What is the largest number

A child asked today: “What is the name of the most big number in the world?" Interesting question. I went online and on the first line of Yandex I found a detailed article in LiveJournal. 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 is 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.

Number	Latin name	Russian prefix
1	unus	an-
2	duo	duo-
3	tres	three-
4	quattuor	quadri-
5	quinque	quinti-
6	sex	sexty
7	septem	septi-
8	octo	octi-
9	novem	noni-
10	decem	deci-

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

The American system is built quite
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:

Name	Number
Unit	10 0
Ten	10 1
One hundred	10 2
Thousand	10 3
Million	10 6
Billion	10 9
Trillion	10 12
Quadrillion	10 15
Quintillion	10 18
Sextillion	10 21
Septillion	10 24
Octillion	10 27
Quintillion	10 30
Decillion	10 33

And now the question arises, what next. What
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.

Name	Number
Myriad	10 4
Google	10 100
Asankheya	10 140
Googolplex	10 10 100
Second Skewes number	10 10 10 1000
Mega	2 (in Moser notation)
Megiston	10 (in Moser notation)
Moser	2 (in Moser notation)
Graham number	G 63 (in Graham notation)
Stasplex	G 100 (in Graham notation)

The smallest such number is myriad
(it’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 the before 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
special systems 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 view 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.

Have you ever thought how many zeros there are in one million? This is a pretty simple question. What about a billion or a trillion? One followed by nine zeros (1000000000) - what is the name of the number?

A short list of numbers and their quantitative designation

Ten (1 zero).
One hundred (2 zeros).
One thousand (3 zeros).
Ten thousand (4 zeros).
One hundred thousand (5 zeros).
Million (6 zeros).
Billion (9 zeros).
Trillion (12 zeros).
Quadrillion (15 zeros).
Quintilion (18 zeros).
Sextillion (21 zeros).
Septillion (24 zeros).
Octalion (27 zeros).
Nonalion (30 zeros).
Decalion (33 zeros).

Grouping of zeros

1000000000 - what is the name of a number that has 9 zeros? This is a billion. For convenience, large numbers are usually grouped into sets of three, separated from each other by a space or punctuation marks such as a comma or period.

This is done to make the quantitative value easier to read and understand. For example, what is the name of the number 1000000000? In this form, it’s worth straining a little and doing the math. And if you write 1,000,000,000, then the task immediately becomes visually easier, since you need to count not zeros, but triples of zeros.

Numbers with a lot of zeros

The most popular are million and billion (1000000000). What is the name of a number that has 100 zeros? This is a Googol number, so called by Milton Sirotta. This is a wildly huge amount. Do you think this number is large? Then what about a googolplex, a one followed by a googol of zeros? This figure is so large that it is difficult to come up with a meaning for it. In fact, there is no need for such giants, except to count the number of atoms in the infinite Universe.

Is 1 billion a lot?

There are two measurement scales - short and long. Around the world in science and finance, 1 billion is 1,000 million. This is on a short scale. According to it, this is a number with 9 zeros.

There is also a long scale that is used in some European countries, including France, and was formerly used in the UK (until 1971), where a billion was 1 million million, that is, a one followed by 12 zeros. This gradation is also called the long-term scale. The short scale is now predominant in financial and scientific matters.

Some European languages, such as Swedish, Danish, Portuguese, Spanish, Italian, Dutch, Norwegian, Polish, German, use billion (or billion) in this system. In Russian, a number with 9 zeros is also described for the short scale of a thousand million, and a trillion is a million million. This avoids unnecessary confusion.

Conversational options

In Russian colloquial speech after the events of 1917 - the Great October revolution- and the period of hyperinflation in the early 1920s. 1 billion rubles was called “limard”. And in the dashing 1990s, a new slang expression “watermelon” appeared for a billion; a million were called “lemon.”

The word "billion" is now used internationally. This is a natural number, which is represented in the decimal system as 10 9 (one followed by 9 zeros). There is also another name - billion, which is not used in Russia and the CIS countries.

Billion = billion?

A word such as billion is used to designate a billion only in those states in which the “short scale” is adopted as a basis. These are countries like Russian Federation, United Kingdom of Great Britain and Northern Ireland, USA, Canada, Greece and Turkey. In other countries, the concept of a billion means the number 10 12, that is, one followed by 12 zeros. In countries with a “short scale”, including Russia, this figure corresponds to 1 trillion.

Such confusion appeared in France at a time when the formation of such a science as algebra was taking place. Initially, a billion had 12 zeros. However, everything changed after the appearance of the main manual on arithmetic (author Tranchan) in 1558), where a billion is already a number with 9 zeros (a thousand millions).

For several subsequent centuries, these two concepts were used on an equal basis with each other. In the mid-20th century, namely in 1948, France switched to a long scale numerical naming system. In this regard, the short scale, once borrowed from the French, is still different from the one they use today.

Historically, the United Kingdom used the long-term billion, but since 1974 official UK statistics have used the short-term scale. Since the 1950s, the short-term scale has been increasingly used in the fields of technical writing and journalism, although the long-term scale still persists.

June 17th, 2015

“I see clusters of vague numbers that are hidden there in the darkness, behind the small spot of light that the candle of reason gives. They whisper to each other; conspiring about who knows what. Perhaps they don't like us very much for capturing their little brothers in our minds. Or perhaps they simply lead a single-digit life, out there, beyond our understanding.
Douglas Ray

We continue ours. Today we have numbers...

Sooner or later, everyone is tormented by the question, what is the largest number. There are a million answers to a child's question. What's next? Trillion. And even further? In fact, the answer to the question of what are the largest numbers is simple. Just add one to the largest number, and it will no longer be the largest. This procedure can be continued indefinitely.

But if you ask the question: what is the largest number that exists, and what is its proper name?

Now we will find out everything...

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

The American system is built quite simply. All names of large numbers are constructed like this: at the beginning there is a Latin ordinal number, and at the end the suffix -million is added to it. 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 is absolutely different numbers! You can find out the number of zeros in a number written according to the English system and ending with the suffix -million, using the formula 6 x + 3 (where x is a Latin numeral) and using the formula 6 x + 6 for numbers ending in - billion.

Only the number billion (10 9) passed from the English system into the Russian language, which would still be more correct to be called as the Americans call it - billion, since we have adopted the American system. But who in our country does anything according to the rules! ;-) By the way, sometimes the word trillion is used in Russian (you can see this for yourself by running a search in Google or Yandex) and, apparently, it means 1000 trillion, i.e. quadrillion.

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

Let's return to writing using Latin numerals. It would seem that they can write down numbers to infinity, but this is not entirely true. Now I will explain why. Let's first see what the numbers from 1 to 10 33 are called:

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

Thus, according to such a system, numbers are greater than 10 3003 , which would have its own, non-compound name is impossible to obtain! But nevertheless, numbers greater than a million are known - these are the same non-systemic numbers. Let's finally talk about them.

The smallest such number is a myriad (it is even in Dahl’s dictionary), which means a hundred hundreds, that is, 10,000. This word, however, is outdated and practically not used, but it is curious that the word “myriads” is widely used, does not mean a definite number at all, but an uncountable, uncountable multitude of something. It is believed that the word myriad came into European languages from ancient Egypt.

There are different opinions about the origin of this number. Some believe that it originated in Egypt, while others believe that it was born only in Ancient Greece. Be that as it may in fact, the myriad gained fame precisely thanks to the Greeks. Myriad was the name for 10,000, but there were no names for numbers greater than ten thousand. However, in his note “Psammit” (i.e., calculus of sand), Archimedes showed how to systematically construct and name arbitrarily large numbers. In particular, placing 10,000 (myriad) grains of sand in a poppy seed, he finds that in the Universe (a ball with a diameter of a myriad of Earth diameters) there would fit (in our notation) no more than 10 63 grains of sand It is curious that modern calculations of the number of atoms in the visible Universe lead to the number 10 67 (in total a myriad of times more). Archimedes suggested the following names for the numbers:
1 myriad = 10 4 .
1 di-myriad = myriad of myriads = 10 8 .
1 tri-myriad = di-myriad di-myriad = 10 16 .
1 tetra-myriad = three-myriad three-myriad = 10 32 .
etc.

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

Edward Kasner.

On the Internet you can often find it mentioned that - but this is not true...

In the famous Buddhist treatise Jaina Sutra, dating back to 100 BC, the number asankheya (from Chinese. asenzi- uncountable), equal to 10 140. It is believed that this number is equal to the number of cosmic cycles required to achieve nirvana.

Googolplex (English) googolplex) - a number also invented by Kasner and his nephew and meaning one with a googol of zeros, that is, 10 10100 . This is how Kasner himself describes this “discovery”:

Words of wisdom are spoken by children at least as often as by scientists. The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex." A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out.
Mathematics and the Imagination(1940) by Kasner and James R. Newman.

An even larger number than the googolplex, the Skewes number, was proposed by Skewes in 1933. J. London Math. Soc. 8, 277-283, 1933.) in proving the Riemann hypothesis concerning prime numbers. It means e to a degree e to a degree e to the power of 79, that is, ee e 79 . Later, te Riele, H. J. J. "On the Sign of the Difference P(x)-Li(x)." Math. Comput. 48, 323-328, 1987) reduced the Skuse number to ee 27/4 , which is approximately equal to 8.185·10 370. It is clear that since the value of the Skuse number depends on the number e, then it is not an integer, so we will not consider it, otherwise we would have to remember other non-natural numbers - the number pi, the number e, etc.

But it should be noted that there is a second Skuse number, which in mathematics is denoted as Sk2, which is even greater than the first Skuse number (Sk1). Second Skewes number, was introduced by J. Skuse in the same article to denote a number for which the Riemann hypothesis does not hold. Sk2 equals 1010 10103 , that is 1010 101000 .

As you understand, the more degrees there are, the more difficult it is to understand which number is greater. For example, looking at Skewes numbers, without special calculations, it is almost impossible to understand which of these two numbers is larger. Thus, for super-large numbers it becomes inconvenient to use powers. Moreover, you can come up with such numbers (and they have already been invented) when the degrees of degrees simply do not fit on the page. Yes, that's on the page! They won’t fit even into a book the size of the entire Universe! In this case, the question arises of how to write them down. The problem, as you understand, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who asked about this problem came up with his own way of writing, which led to the existence of several, unrelated to each other, methods for writing numbers - these are the notations of Knuth, Conway, Steinhouse, etc.

Consider the notation of Hugo Stenhouse (H. Steinhaus. Mathematical Snapshots, 3rd edn. 1983), which is quite simple. Stein House suggested writing large numbers inside geometric shapes - triangle, square and circle:

Steinhouse came up with two new superlarge numbers. He named the number - Mega, and the number - Megiston.

Mathematician Leo Moser refined Stenhouse's notation, which was limited by the fact that if it was necessary to write down numbers much larger than a megiston, difficulties and inconveniences arose, since many circles had to be drawn one inside the other. Moser suggested that after the squares, draw not circles, but pentagons, then hexagons, and so on. He also proposed a formal notation for these polygons so that numbers could be written without drawing complex pictures. Moser notation looks like this:

Thus, according to Moser's notation, Steinhouse's mega is written as 2, and megiston as 10. In addition, Leo Moser proposed calling a polygon with the number of sides equal to mega - megagon. And he proposed the number “2 in Megagon,” that is, 2. This number became known as Moser’s number or simply as Moser.

But Moser is not the largest number. The largest number ever used in a mathematical proof is the limiting quantity known as Graham's number, first used in 1977 in the proof of an estimate in Ramsey theory. It is associated with bichromatic hypercubes and cannot be expressed without the special 64-level system of special mathematical symbols introduced by Knuth in 1976.

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

In general it looks like this:

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

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

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

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

G63 = ..3, where the number of superpower arrows is G62.

The G63 number came to be called the Graham number (it is often designated simply as G). This number is the largest known number in the world and is even listed in the Guinness Book of Records. And here

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 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.

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

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 appears asankheya(from China asenzi- uncountable), equal to 10 140. It is believed that this number is equal to the number of cosmic cycles required to achieve nirvana.

Googolplex(English) googolplex) - a number also invented by Kasner and his nephew and meaning one with a googol of zeros, that is, 10 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 a googolplex - Skewes number (Skewes" number) was proposed by Skewes in 1933 (Skewes. 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.

Steinhouse came up with two new superlarge numbers. He named the number - Mega, and the number is Megiston.

Thus, according to Moser's notation, Steinhouse's mega is written as 2, and megiston as 10. In addition, Leo Moser proposed calling a polygon with the number of sides equal to mega - megagon. And he proposed the number “2 in Megagon”, that is, 2. This number became known as Moser’s number or simply as Moser

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

In general it looks like this:

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

The number G63 began to be called Graham number(it is often designated simply as G). This number is the largest known number in the world and is even listed in the Guinness Book of Records. Well, the Graham number is greater than the Moser number.

P.S. In order to bring great benefit to all humanity and become famous throughout the centuries, I decided to come up with and name the largest number myself. This number will be called stasplex and it is equal to the number G100. Remember it, and when your children ask what is the largest number in the world, tell them that this number is called stasplex

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

It is impossible to answer this question correctly, since the number series has no upper limit. So, to any number you just need to add one to get an even larger number. Although the numbers themselves are infinite, they do not have many proper names, since most of them are content with names made up of smaller numbers. So, for example, numbers have their own names “one” and “one hundred”, and the name of the number 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 at the same time find out how large numbers mathematicians came up with.

"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 (ca. 1450 - ca. 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 Chuquet system, a number between a million and a billion did not have its own name and was simply called “a thousand millions”, similarly called “a thousand billion”, “a thousand trillion”, etc. This was not very convenient, and in 1549 the French writer and scientist Jacques Peletier du Mans (1517–1582) proposed naming such “intermediate” numbers using the same Latin prefixes, but with the ending “-billion”. So, it began to be called “billion”, - “billiard”, - “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 not “billion” or “thousand millions”, but “billion”. Soon this error quickly spread, and a paradoxical situation arose - “billion” became simultaneously synonymous with “billion” () and “million millions” ().

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 Schuquet 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 () began to be called a “billion”, () - a “trillion”, () - 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:

Number name	Short scale value	Long scale value
Million
Billion
Billion
Billiards	-
Trillion
trillion	-
Quadrillion
Quadrillion	-
Quintillion
Quintilliard	-
Sextillion
Sextillion	-
Septillion
Septilliard	-
Octillion
Octilliard	-
Quintillion
Nonilliard	-
Decillion
Decilliard	-
Vigintillion
Wigintilliard	-
Centillion
Centilliard	-
Million
Millebillion	-

The short naming scale is currently used in the USA, UK, Canada, Ireland, Australia, Brazil and Puerto Rico. Russia, Denmark, Turkey and Bulgaria also use a short scale, except that the number is called “billion” rather than “billion.” The long scale continues to be used in most other countries.

It is curious that in our country the final transition to a short scale occurred only in the second half of the 20th century. For example, Yakov Isidorovich Perelman (1882–1942) in his “Entertaining Arithmetic” mentions the parallel existence of two scales in the USSR. The short scale, according to Perelman, was used in everyday life and financial calculations, and the long scale was used in scientific books on astronomy and physics. However, now it is wrong to use a long scale in Russia, although the numbers there are large.

But let's return to the search for the largest number. After decillion, the names of numbers are obtained by combining prefixes. This produces numbers such as undecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion, novemdecillion, etc. However, these names are no longer interesting to us, since we agreed to find the largest number with its own non-composite name.

If we turn to Latin grammar, we will find that the Romans had only three non-compound names for numbers greater than ten: viginti - “twenty”, centum - “hundred” and mille - “thousand”. The Romans did not have their own names for numbers greater than a thousand. For example, a million () The Romans called it "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” (). If Russia adopted a “long scale” for naming numbers, then the largest number with its own name would be “billion” ().

However, there are names for even larger numbers.

Numbers outside the system

Some numbers have their own name, without any connection with the naming system using Latin prefixes. And there are many such numbers. You can, for example, recall the number e, the number “pi”, dozen, the number of the beast, etc. However, since we are now interested in large numbers, we will consider only those numbers with their own non-composite name that are greater than a million.

Until the 17th century, Rus' used its own system for naming numbers. Tens of thousands were called "darkness", hundreds of thousands were called "legions", millions were called "leoders", tens of millions were called "ravens", and hundreds of millions were called "decks". This count 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 () , “legion” - the darkness of those () ; "leodr" - legion of legions () , "raven" - leodr leodrov (). For some reason, “deck” in the great Slavic counting was not called “raven of ravens” () , but only ten “ravens”, that is (see table).

Number name	Meaning in "small count"	Meaning in the "great count"	Designation
Dark
Legion
Leodre
Raven (corvid)
Deck
Darkness of topics

The number 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 originated 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 of moves and on each move the player makes a choice on average from the options, which corresponds to (approximately equal to) the game options. This work became widely known and given 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 . 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 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) in his proof of the Riemann hypothesis. The first number, which later became known as the "Skuse number", is equal to the power to the power to the power of , that is, . However, the “second Skewes number” is even larger and amounts to .

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 wondered about this problem came up with his own way of writing, which led to the existence of several unrelated methods for writing large numbers - these are the notations of Knuth, Conway, Steinhaus, etc. We now have to deal with some of them.

Other notations

In 1938, the same year that nine-year-old Milton Sirotta invented the numbers googol and googolplex, a book about entertaining mathematics, A Mathematical Kaleidoscope, written by Hugo Dionizy Steinhaus (1887–1972), was published in Poland. This book became very popular, went through many editions and was translated into many languages, including English and Russian. In it, Steinhaus, discussing large numbers, offers a simple way to write them using three geometric figures- triangle, square and circle:

"in a triangle" means "",
"squared" means "in triangles"
"in a circle" means "in squares".

Explaining this method of notation, Steinhaus comes up with the number “mega”, which is equal in a circle and shows that it is equal in a “square” or in triangles. To calculate it, you need to raise it to the power of , raise the resulting number to the power of , then raise the resulting number to the power of the resulting number, and so on, raise it to the power of times. For example, a calculator in MS Windows cannot calculate due to overflow even in two triangles. This huge number is approximately .

Having determined the “mega” number, Steinhaus invites readers to independently estimate another number - “medzon”, equal in a circle. In another edition of the book, Steinhaus, instead of the medzone, suggests estimating an even larger number - “megiston”, equal 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 large 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:

"triangle" = = ;
"squared" = = "triangles" = ;
"in a pentagon" = = "in squares" = ;
"in -gon" = = "in -gon" = .

Thus, according to Moser’s notation, Steinhaus’s “mega” is written as , “medzone” as , and “megiston” as . In addition, Leo Moser proposed calling a polygon with the number of sides equal to mega - “megagon”. And suggested a number « in megagon", that is. This number became known as the Moser number or simply "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 -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.

The ordinary arithmetic operations—addition, multiplication, and exponentiation—can naturally be extended into a sequence of hyperoperators as follows.

Multiplication natural numbers can be defined through a repeated addition operation (“add copies of a number”):

For example,

Raising a number to a power can be defined as a repeated multiplication operation ("multiplying copies of a number"), and in Knuth's notation this notation looks like a single arrow pointing up:

For example,

This single up arrow was used as the degree icon in the Algol programming language.

For example,

Here and below, the expression is always evaluated from right to left, and Knuth's arrow operators (as well as the operation of exponentiation) by definition have right associativity (order from right to left). According to this definition,

This already leads to quite large numbers, but the notation system does not end there. The triple arrow operator is used to write the repeated exponentiation of the double arrow operator (also known as pentation):

Then the “quad arrow” operator:

Etc. General rule operator "-I arrow", in accordance with right associativity, continues to the right in a sequential series of operators « arrow." Symbolically, this can be written as follows,

For example:

The notation form is usually used for notation with arrows.

Some numbers are so large that even writing with Knuth's arrows becomes too cumbersome; in this case, the use of the -arrow operator is preferable (and also for descriptions with a variable number of arrows), or is equivalent to hyperoperators. But some numbers are so huge that even such a notation is insufficient. For example, Graham's number.

Using Knuth's Arrow notation, the Graham number can be written as

Where the number of arrows in each layer, starting from the top, is determined by the number in the next layer, that is, where , where the superscript of the arrow indicates the total number of arrows. In other words, it is calculated in steps: in the first step we calculate with four arrows between threes, in the second - with arrows between threes, in the third - with arrows between threes, and so on; at the end we calculate with the arrows between the triplets.

This can be written as , where , where the superscript y denotes function iterations.

If other numbers with “names” can be matched to the corresponding number of objects (for example, the number of stars in the visible part of the Universe is estimated at sextillions - , and the number of atoms that make up the globe is on the order of dodecalions), then the googol is already “virtual”, not to mention about Graham's number. The scale of the first term alone is so large that it is almost impossible to comprehend, although the notation above is relatively easy to understand. Although this is just the number of towers in this formula for , this number is already much larger than the number of Planck volumes (the smallest possible physical volume) that are contained in the observable universe (approximately). After the first member, we are expecting another member of the rapidly growing sequence.