What is the largest unit called? The largest numbers in mathematics

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 most big 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 titles large numbers are constructed like this: at the beginning there is a Latin ordinal number, and at the end the suffix -illion 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 (English: myriad) came into European languages ​​from ancient Egypt.

Regarding the origin of this number, there are different opinions. 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 so...

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 drawings. 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 a special 64-level special system 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 view it looks like this:

I think everything is clear, so let’s return to Graham’s number. Graham proposed so-called G-numbers:

1. G1 = 3..3, where the number of superpower arrows is 33.


2. G2 = ..3, where the number of superpower arrows is equal to G1.


3. G3 = ..3, where the number of superpower arrows is equal to G2.



4. 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

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 high 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 we're talking about about 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 G number made it onto the pages famous Book records.

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 of up to hundreds of millions was called the “small count,” and in some manuscripts the authors considered “ great score”, 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 invented 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 - 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.

Partner news

The question “What is the largest number in the world?” is, to say the least, incorrect. There are different number systems - decimal, binary and hexadecimal, as well as various categories of numbers - semi-prime and simple, the latter being divided into legal and illegal. In addition, there are Skewes numbers, Steinhouse and other mathematicians who, either as a joke or seriously, invent and present to the public such exotics as “Megiston” or “Moser”.

What is the largest number in the world in decimal system

Of the decimal system, most “non-mathematicians” are familiar with million, billion and trillion. Moreover, if Russians generally associate a million with a dollar bribe that can be carried away in a suitcase, then where to stuff a billion (not to mention a trillion) North American banknotes - most people lack imagination. However, in the theory of large numbers there are such concepts as quadrillion (ten to the fifteenth power - 1015), sextillion (1021) and octillion (1027).

In English, the most widely spoken in the world decimal system The maximum number is considered to be a decillion - 1033.

In 1938, in connection with the development of applied mathematics and the expansion of the micro- and macrocosm, professor at Columbia University (USA), Edward Kasner published in the pages of the journal Scripta Mathematica his nine-year-old nephew’s proposal to use the decimal system as the most the large number "googol" - representing ten to the hundredth power (10100), which on paper is expressed as one followed by one hundred zeros. However, they did not stop there and a few years later proposed introducing a new largest number in the world - the “googolplex”, which represents ten raised to the tenth power and again raised to the hundredth power - (1010)100, expressed by a unit, to which a googol of zeros is assigned to the right. However, for the majority of even professional mathematicians, both “googol” and “googolplex” are of purely speculative interest, and it is unlikely that they can be applied to anything in everyday practice.

Exotic numbers

What is the largest number in the world among prime numbers - those that can only be divided by themselves and one. One of the first to record the largest prime number, equal to 2,147,483,647, was the great mathematician Leonhard Euler. As of January 2016, this number is recognized as the expression calculated as 274,207,281 – 1.

Many people are interested in questions about what large numbers are called and what number is the largest in the world. With these interesting questions and we will look into this in this article.

Story

Southern and eastern Slavic peoples Alphabetical numbering was used to record numbers, and only those letters that are in the Greek alphabet. A special “title” icon was placed above the letter that designated the number. The numerical values ​​of the letters increased in the same order as the letters in the Greek alphabet (in the Slavic alphabet the order of the letters was slightly different). In Russia, Slavic numbering was preserved until the end of the 17th century, and under Peter I they switched to “Arabic numbering,” which we still use today.

The names of the numbers also changed. Thus, until the 15th century, the number “twenty” was designated as “two tens” (two tens), and then it was shortened for faster pronunciation. The number 40 was called “fourty” until the 15th century, then it was replaced by the word “forty,” which originally meant a bag containing 40 squirrel or sable skins. The name “million” appeared in Italy in 1500. It was formed by adding an augmentative suffix to the number “mille” (thousand). Later this name came to the Russian language.

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. the book “Entertaining Arithmetic” gives the names of large numbers of that time, slightly different from today: septillion (10^42), octalion (10^48), nonalion (10^54), decalion (10^60), endecalion (10^ 66), dodecalion (10^72) and it is written that “there are no further names.”

Ways to construct names for large numbers

There are 2 main ways to name large numbers:

• American system, which is used in the USA, Russia, France, Canada, Italy, Turkey, Greece, Brazil. The names of large numbers are constructed quite simply: the Latin ordinal number comes first, and the suffix “-million” is added to it at the end. An exception is the number “million”, which is the name of the number thousand (mille) and the augmentative suffix “-million”. The number of zeros in a number, which is written according to the American system, can be found out by the formula: 3x+3, where x is the Latin ordinal number
• English system most common in the world, it is used in Germany, Spain, Hungary, Poland, Czech Republic, Denmark, Sweden, Finland, Portugal. The names of numbers according to this system are constructed as follows: the suffix “-million” is added to the Latin numeral, the next number (1000 times larger) is the same Latin numeral, but the suffix “-billion” is added. The number of zeros in a number, which is written according to the English system and ends with the suffix “-million,” can be found out by the formula: 6x+3, where x is the Latin ordinal number. The number of zeros in numbers ending with the suffix “-billion” can be found using the formula: 6x+6, where x is the Latin ordinal number.

Only the word billion passed from the English system into the Russian language, which is still more correctly called as the Americans call it - billion (since the Russian language uses the American system for naming numbers).

In addition to numbers that are written according to the American or English system using Latin prefixes, non-system numbers are known that have their own names without Latin prefixes.

Proper names for large numbers

• Vigintillion (from Latin viginti - twenty) - 10 63
• Centillion (from Latin centum - one hundred) - 10,303
• Million (from Latin mille - thousand) - 10 3003

For numbers greater than a thousand, the Romans did not have their own names (all names for numbers were then composite).

Compound names of large numbers

In addition to proper names, for numbers greater than 10 33 you can get compound names by combining prefixes.

Compound names of large numbers

• 10 123 - quadragintillion
• 10 153 — quinquagintillion
• 10 183 — sexagintillion
• 10,213 - septuagintillion
• 10,243 — octogintillion
• 10,273 — nonagintillion
• 10 303 - centillion

Further names can be obtained by direct or reverse order of Latin numerals (which is correct is not known):

• 10 306 - ancentillion or centunillion
• 10 309 - duocentillion or centullion
• 10 312 - trcentillion or centtrillion
• 10 315 - quattorcentillion or centquadrillion
• 10 402 - tretrigyntacentillion or centretrigintillion

The second spelling is more consistent with the construction of numerals in Latin and avoids ambiguities (for example, in the number trcentillion, which according to the first spelling is both 10,903 and 10,312).

• 10 603 - decentillion
• 10,903 - trcentillion
• 10 1203 — quadringentillion
• 10 1503 — quingentillion
• 10 1803 - sescentillion
• 10 2103 - septingentillion
• 10 2403 - octingentillion
• 10 2703 — nongentillion
• 10 3003 - million
• 10 6003 - duo-million
• 10 9003 - three million
• 10 15003 — quinquemillillion
• 10 308760 -ion
• 10 3000003 — mimiliaillion
• 10 6000003 - duomimiliaillion

Myriad– 10,000. The name is outdated and practically not used. However, the word “myriads” is widely used, which means not certain number, but an uncountable, uncountable set of something.

Googol ( English . googol) — 10 100. The American mathematician Edward Kasner first wrote about this number in 1938 in the journal Scripta Mathematica in the article “New Names in Mathematics.” According to him, his 9-year-old nephew Milton Sirotta suggested calling the number this way. This number became publicly known thanks to the Google search engine named after it.

Asankheya(from Chinese asentsi - uncountable) - 10 1 4 0 . This number is found in the famous Buddhist treatise Jaina Sutra (100 BC). It is believed that this number is equal to the number of cosmic cycles required to achieve nirvana.

Googolplex ( English . Googolplex) — 10^10^100. This number was also invented by Edward Kasner and his nephew; it means one followed by a googol of zeros.

Skewes number (Skewes' number Sk 1) means e to the power of e to the power of e to the power of 79, that is, e^e^e^79. This number was proposed by 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. However, this number is not an integer, so it is not included in the table of large numbers.

Second Skuse number (Sk2) equals 10^10^10^10^3, that is, 10^10^10^1000. This number was introduced by J. Skuse in the same article to indicate the number up to which the Riemann hypothesis is valid.

For super-large numbers it is inconvenient to use powers, so there are several ways to write numbers - Knuth, Conway, Steinhouse notations, etc.

Hugo Steinhouse proposed writing large numbers inside geometric shapes (triangle, square and circle).

Mathematician Leo Moser improved Steinhouse's notation, proposing to draw pentagons, then hexagons, etc. after the squares. Moser also proposed a formal notation for these polygons so that the numbers could be written without drawing complex pictures.

Steinhouse came up with two new super-large numbers: Mega and Megiston. In Moser notation they are written as follows: Mega – 2, Megiston– 10. Leo Moser also proposed to call a polygon with the number of sides equal to mega – megagon, and also suggested the number “2 in Megagon” - 2. Last number known as Moser's number or just like Moser.

There are numbers larger than Moser. The largest number that has been used in a mathematical proof is number Graham(Graham's number). It was first used in 1977 to prove an estimate in Ramsey theory. This number is associated with bichromatic hypercubes and cannot be expressed without a special 64-level system of special mathematical symbols introduced by Knuth in 1976. Donald Knuth (who wrote “The Art of Programming” and created the TeX editor) came up with the concept of superpower, which he suggested writing with arrows pointing up:

In general

Graham proposed G-numbers:

The number G 63 is called the Graham number, often denoted simply G. This number is the largest known number in the world and is listed in the Guinness Book of Records.