What numbers are greater than a trillion. Not included in the collection of essays

Naming systems for large numbers

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

In the American system, 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. An exception is the name "million", which is the name of the number thousand (Latin mille) and the magnifying suffix "illion". This is how numbers are obtained - trillion, quadrillion, quintillion, sextillion, etc. The American system is used in the USA, Canada, France and Russia. The number of zeros in a number written according to the American system is determined by the formula 3 x + 3 (where x is a Latin numeral).

The European (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 constructed as follows: the suffix “million” is added to the Latin numeral, the name of the next number (1,000 times larger) is formed from the same Latin numeral, but with the suffix “billion”. That is, after a trillion in this system there is a trillion, and only then a quadrillion, followed by a quadrillion, etc. The number of zeros in a number written according to the European system and ending with the suffix “million” is determined by the formula 6 x + 3 (where x is a Latin numeral) and by the formula 6 x + 6 for numbers ending in “billion”. In some countries that use the American system, for example, in Russia, Turkey, Italy, the word “billion” is used instead of the word “billion”.

Both systems originate from France. French physicist and mathematician Nicolas Chuquet coined the words "billion" and "trillion" and used them to represent the numbers 10 12 and 10 18 respectively, which served as the basis for the European system.

But some French mathematicians in the 17th century used the words "billion" and "trillion" for the numbers 10 9 and 10 12, respectively. This naming system took hold in France and America, and became known as American, while the original Choquet system continued to be used in Great Britain and Germany. France returned to the Choquet system (i.e. European) in 1948.

IN last years The American system is replacing the European one, partially in Great Britain and so far hardly noticeably in the rest European countries. This is mainly due to the fact that Americans insist in financial transactions that $1,000,000,000 should be called a billion dollars. In 1974, Prime Minister Harold Wilson's government announced that the word billion would be 10 9 rather than 10 12 in UK official records and statistics.

12 - Dozen(from French douzaine or Italian dozzina, which in turn came from Latin duodecim.)
A measure of piece counting of homogeneous objects. Widely used before the introduction of the metric system. For example, a dozen scarves, a dozen forks. 12 dozen make a gross. The word “dozen” was mentioned for the first time in Russian in 1720. It was originally used by sailors.

13 - Baker's dozen

The number is considered unlucky. Many Western hotels do not have rooms numbered 13, and in office buildings 13th floors. IN opera houses There are no places with this number in Italy. On almost all ships, after the 12th cabin comes the 14th.

144 - Gross- “big dozen” (from German Gro? - big)

A counting unit equal to 12 dozen. It was usually used when counting small haberdashery and stationery items - pencils, buttons, writing pens, etc. A dozen gross makes a mass.

1728 - Weight

Mass (obsolete) - a measure equal to a dozen gross, i.e. 144 * 12 = 1728 pieces. Widely used before the introduction of the metric system.

666 or 616 - Number of the beast

A special number mentioned in the Bible (Revelation 13:18, 14:2). It is assumed that in connection with the assignment of a numerical value to the letters of ancient alphabets, this number can mean any name or concept, the sum of the numerical values ​​of the letters of which is 666. Such words could be: "Lateinos" (meaning in Greek everything Latin; suggested by Jerome ), "Nero Caesar", "Bonaparte" and even "Martin Luther". In some manuscripts the number of the beast is read as 616.

Myriad - the word is outdated and practically not used, but the word "myriads" - (astronomer) is widely used, which means an uncountable, uncountable multitude of something.

Myriad was the largest number for which the ancient Greeks had a name. However, in his work "Psammit" ("Calculus of grains of sand"), Archimedes showed how to systematically construct and name arbitrarily large numbers. Archimedes called all the numbers from 1 to the myriad (10,000) the first numbers, he called the myriad of myriads (10 8) the unit of second numbers (dimyriad), he called the myriad of myriads of second numbers (10 16) the unit of third numbers (trimyriad), etc. .

10 000 - dark
100 000 - legion
1 000 000 - Leodr
10 000 000 - raven or corvid
100 000 000 - deck

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 " great score", 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.” Names used in the “small count” were transferred to the “great count,” but with a different meaning. Thus, darkness meant not 10,000, but a million, legion - the darkness of those (a million millions); leodr - legion of legions - 10 24, then it said - ten leodres, one hundred leodres, ..., and, finally, one hundred thousand those legion of leodres - 10 47 ; leodr leodrov -10 48 was called a raven and, finally, a deck -10 49 .

10 140 - Asankhey I (from Chinese asentsi - innumerable)

Mentioned in the famous Buddhist treatise Jaina Sutra, dating back to 100 BC. It is believed that this number is equal to the number of cosmic cycles required to achieve nirvana.

Google(from English googol) - 10 100 , 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. Note that " Google" - This trademark, A googol - number.

Googolplex(English googolplex) 10 10 100 - 10 to the power of googol.

The number was also invented by Kasner and his nephew and means one with a googol of zeros, that is, 10 to the power of a googol. 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.

Skewes number(Skewes` number) - Sk 1 e e e 79 - means e to the power of e to the power of e to the power of 79.

It was proposed by J. Skewes in 1933 (Skewes. J. London Math. Soc. 8, 277-283, 1933.) when proving the Riemann hypothesis concerning prime numbers. Later, Riele (te Riele, H. J. J. "On the Sign of the Difference П(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 was introduced by J. Skuse in the same article to denote the number up to which the Riemann hypothesis is not valid. Sk 2 is equal to 10 10 10 10 3 .

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.

Hugo Stenhouse notation(H. Steinhaus. Mathematical Snapshots, 3rd edn. 1983) is quite simple. Steinhaus (German: Steihaus) proposed writing large numbers inside geometric figures - triangle, square and circle.

Steinhouse came up with super-large numbers and called the number 2 in a circle - Mega, 3 in a circle - Medzone, and the number 10 in a circle is Megiston.

Mathematician Leo Moser modified Stenhouse's notation, which was limited by the fact that if it was necessary to write numbers much larger than megiston, difficulties and inconveniences arose, since it was necessary to draw many circles 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 drawings. Moser notation looks like this:

• "n triangle" = nn = n.
• "n squared" = n = "n in n triangles" = nn.
• "n in a pentagon" = n = "n in n squares" = nn.
• n = "n in n k-gons" = n[k]n.

In Moser's notation, Steinhouse's mega is written as 2, and megiston as 10. Leo Moser proposed calling a polygon with the number of sides equal to mega - megagon. He also proposed the number “2 in Megagon”, that is, 2. This number became known as Moser number(Moser`s number) or just like Moser. But the Moser number 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's theory. It is related to bichromatic hypercubes and cannot be expressed without a special 64-level system of special mathematical symbols, introduced by D. Knuth in 1976.

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

A little history

Southern and eastern Slavic peoples Alphabetical numbering was used to record numbers. Moreover, for the Russians, not all letters played the role of numbers, but only those that are in the Greek alphabet. A special “title” icon was placed above the letter indicating the number. At the same time, the numerical values ​​of the letters increased in the same order as the letters in the Greek alphabet (the order of the letters of the Slavic alphabet was slightly different).

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

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

The name “million” first appeared in Italy in 1500 and was formed by adding an augmentative suffix to the number “mille” - a thousand (i.e., it meant “big thousand”), it penetrated into the Russian language later, and before that the same meaning in in Russian it was designated by the number "leodr". The word “billion” came into use only since the Franco-Prussian War (1871), when the French had to pay Germany an indemnity of 5,000,000,000 francs. Like "million," the word "billion" comes from the root "thousand" with the addition of an Italian magnifying suffix. In Germany and America for some time the word “billion” meant the number 100,000,000; This explains that the word billionaire was used in America before any of the rich people had $1,000,000,000. In the ancient (18th century) “Arithmetic” of Magnitsky, a table of the names of numbers is given, brought to the “quadrillion” (10^24, according to the system through 6 digits). Perelman Ya.I. in the book "Entertaining Arithmetic" the names of large numbers of that time are given, slightly different from today's: septillion (10^42), octalion (10^48), nonalion (10^54), decalion (10^60), endecalion (10^ 66), dodecalion (10^72) and it is written that “there are no further names.”

Principles for constructing names and a list of large numbers

All names of large numbers are constructed quite in a simple way: the Latin ordinal number comes at the beginning, and the suffix -million is added to it at the end. An exception is the name "million" which is the name of the number thousand (mille) and the augmentative suffix -million. There are two main types of names for large numbers in the world:
system 3x+3 (where x is a Latin ordinal number) - this system is used in Russia, France, USA, Canada, Italy, Turkey, Brazil, Greece
and the 6x system (where x is a Latin ordinal number) - this system is most common in the world (for example: Spain, Germany, Hungary, Portugal, Poland, Czech Republic, Sweden, Denmark, Finland). In it, the missing intermediate 6x+3 end with the suffix -billion (from it we borrowed billion, which is also called billion).

Below is a general list of numbers used in Russia:

...

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



• 10 123 - quadragintillion (quadraginta, XL)


• 10 153 - quinquagintillion (quinquaginta, L)


• 10 183 - sexagintillion (sexaginta, LX)


• 10,213 - septuagintillion (septuaginta, LXX)


• 10,243 - octogintillion (octoginta, LXXX)


• 10,273 - nonagintillion (nonaginta, XC)


• 10 303 - centillion (Centum, C)

• 10 306 - ancentillion or centunillion


• 10 309 - duocentillion or centullion


• 10 312 - trecentillion or centtrillion


• 10 315 - quattorcentillion or centquadrillion


• 10 402 - tretrigyntacentillion or centretrigyntillion

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


2. Vygodsky M.Ya. "Handbook of Elementary Mathematics". - St. Petersburg, 1994, pp. 64-65


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


4. “Interesting about physics and mathematics.” - Quantum Library. issue 50. - M.: Nauka, 1988, p. 50

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 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 like Russian Federation, United Kingdom of Great Britain and Northern Ireland, USA, Canada, Greece and Türkiye. 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.

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 seals and seals, but writes and writes 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 finite set of numbers that humanity has awarded 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!

"Short" and "long" scale

Story modern system The names of large numbers date 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 French writer and the 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:

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 one - in scientific books in 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 in Rus' it was used own system names of 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).

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 possible options 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"Mathematical Kaleidoscope", written by Hugo Dionizy Steinhaus, 1887-1972. 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- triangle, square and 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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It is known that an infinite number of numbers and only a few have their own names, because most numbers received names consisting of small numbers. Largest numbers needs to be designated somehow.

"Short" and "long" scale

Number names used today began to receive in the fifteenth century, then the Italians first used the word million, meaning " big thousand", bimillion (million squared) and trimillion (million cubed).

This system was described in his monograph by the Frenchman Nicolas Chuquet, he recommended using numerals Latin language, adding the inflection “-million” to them, so bimillion became billion, and three million became trillion, and so on.

But according to the proposed system, he called the numbers between a million and a billion “a thousand millions.” It was not comfortable to work with such a gradation and in 1549 by the Frenchman Jacques Peletier advised to name the numbers located in the indicated interval, again using Latin prefixes, while introducing another ending - “-billion”.

So 109 was called billion, 1015 - billiard, 1021 - trillion.

Gradually this system began to be used in Europe. But some scientists confused the names of the numbers, this created a paradox when the words billion and billion became synonymous. Subsequently, the United States created its own procedure for naming large numbers. According to him, the construction of names is carried out in a similar way, but only the numbers differ.

The previous system continued to be used in Great Britain, which is why it was called British, although it was originally created by the French. But already in the seventies of the last century, Great Britain also began to apply the system.

Therefore, in order to avoid confusion, the concept created by American scientists is usually called short scale, while the original French-British - long scale.

The short scale has found active use in the USA, Canada, Great Britain, Greece, Romania, and Brazil. In Russia it is also used, with only one difference - the number 109 is traditionally called a billion. But the French-British version was preferred in many other countries.

In order to denote numbers larger than a decillion, scientists decided to combine several Latin prefixes, so undecillion, quattordecillion and others were named. If you use Schuke system, then, according to it, giant numbers will receive the names “vigintillion”, “centillion” and “million” (103003), respectively, according to the long scale, such a number will receive the name “billion” (106003).

Numbers with unique names

Many numbers were named without reference to various systems and parts of words. There are a lot of these numbers, for example, this Pi", a dozen, and numbers over a million.

IN Ancient Rus' its own numerical system has been used for a long time. Hundreds of thousands were designated by the word legion, a million were called leodromes, tens of millions were ravens, hundreds of millions were called a deck. This was the “small count,” but the “great count” used the same words, only they had a different meaning, for example, leodr could mean a legion of legions (1024), and a deck could mean ten ravens (1096).

It happened that children came up with names for numbers, so the mathematician Edward Kasner gave the idea young Milton Sirotta, who proposed to name the number with a hundred zeros (10100) simply "googol". This number received the greatest publicity in the nineties of the twentieth century, when the Google search engine was named in its honor. The boy also suggested the name “googloplex,” a number with a googol of zeros.

But Claude Shannon in the middle of the twentieth century, evaluating moves in a chess game, calculated that there were 10,118 of them, now this "Shannon number".

In the ancient work of Buddhists "Jaina Sutras", written almost twenty-two centuries ago, notes the number “asankheya” (10140), which is exactly how many cosmic cycles, according to Buddhists, are necessary to achieve nirvana.

Stanley Skuse described large quantities as "first Skewes number" equal to 10108.85.1033, and the “second Skewes number” is even more impressive and equals 1010101000.

Notations

Of course, depending on the number of degrees contained in a number, it becomes problematic to record it in writing, and even in reading, error databases. Some numbers cannot be contained on several pages, so mathematicians have come up with notations to capture large numbers.

It is worth considering that they are all different, each has its own principle of fixation. Among these it is worth mentioning Steinhaus and Knuth notations.

However, most large number- “Graham number”, used Ronald Graham in 1977 when performing mathematical calculations, and this is the number G64.