For convenience, large numbers are usually grouped into sets of three, separated from each other by a space or punctuation marks such as a comma or period.
This is done to make the quantitative value easier to read and understand. For example, what is the name of the number 1000000000? In this form, it’s worth straining a little and doing the math. And if you write 1,000,000,000, then the task immediately becomes visually easier, since you need to count not zeros, but triples of zeros.
Numbers with a lot of zeros
The most popular are million and billion (1000000000). What is the name of a number that has 100 zeros? This is a Googol number, so called by Milton Sirotta. This is a wildly huge amount. Do you think this number is large? Then what 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 the infinite Universe.
Is 1 billion a lot?
There are two measurement scales - short and long. Around the world in 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 formerly used in the UK (until 1971), where a billion was 1 million million, that is, a one followed by 12 zeros. This gradation is also called the long-term scale. The short scale is now predominant in financial and scientific matters.
Some European languages, such as Swedish, Danish, Portuguese, Spanish, Italian, Dutch, Norwegian, Polish, German, use billion (or billion) 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 were called “lemon.”
The word "billion" is now used internationally. This is a natural number, which is represented in the decimal system as 10 9 (one followed by 9 zeros). There is also another name - billion, which is not used in Russia and the CIS countries.
Billion = billion?
A word such as billion is used to designate a billion only in those states in which the “short scale” is adopted as a basis. These are countries such as the Russian Federation, the United Kingdom of Great Britain and Northern Ireland, the USA, Canada, Greece and Turkey. In other countries, the concept of a billion means the number 10 12, that is, one followed by 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, a billion had 12 zeros. However, everything changed after the appearance of the main manual on arithmetic (author Tranchan) in 1558), where a billion is already a number with 9 zeros (a thousand millions).
For several subsequent centuries, these two concepts were used on an equal basis with each other. In the mid-20th century, namely in 1948, France switched to a long scale numerical naming 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 used the long-term billion, but since 1974 official UK statistics have used the 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 persists.
June 17th, 2015
“I see clusters of vague numbers that are hidden there in the darkness, behind the small spot of light that the candle of reason gives. They whisper to each other; conspiring about who knows what. Perhaps they don't like us very much for capturing their little brothers in our minds. Or perhaps they simply lead a single-digit life, out there, beyond our understanding.
Douglas Ray
We continue ours. Today we have numbers...
Sooner or later, everyone is tormented by the question, what is the largest number. There are a million answers to a child's question. What's next? Trillion. And even further? In fact, the answer to the question of what are the largest numbers is simple. Just add one to the largest number, and it will no longer be the largest. This procedure can be continued indefinitely.
But if you ask the question: what is the largest number that exists, and what is its proper name?
Now we will find out everything...
There are two systems for naming numbers - American and English.
The American system is built quite simply. All names of large numbers are constructed like this: at the beginning there is a Latin ordinal number, and at the end the suffix -million is added to it. The exception is the name "million" which is the name of the number thousand (lat. mille) and the magnifying suffix -illion (see table). This is how we get the numbers trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion and decillion. The American system is used in the USA, Canada, France and Russia. You can find out the number of zeros in a number written 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 former English and Spanish colonies. The names of numbers in this system are built like this: like this: the suffix -million is added to the Latin numeral, the next number (1000 times larger) is built according to the principle - the same Latin numeral, but the suffix - billion. That is, after a trillion in the English system there is a trillion, and only then a quadrillion, followed by a quadrillion, etc. Thus, a quadrillion according to the English and American systems are completely different numbers! You can find out the number of zeros in a number written according to the English system and ending with the suffix -million, using the formula 6 x + 3 (where x is a Latin numeral) and using the formula 6 x + 6 for numbers ending in - billion.
Only the number billion (10 9) passed from the English system into the Russian language, which would still be more correct to be called as the Americans call it - billion, since we have adopted the American system. But who in our country does anything according to the rules! ;-) By the way, sometimes the word trillion is used in Russian (you can see this for yourself by running a search in Google or Yandex) and, apparently, it means 1000 trillion, i.e. quadrillion.
In addition to numbers written using Latin prefixes according to the American or English system, so-called non-system numbers are also known, i.e. numbers that have their own names without any Latin prefixes. There are several such numbers, but I will tell you more about them a little later.
Let's return to writing using Latin numerals. It would seem that they can write down numbers to infinity, but this is not entirely true. Now I will explain why. Let's first see what the numbers from 1 to 10 33 are called:
And now the question arises, what next. What's behind the decillion? In principle, it is, of course, possible, by combining prefixes, to generate such monsters as: andecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion and novemdecillion, but these will already be compound names, and we were interested in our own names numbers. Therefore, according to this system, in addition to those indicated above, you can still get only three proper names - vigintillion (from Lat.viginti- twenty), centillion (from lat.centum- one hundred) and million (from lat.mille- thousand). The Romans did not have more than a thousand proper names for numbers (all numbers over a thousand were composite). For example, the Romans called a million (1,000,000)decies centena milia, that is, "ten hundred thousand." And now, actually, the table:
Thus, according to such a system, numbers are greater than 10 3003 , which would have its own, non-compound name is impossible to obtain! But nevertheless, numbers greater than a million are known - these are the same non-systemic numbers. Let's finally talk about them.
The smallest such number is a myriad (it is even in Dahl’s dictionary), which means a hundred hundreds, that is, 10,000. This word, however, is outdated and practically not used, but it is curious that the word “myriads” is widely used, does not mean a definite number at all, but an uncountable, uncountable multitude 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 fact, the myriad gained fame precisely thanks to the Greeks. Myriad was the name for 10,000, but there were no names for numbers greater than ten thousand. However, in his note “Psammit” (i.e., calculus of sand), Archimedes showed how to systematically construct and name arbitrarily large numbers. In particular, placing 10,000 (myriad) grains of sand in a poppy seed, he finds that in the Universe (a ball with a diameter of a myriad of Earth diameters) there would fit (in our notation) no more than 10 63
grains of sand It is curious that modern calculations of the number of atoms in the visible Universe lead to the number 10 67
(in total a myriad of times more). Archimedes suggested the following names for the numbers:
1 myriad = 10 4.
1 di-myriad = myriad of myriads = 10 8
.
1 tri-myriad = di-myriad di-myriad = 10 16
.
1 tetra-myriad = three-myriad three-myriad = 10 32
.
etc.
Googol (from the English googol) is the number ten to the hundredth power, that is, one followed by one hundred zeros. The “googol” was first written about in 1938 in the article “New Names in Mathematics” in the January issue of the journal Scripta Mathematica by the American mathematician Edward Kasner. According to him, it was his nine-year-old nephew Milton Sirotta who suggested calling the large number a “googol”. This number became generally known thanks to the search engine named after it. Google. Please note that "Google" is a brand name and googol is a number.
Edward Kasner.
On the Internet you can often find it mentioned that - but this is not true...
In the famous Buddhist treatise Jaina Sutra, dating back to 100 BC, the number asankheya (from Chinese. asenzi- uncountable), equal to 10 140. It is believed that this number is equal to the number of cosmic cycles required to achieve nirvana.
Googolplex (English) googolplex) - a number also invented by Kasner and his nephew and meaning one with a googol of zeros, that is, 10 10100 . This is how Kasner himself describes this “discovery”:
Words of wisdom are spoken by children at least as often as by scientists. The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex." A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out.
Mathematics and the Imagination(1940) by Kasner and James R. Newman.
An even larger number than the googolplex, the Skewes number, was proposed by Skewes in 1933. J. London Math. Soc. 8, 277-283, 1933.) in proving the Riemann hypothesis concerning prime numbers. It means e to a degree e to a degree e to the power of 79, that is, ee e 79 . Later, te Riele, H. J. J. "On the Sign of the Difference P(x)-Li(x)." Math. Comput. 48, 323-328, 1987) reduced the Skuse number to ee 27/4 , which is approximately equal to 8.185·10 370. It is clear that since the value of the Skuse number depends on the number e, then it is not an integer, so we will not consider it, otherwise we would have to remember other non-natural numbers - the number pi, the number e, etc.
But it should be noted that there is a second Skuse number, which in mathematics is denoted as Sk2, which is even greater than the first Skuse number (Sk1). Second Skewes number, was introduced by J. Skuse in the same article to denote a number for which the Riemann hypothesis does not hold. Sk2 equals 1010 10103 , that is 1010 101000 .
As you understand, the more degrees there are, the more difficult it is to understand which number is greater. For example, looking at Skewes numbers, without special calculations, it is almost impossible to understand which of these two numbers is larger. Thus, for super-large numbers it becomes inconvenient to use powers. Moreover, you can come up with such numbers (and they have already been invented) when the degrees of degrees simply do not fit on the page. Yes, that's on the page! They won’t fit even into a book the size of the entire Universe! In this case, the question arises of how to write them down. The problem, as you understand, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who asked about this problem came up with his own way of writing, which led to the existence of several, unrelated to each other, methods for writing numbers - these are the notations of Knuth, Conway, Steinhouse, etc.
Consider the notation of Hugo Stenhouse (H. Steinhaus. Mathematical Snapshots, 3rd edn. 1983), which is quite simple. Stein House suggested writing large numbers inside geometric shapes - triangle, square and circle:
Steinhouse came up with two new superlarge numbers. He named the number - Mega, and the number - Megiston.
Mathematician Leo Moser refined Stenhouse's notation, which was limited by the fact that if it was necessary to write down numbers much larger than a megiston, difficulties and inconveniences arose, since many circles had to be drawn one inside the other. Moser suggested that after the squares, draw not circles, but pentagons, then hexagons, and so on. He also proposed a formal notation for these polygons so that numbers could be written without drawing complex pictures. Moser notation looks like this:
Thus, according to Moser's notation, Steinhouse's mega is written as 2, and megiston as 10. In addition, Leo Moser proposed calling a polygon with the number of sides equal to mega - megagon. And he proposed the number “2 in Megagon,” that is, 2. This number became known as Moser’s number or simply as Moser.
But Moser is not the largest number. The largest number ever used in a mathematical proof is the limiting quantity known as Graham's number, first used in 1977 in the proof of an 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, a number written in Knuth's notation cannot be converted into notation in the Moser system. Therefore, we will have to explain this system too. In principle, there is nothing complicated about it either. Donald Knuth (yes, yes, this is the same Knuth who wrote “The Art of Programming” and created the TeX editor) came up with the concept of superpower, which he proposed to write with arrows pointing upward:
In general it looks like this:
I think everything is clear, so let’s return to Graham’s number. Graham proposed so-called G-numbers:
- G1 = 3..3, where the number of superpower arrows is 33.
- G2 = ..3, where the number of superpower arrows is equal to G1.
- G3 = ..3, where the number of superpower arrows is equal to G2.
- G63 = ..3, where the number of superpower arrows is G62.
The G63 number came to be called the Graham number (it is often designated simply as G). This number is the largest known number in the world and is even listed in the Guinness Book of Records. And here
As a child, I was tormented by the question of what the largest number exists, and I tormented almost everyone with this stupid question. Having learned the number one million, I asked if there was a number greater than a million. Billion? How about more than a billion? Trillion? How about more than a trillion? Finally, there was someone smart who explained to me that the question was stupid, since it is enough just to add one to the largest number, and it turns out that it was never the largest, since there are even larger numbers.
And so, many years later, I decided to ask myself another question, namely: What is the largest number that has its own name? Fortunately, now there is the Internet and you can puzzle patient search engines with it, which will not call my questions idiotic ;-). Actually, that’s what I did, and this is what I found out as a result.
| Number | Latin name | Russian prefix |
| 1 | unus | an- |
| 2 | duo | duo- |
| 3 | tres | three- |
| 4 | quattuor | quadri- |
| 5 | quinque | quinti- |
| 6 | sex | sexty |
| 7 | septem | septi- |
| 8 | octo | octi- |
| 9 | novem | noni- |
| 10 | decem | deci- |
There are two systems for naming numbers - American and English.
The American system is built quite simply. All names of large numbers are constructed like this: at the beginning there is a Latin ordinal number, and at the end the suffix -million is added to it. The exception is the name "million" which is the name of the number thousand (lat. mille) and the magnifying suffix -illion (see table). This is how we get the numbers trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion and decillion. The American system is used in the USA, Canada, France and Russia. You can find out the number of zeros in a number written 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 former English and Spanish colonies. The names of numbers in this system are built like this: like this: the suffix -million is added to the Latin numeral, the next number (1000 times larger) is built according to the principle - the same Latin numeral, but the suffix - billion. That is, after a trillion in the English system there is a trillion, and only then a quadrillion, followed by a quadrillion, etc. Thus, a quadrillion according to the English and American systems are completely different numbers! You can find out the number of zeros in a number written according to the English system and ending with the suffix -million, using the formula 6 x + 3 (where x is a Latin numeral) and using the formula 6 x + 6 for numbers ending in - billion.
Only the number billion (10 9) passed from the English system into the Russian language, which would still be more correct to be called as the Americans call it - billion, since we have adopted the American system. But who in our country does anything according to the rules! ;-) By the way, sometimes the word trillion is used in Russian (you can see this for yourself by running a search in Google or Yandex) and 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 non-system numbers are also known, i.e. numbers that have their own names without any Latin prefixes. There are several such numbers, but I will tell you more about them a little later.
Let's return to writing using Latin numerals. It would seem that they can write down numbers to infinity, but this is not entirely true. Now I will explain why. Let's first see what the numbers from 1 to 10 33 are called:
| Name | Number |
| Unit | 10 0 |
| Ten | 10 1 |
| One hundred | 10 2 |
| Thousand | 10 3 |
| Million | 10 6 |
| Billion | 10 9 |
| Trillion | 10 12 |
| Quadrillion | 10 15 |
| Quintillion | 10 18 |
| Sextillion | 10 21 |
| Septillion | 10 24 |
| Octillion | 10 27 |
| Quintillion | 10 30 |
| Decillion | 10 33 |
And now the question arises, what next. What's behind the decillion? In principle, it is, of course, possible, by combining prefixes, to generate such monsters as: andecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion and novemdecillion, but these will already be compound names, and we were interested in our own names numbers. Therefore, according to this system, in addition to those indicated above, you can still get only three proper names - vigintillion (from Lat. viginti- twenty), centillion (from lat. centum- one hundred) and million (from lat. mille- thousand). The Romans did not have more than a thousand proper names for numbers (all numbers over a thousand were composite). For example, the Romans called a million (1,000,000) decies centena milia, that is, "ten hundred thousand." And now, actually, the table:
Thus, according to such a system, it is impossible to obtain numbers greater than 10 3003, which would have its own, non-compound name! But nevertheless, numbers greater than a million are known - these are the same non-systemic numbers. Let's finally talk about them.
| Name | Number |
| Myriad | 10 4 |
| 10 100 | |
| Asankheya | 10 140 |
| Googolplex | 10 10 100 |
| Second Skewes number | 10 10 10 1000 |
| Mega | 2 (in Moser notation) |
| Megiston | 10 (in Moser notation) |
| Moser | 2 (in Moser notation) |
| Graham number | G 63 (in Graham notation) |
| Stasplex | G 100 (in Graham notation) |
The smallest such number is myriad(it is even in Dahl’s dictionary), which means a hundred hundreds, that is, 10,000. This word, however, is outdated and practically not used, but it is curious that the word “myriads” is widely used, which does not mean a specific number at all, but countless, uncountable multitudes of something. It is believed that the word myriad came into European languages from ancient Egypt.
Google(from the English googol) is the number ten to the hundredth power, that is, one followed by one hundred zeros. The “googol” was first written about in 1938 in the article “New Names in Mathematics” in the January issue of the journal Scripta Mathematica by the American mathematician Edward Kasner. According to him, it was his nine-year-old nephew Milton Sirotta who suggested calling the large number a “googol”. This number became generally known thanks to the search engine named after it. Google. Please note that "Google" is a brand name and googol is a number.
In the famous Buddhist treatise Jaina Sutra, dating back to 100 BC, the number appears asankheya(from China asenzi- uncountable), equal to 10 140. It is believed that this number is equal to the number of cosmic cycles required to achieve nirvana.
Googolplex(English) googolplex) - a number also invented by Kasner and his nephew and meaning one with a googol of zeros, that is, 10 10 100. This is how Kasner himself describes this “discovery”:
Words of wisdom are spoken by children at least as often as by scientists. The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex." A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out.
Mathematics and the Imagination(1940) by Kasner and James R. Newman.
An even larger number than the googolplex, the Skewes number, was proposed by Skewes in 1933. J. London Math. Soc. 8 , 277-283, 1933.) in proving the Riemann hypothesis concerning prime numbers. It means e to a degree e to a degree e to the power of 79, that is, e e e 79. Later, te Riele, H. J. J. "On the Sign of the Difference P(x)-Li(x)." Math. Comput. 48 , 323-328, 1987) reduced the Skuse number to e e 27/4, which is approximately equal to 8.185 10 370. It is clear that since the value of the Skuse number depends on the number e, then it is not an integer, so we will not consider it, otherwise we would have to remember other non-natural numbers - pi, e, Avogadro's number, etc.
But it should be noted that there is a second Skuse number, which in mathematics is denoted as Sk 2, which is even greater than the first Skuse number (Sk 1). Second Skewes number, was introduced by J. Skuse in the same article to denote the number up to which the Riemann hypothesis is valid. Sk 2 is equal to 10 10 10 10 3, that is, 10 10 10 1000.
As you understand, the more degrees there are, the more difficult it is to understand which number is greater. For example, looking at Skewes numbers, without special calculations, it is almost impossible to understand which of these two numbers is larger. Thus, for super-large numbers it becomes inconvenient to use powers. Moreover, you can come up with such numbers (and they have already been invented) when the degrees of degrees simply do not fit on the page. Yes, that's on the page! They won’t fit even into a book the size of the entire Universe! In this case, the question arises of how to write them down. The problem, as you understand, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who wondered about this problem came up with his own way of writing, which led to the existence of several, unrelated to each other, methods for writing numbers - these are the notations of Knuth, Conway, Steinhouse, etc.
Consider the notation of Hugo Stenhouse (H. Steinhaus. Mathematical Snapshots, 3rd edn. 1983), which is quite simple. Stein House suggested writing large numbers inside geometric shapes - triangle, square and circle:
Steinhouse came up with two new superlarge numbers. He named the number - Mega, and the number is Megiston.
Mathematician Leo Moser refined Stenhouse's notation, which was limited by the fact that if it was necessary to write down numbers much larger than a megiston, difficulties and inconveniences arose, since many circles had to be drawn one inside the other. Moser suggested that after the squares, draw not circles, but pentagons, then hexagons, and so on. He also proposed a formal notation for these polygons so that numbers could be written without drawing complex pictures. Moser notation looks like this:
Thus, according to Moser's notation, Steinhouse's mega is written as 2, and megiston as 10. In addition, Leo Moser proposed calling a polygon with the number of sides equal to mega - megagon. And he proposed the number “2 in Megagon”, that is, 2. This number became known as Moser’s number or simply as moser.
But Moser is not the largest number. The largest number ever used in mathematical proof is the limit known as Graham number(Graham's number), first used in 1977 in the proof of one estimate in Ramsey theory. It is associated with bichromatic hypercubes and cannot be expressed without a special 64-level system of special mathematical symbols introduced by Knuth in 1976.
Unfortunately, a number written in Knuth's notation cannot be converted into notation in the Moser system. Therefore, we will have to explain this system too. In principle, there is nothing complicated about it either. Donald Knuth (yes, yes, this is the same Knuth who wrote “The Art of Programming” and created the TeX editor) came up with the concept of superpower, which he proposed to write with arrows pointing upward:
In general it looks like this:
I think everything is clear, so let’s return to Graham’s number. Graham proposed so-called G-numbers:
The number G 63 began to be called Graham number(it is often designated simply as G). This number is the largest known number in the world and is even listed in the Guinness Book of Records. Well, the Graham number is greater than the Moser number.
P.S. In order to bring great benefit to all humanity and become famous throughout the centuries, I decided to come up with and name the largest number myself. This number will be called stasplex and it is equal to the number G 100. Remember it, and when your children ask what is the largest number in the world, tell them that this number is called stasplex.
Update (4.09.2003): Thank you all for the comments. It turned out that I made several mistakes when writing the text. I'll try to fix it now.
- I made several mistakes just by mentioning Avogadro's number. First, several people pointed out to me that 6.022 10 23 is, in fact, the most natural number. And secondly, there is an opinion, and it seems correct to me, that Avogadro’s number is not a number at all in the proper, mathematical sense of the word, since it depends on the system of units. Now it is expressed in “mol -1”, but if it is expressed, for example, in moles or something else, then it will be expressed as a completely different number, but this will not cease to be Avogadro’s number at all.
- 10,000 - darkness
100,000 - legion
1,000,000 - leodr
10,000,000 - raven or corvid
100,000,000 - deck
Interestingly, the ancient Slavs also loved large numbers and were able to count to a billion. Moreover, they called such an account a “small account.” In some manuscripts, the authors also considered the “great count”, reaching the number 10 50. About numbers greater than 10 50 it was said: “And more than this cannot be understood by the human mind.” The names used in the “small count” were transferred to the “great count”, but with a different meaning. So, darkness no longer meant 10,000, but a million, legion - the darkness of those (a million millions); leodre - legion of legions (10 to the 24th degree), then it was said - ten leodres, one hundred leodres, ..., and finally, one hundred thousand those legion of leodres (10 to 47); leodr leodrov (10 in 48) was called a raven and, finally, a deck (10 in 49). - The topic of national names of numbers can be expanded if we remember about the Japanese system of naming numbers that I had forgotten, which is very different from the English and American systems (I won’t draw hieroglyphs, if anyone is interested, they are):
10 0 - ichi
10 1 - jyuu
10 2 - hyaku
10 3 - sen
10 4 - man
10 8 - oku
10 12 - chou
10 16 - kei
10 20 - gai
10 24 - jyo
10 28 - jyou
10 32 - kou
10 36 - kan
10 40 - sei
10 44 - sai
10 48 - goku
10 52 - gougasya
10 56 - asougi
10 60 - nayuta
10 64 - fukashigi
10 68 - muryoutaisuu - Regarding the numbers of Hugo Steinhaus (in Russia for some reason his name was translated as Hugo Steinhaus). botev assures that the idea of writing superlarge numbers in the form of numbers in circles belongs not to Steinhouse, but to Daniil Kharms, who long before him published this idea in the article “Raising a Number.” I also want to thank Evgeniy Sklyarevsky, the author of the most interesting site on entertaining mathematics on the Russian-language Internet - Arbuza, for the information that Steinhouse came up with not only the numbers mega and megiston, but also suggested another number medical zone, equal (in his notation) to "3 in a circle".
- Now about the number myriad or mirioi. 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, but there were no names for numbers greater than ten thousand. However, in his note “Psammit” (i.e., calculus of sand), Archimedes showed how to systematically construct and name arbitrarily large numbers. In particular, placing 10,000 (myriad) grains of sand in a poppy seed, he finds that in the Universe (a ball with a diameter of a myriad of the diameters of the Earth) no more than 10 63 grains of sand could fit (in our notation). It is curious that modern calculations of the number of atoms in the visible Universe lead to the number 10 67 (in total a myriad of times more). Archimedes suggested the following names for the numbers:
1 myriad = 10 4.
1 di-myriad = myriad of myriads = 10 8 .
1 tri-myriad = di-myriad di-myriad = 10 16 .
1 tetra-myriad = three-myriad three-myriad = 10 32 .
etc.
If you have any comments -
Countless different numbers surround us every day. Surely many people have at least once wondered what number is considered the largest. You can simply say to a child that this is a million, but adults understand perfectly well that other numbers follow a million. For example, all you have to do is add one to a number each time, and it will become larger and larger - this happens ad infinitum. But if you look at the numbers that have names, you can find out what the largest number in the world is called.
The appearance of number names: what methods are used?
Today 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 throughout the world. The American one allows you to give names to large numbers as follows: first, the ordinal number in Latin is indicated, and then the suffix “million” is added (the exception here is 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, 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” “billion”. For example, the trillion comes first, the trillion comes after it, the quadrillion comes after the quadrillion, etc.
Thus, the same number in different systems can mean different things; for example, an American billion in the English system is called a billion.
Extra-system numbers
In addition to the numbers that are written according to the known systems (given 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 according to 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 a googol, denoting 10 to the power of 100. This name was first used in 1938 by the American mathematician E. Kasner, who noted that this name was invented by his nephew.
Google (search engine) got its name in honor of googol. Then 1 with a googol of zeros (1010100) represents a googolplex - Kasner also came up with this name.
Even larger than the googolplex is the Skuse number (e to the power of e to the power of e79), proposed by Skuse in his proof of the Rimmann conjecture about prime numbers (1933). There is another Skuse number, but it is used when the Rimmann hypothesis is not valid. Which one is greater is quite difficult to say, especially when it comes to large degrees. However, this number, despite its “hugeness,” cannot be considered the very best 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 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 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 proposed the use of up arrows. So we found out what the largest number in the world is called. It is worth noting that this number G was included in the pages of the famous Book of Records.