What is the largest known number. What are large numbers called?

A child asked today: “What is the name of the largest number in the world?” Interesting question. I went online and found a detailed article in LiveJournal on the first line of Yandex. Everything is described there in detail. It turns out that there are two systems for naming numbers: English and American. And, for example, a quadrillion according to the English and American systems are completely different numbers! The largest non-composite number is Million = 10 to the 3003rd power.
As a result, the son came to a completely reasonable conclusion that it is possible to count endlessly.

Original taken from ctac in The largest number in the world

As a child, I was tormented by the question of what kind of
the largest number, and I was tormented by this stupid
a question for almost everyone. Having learned the number
million, I asked if there was a higher number
million. Billion? How about more than a billion? Trillion?
How about more than a trillion? Finally, someone smart was found
who explained to me that the question is stupid, because
it is enough just to add to itself
a large number is one, and it turns out that it
has never been the biggest since there are
the number is even greater.

And so, many years later, I decided to ask myself something else
question, namely: what is the most
a large number that has its own
Name? Fortunately, now there is an Internet and it’s puzzling
they can patient search engines that do not
they will call my questions idiotic ;-).
Actually, that's what I did, and this is the result
found out.

There are two systems for naming numbers −
American and English.

The American system is built quite
Just. 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 the numbers come out - trillion, quadrillion,
quintillion, sextillion, septillion, octillion,
nonillion and decillion. American system
used in the USA, Canada, France and Russia.
Find out the number of zeros in a number written by
American system, using a simple formula
3 x+3 (where x is a Latin numeral).

The English system of naming the most
widespread in the world. It is used, for example, in
Great Britain and Spain, as well as most
former English and Spanish colonies. Titles
numbers in this system are constructed like this: like this: to
a suffix is ​​added to the Latin numeral
-million, the next number (1000 times larger)
is built on the same principle
Latin numeral, but the suffix is ​​-billion.
That is, after a trillion in the English system
there is a trillion, and only then a quadrillion, after
followed by quadrillion, etc. So
Thus, quadrillion in English and
American systems are completely different
numbers! Find out the number of zeros in a number
written according to the English system and
ending with the suffix -illion, you can
formula 6 x+3 (where x is a Latin numeral) and
using the formula 6 x + 6 for numbers ending in
-billion

Passed from the English system to the Russian language
only the number billion (10 9), which is still
it would be more correct to call it what it is called
Americans - a billion, as we have adopted
namely the American system. But who is in our
the country is doing something according to the rules! ;-) By the way,
sometimes in Russian they use the word
trillion (you can see this for yourself,
by running a search in Google or Yandex) and it means, judging by
in total, 1000 trillion, i.e. quadrillion.

In addition to numbers written using Latin
prefixes according to the American or English system,
the so-called non-system numbers are also known,
those. numbers that have their own
names without any Latin prefixes. Such
There are several numbers, but I will tell you more about them
I'll tell you a little later.

Let's return to recording using Latin
numerals. It would seem that they can
write down numbers to infinity, but this is not
quite like that. Now I will explain why. Let's see for
beginning of what the numbers from 1 to 10 33 are called:

And now the question arises, what next. What
there behind a decillion? In principle, you can, of course,
by combining prefixes to generate such
monsters like: andecillion, duodecillion,
tredecillion, quattordecillion, quindecillion,
sexdecillion, septemdecillion, octodecillion and
newdecillion, but these will already be composite
names, but we were interested specifically
proper names for numbers. Therefore, own
names according to this system, in addition to those indicated above, more
you can only get three
- vigintillion (from lat. viginti —
twenty), centillion (from lat. centum- one hundred) and
million million (from lat. mille- thousand). More
thousands of proper names for numbers among the Romans
did not have (all numbers over a thousand they had
compound). For example, a million (1,000,000) Romans
called decies centena milia, that is, "ten hundred
thousand." And now, actually, the table:

Thus, according to a similar number system
greater than 10 3003 which would have
get your own, non-compound name
impossible! But still the numbers are higher
million are known - these are the same
non-system numbers. Let's finally talk about them.

The smallest such number is myriad
(it’s even in Dahl’s dictionary), which means
a hundred hundreds, that is, 10,000. This word, however,
outdated and practically not used, but
It's interesting that the word is widely used
"myriads", which does not mean at all
a certain number, but an innumerable, uncountable
a lot of something. It is believed that the word myriad
(eng. myriad) came to European languages ​​from ancient
Egypt.

Google(from English googol) is the number ten in
hundredth power, that is, one followed by one hundred zeros. ABOUT
"googole" was first written in 1938 in an article
"New Names in Mathematics" in the January issue of the magazine
Scripta Mathematica American mathematician Edward Kasner
(Edward Kasner). According to him, call it "googol"
a large number was suggested by his nine-year-old
nephew Milton Sirotta.
This number became generally known thanks to
the search engine named after him Google. note that
"Google" is a brand name and googol is a number.

In the famous Buddhist treatise Jaina Sutra,
dating back to 100 BC, there is a number asankheya
(from China asenzi- uncountable), equal to 10 140.
It is believed that this number is equal to the number
cosmic cycles necessary to obtain
nirvana.

Googolplex(English) googolplex) - number also
invented by Kasner with his nephew and
meaning one followed by 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 a googolplex is a number
Skewes "number" was proposed by Skewes in 1933
year (Skewes. J. London Math. Soc. 8 , 277-283, 1933.) with
proof of hypothesis
Riemann concerning prime numbers. It
means e to a degree e to a degree e V
degrees 79, that is, e e e 79. Later,
Riele (te Riele, H. J. J. "On the Sign of the Difference P(x)-Li(x)."
Math. Comput. 48 , 323-328, 1987) reduced the Skuse number to e e 27/4,
which is approximately equal to 8.185 10 370. Understandable
the point is that since the value of the Skewes number depends on
numbers e, then it is not whole, therefore
we will not consider it, otherwise we would have to
remember other non-natural numbers - number
pi, number e, Avogadro's number, etc.

But it should be noted that there is a second number
Skuse, 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 true. Sk 2
equals 10 10 10 10 3, that is, 10 10 10 1000
.

As you understand, the greater the number of degrees,
the more difficult it is to understand which number is greater.
For example, looking at the Skewes numbers, without
special calculations are almost impossible
understand which of these two numbers is greater. So
Thus, for super-large numbers use
degrees becomes uncomfortable. Moreover, you can
come up with such numbers (and they have already been invented) when
degrees of degrees just don't fit on the page.
Yes, that's on the page! They won't fit even in a book,
the size of the entire Universe! In this case it gets up
The question is how to write them down. The problem is how you
you understand, it is solvable, and mathematicians have developed
several principles for writing such numbers.
True, every mathematician who asked this question
problem I came up with my own way of recording that
led to the existence of several unrelated
with each other, ways to write numbers are
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 extra-large
numbers. He named the number - Mega, and the number is Megiston.

Mathematician Leo Moser refined the notation
Stenhouse, which was limited to what if
it was necessary to write down much larger numbers
megiston, difficulties and inconveniences arose, so
how I had to draw many circles alone
inside another. Moser suggested after squares
draw pentagons rather than circles, then
hexagons and so on. He also suggested
formal notation for these polygons,
so you can write numbers 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 suggested
call a polygon with the same number of sides
mega - megagon. And suggested the number "2 in
Megagone", that is, 2. This number became
known as Moser's number or simply
How Moser.

But Moser is not the largest number. The biggest
number ever used in
mathematical proof is
limit value known as Graham number
(Graham's number), first used in 1977
proof of one estimate in Ramsey theory. It
related to bichromatic hypercubes and not
can be expressed without special 64-level
systems of special mathematical symbols,
introduced by Knuth in 1976.

Unfortunately, the number written in Knuth notation
cannot be converted into a Moser entry.
Therefore, we will have to explain this system too. IN
In principle, there is nothing complicated about it either. Donald
Knut (yes, yes, this is the same Knut who wrote
"The Art of Programming" and created
TeX editor) came up with the concept of superpower,
which he proposed to write down with arrows,
upward:

In general it looks like this:

I think everything is clear, so let's go back to the number
Graham. Graham proposed so-called G-numbers:

The number G 63 began to be called number
Graham(it is often designated simply as G).
This number is the largest known in
number in the world and is even included in the Book of Records
Guinness". Ah, that Graham number is greater than the number
Moser.

P.S. To bring great benefit
to all mankind and to be glorified throughout the ages, I
I decided to come up with and name the biggest
number. This number will be called stasplex And
it is equal to the number G 100. Remember it and when
your children will ask what is the biggest
number in the world, tell them what this number is called stasplex.

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. Those. It turns out there is not the largest number in the world? Is this infinity?

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. An 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 according to 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, 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.

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, which 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 the diameters of the Earth) no more than 1063 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 1067 (in total a myriad of times more). Archimedes suggested the following names for the numbers:
1 myriad = 104.
1 di-myriad = myriad of myriads = 108.
1 tri-myriad = di-myriad di-myriad = 1016.
1 tetra-myriad = three-myriad three-myriad = 1032.
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 Google search engine named after it. Please note that "Google" is a brand name and googol is a number.

Edward Kasner.

On the Internet you can often find mention that Google is the largest number in the world, but this is not true...

In the famous Buddhist treatise Jaina Sutra, dating back to 100 BC, the number asankheya (from Chinese. asenzi- innumerable), equal to 10,140. It is believed that this number is equal to the number of cosmic cycles necessary 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 the proof of the Riemann hypothesis concerning prime numbers. It means e to a degree e to a degree e to the power of 79, that is eee79. 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 ee27/4, which is approximately 8.185 10370. 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). The second Skuse number was introduced by J. Skuse in the same article to designate a number for which the Riemann hypothesis does not hold. Sk2 is equal to 101010103, that is, 1010101000.

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 - 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:

• • n[k+1] = "n V n k-gons" = n[k]n.

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:

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.

So are there numbers greater than Graham's number? There is, of course, for starters there is the Graham number + 1. As for the significant number... well, there are some fiendishly complex areas of mathematics (specifically the area known as combinatorics) and computer science in which numbers even larger than the Graham number occur. But we have almost reached the limit of what can be rationally and clearly explained.

sources http://ctac.livejournal.com/23807.html
http://www.uznayvse.ru/interesting-facts/samoe-bolshoe-chislo.html
http://www.vokrugsveta.ru/quiz/310/

https://masterok.livejournal.com/4481720.html

10 to the 3003rd power

Disputes about what is the largest figure in the world are ongoing. Different calculus systems offer different options and people don’t know what to believe and which number to consider as the largest.

This question has interested scientists since the times of the Roman Empire. The biggest problem lies in the definition of what a “number” is and what a “digit” is. At one time, people for a long time considered the largest number to be a decillion, that is, 10 to the 33rd power. But, after scientists began to actively study the American and English metric systems, it was discovered that the largest number in the world is 10 to the 3003rd power - a million. People in everyday life believe that the largest number is a trillion. Moreover, this is quite formal, since after a trillion, names are simply not given, because the counting begins to be too complex. However, purely theoretically, the number of zeros can be added indefinitely. Therefore, it is almost impossible to imagine even purely visually a trillion and what follows it.

In Roman numerals

On the other hand, the definition of “number” as understood by mathematicians is a little different. A number means a sign that is universally accepted and is used to indicate a quantity expressed in a numerical equivalent. The second concept of “number” means the expression of quantitative characteristics in a convenient form through the use of numbers. It follows from this that numbers are made up of digits. It is also important that the number has symbolic properties. They are conditioned, recognizable, unchangeable. Numbers also have sign properties, but they follow from the fact that numbers consist of digits. From this we can conclude that a trillion is not a figure at all, but a number. Then what is the largest number in the world if it is not a trillion, which is a number?

The important thing is that numbers are used as components of numbers, but not only that. A number, however, is the same number if we are talking about some things, counting them from zero to nine. This system of features applies not only to the familiar Arabic numerals, but also to Roman I, V, X, L, C, D, M. These are Roman numerals. On the other hand, V I I I is a Roman numeral. In Arabic calculus it corresponds to the number eight.

In Arabic numerals

Thus, it turns out that counting units from zero to nine are considered numbers, and everything else is numbers. Hence the conclusion that the largest number in the world is nine. 9 is a sign, and a number is a simple quantitative abstraction. A trillion is a number, and not a number at all, and therefore cannot be the largest number in the world. A trillion can be called the largest number in the world, and that is purely nominally, since numbers can be counted ad infinitum. The number of digits is strictly limited - from 0 to 9.

It should also be remembered that the numerals and numbers of different numerals do not coincide, as we saw from the examples with Arabic and Roman numerals and numerals. This happens because numbers and numbers are simple concepts that are invented by man himself. Therefore, a number in one number system can easily be a number in another and vice versa.

Thus, the largest number is innumerable, because it can continue to be added indefinitely from digits. As for the numbers themselves, in the generally accepted system, 9 is considered the largest number.

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. An 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 according to 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:

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

Sometimes people who are not involved in mathematics wonder: what is the largest number? On the one hand, the answer is obvious - infinity. Bores will even clarify that “plus infinity” or “+∞” is used by mathematicians. But this answer will not convince the most corrosive, especially since this is not a natural number, but a mathematical abstraction. But having understood the issue well, they can discover a very interesting problem.

Indeed, there is no size limit in this case, but there is a limit to human imagination. Each number has a name: ten, one hundred, billion, sextillion, and so on. But where does people's imagination end?

Not to be confused with a trademark of Google Corporation, although they have a common origin. This number is written as 10100, that is, one followed by a hundred zeros. It is difficult to imagine, but it was actively used in mathematics.

It's funny that it was invented by a child - the nephew of the mathematician Edward Kasner. In 1938, my uncle entertained his younger relatives with discussions about very large numbers. To the child’s indignation, it turned out that such a wonderful number had no name, and he gave his own version. Later, my uncle inserted it into one of his books, and the term stuck.

Theoretically, a googol is a natural number, because it can be used for counting. But it’s unlikely that anyone will have the patience to count to the end. Therefore, only theoretically.

As for the name of the company Google, a common mistake has crept in here. The first investor and one of the co-founders was in a hurry when he wrote out the check and missed the letter “O”, but in order to cash it, the company had to be registered with this particular spelling.

Googolplex

This number is a derivative of googol, but is significantly larger than it. The prefix “plex” means raising ten to a power equal to the base number, so guloplex is 10 to the power of 10 to the power of 100 or 101000.

The resulting number exceeds the number of particles in the observable Universe, which is estimated to be about 1080 degrees. But this did not stop scientists from increasing the number by simply adding the prefix “plex” to it: googolplexplex, googolplexplexplex and so on. And for particularly perverted mathematicians, they invented a variant of magnification without the endless repetition of the prefix “plex” - they simply put Greek numbers in front of it: tetra (four), penta (five) and so on, up to deca (ten). The last option sounds like a googoldecaplex and means a tenfold cumulative repetition of the procedure of raising the number 10 to the power of its base. The main thing is not to imagine the result. You still won’t be able to realize it, but it’s easy to get mentally injured.

48th Mersen number

Main characters: Cooper, his computer and a new prime number

Relatively recently, about a year ago, we managed to discover the next, 48th Mersen number. It is currently the largest prime number in the world. Let us recall that prime numbers are those that are divisible without a remainder only by one and themselves. The simplest examples are 3, 5, 7, 11, 13, 17 and so on. The problem is that the further into the wilds, the less common such numbers are. But the more valuable is the discovery of each next one. For example, the new prime number consists of 17,425,170 digits if represented in the form of the decimal number system familiar to us. The previous one had about 12 million characters.

It was discovered by the American mathematician Curtis Cooper, who delighted the mathematical community with a similar record for the third time. It took 39 days of running his personal computer just to check his result and prove that this number was indeed prime.

This is what the Graham number looks like in Knuth arrow notation. It’s difficult to say how to decipher this without having a completed higher education in theoretical mathematics. It is also impossible to write it down in our usual decimal form: the observable Universe is simply not able to accommodate it. Building one degree at a time, as is the case with googolplexes, is also not a solution.

Good formula, just unclear

So why do we need this seemingly useless number? Firstly, for the curious, it was placed in the Guinness Book of Records, and this is already a lot. Secondly, it was used to solve a problem included in the Ramsey problem, which is also unclear, but sounds serious. Thirdly, this number is recognized as the largest ever used in mathematics, and not in comic proofs or intellectual games, but to solve a very specific mathematical problem.

Attention! The following information is dangerous for your mental health! By reading it, you accept responsibility for all consequences!

For those who want to test their mind and meditate on the Graham number, we can try to explain it (but only try).

Imagine 33. It's pretty easy - it turns out 3*3*3=27. What if we now raise three to this number? The result is 3 3 to the 3rd power, or 3 27. In decimal notation, this is equal to 7,625,597,484,987. A lot, but for now it can be realized.

In Knuth's arrow notation, this number can be displayed somewhat more simply - 33. But if you add only one arrow, it becomes more complicated: 33, which means 33 to the power of 33 or in power notation. If we expand to decimal notation, we get 7,625,597,484,987 7,625,597,484,987. Are you still able to follow your thoughts?

Next stage: 33= 33 33 . That is, you need to calculate this wild number from the previous action and raise it to the same power.

And 33 is only the first of 64 terms of Graham's number. To get the second one, you need to calculate the result of this mind-blowing formula and substitute the corresponding number of arrows into diagram 3(...)3. And so on, another 63 times.

I wonder if anyone other than him and a dozen other supermathematicians will be able to get to at least the middle of the sequence without going crazy?

Did you understand something? We are not. But what a thrill!

Why do we need the largest numbers? This is difficult for the average person to understand and comprehend. But with their help, a few specialists are able to introduce new technological toys to ordinary people: phones, computers, tablets. Ordinary people are also unable to understand how they work, but they are happy to use them for their entertainment. And everyone is happy: ordinary people get their toys, “supernerds” have the opportunity to continue playing their mind games.