The largest number on Google. The largest number in the world
John SommerPut zeros after any digit or multiply with tens, raised to any number greater degree... It will not seem a little. A lot will show. But the bare tapes are still not very impressive. The piling zeros in the humanities cause not so much surprise as a slight yawn. In any case, to any largest number in the world that you can imagine, you can always add one more ... And the number will come out even more.
And yet, in Russian or any other language, there are words for a very large numbers? Those that are more than a million, billion, trillion, billion? And in general, how much is a billion?
It turns out that there are two systems for naming numbers. But not Arab, Egyptian, or any other ancient civilizations, but American and English.
In the American system numbers are called like this: the Latin numeral + - illion (suffix) is taken. Thus, the numbers are obtained:
Trillion - 1,000,000,000,000 (12 zeros)
Quadrillion - 1,000,000,000,000,000 (15 zeros)
Quintillion - 1 and 18 zeros
Sextillion - 1 and 21 zero
Septillion - 1 and 24 zeros
octillion - 1 and 27 zeros
Nonillion - 1 and 30 zeros
Decillion - 1 and 33 zeros
The formula is simple: 3 x + 3 (x is a Latin numeral)
In theory, there should also be anilion numbers (unus in Latin- one) and duolion (duo - two), but, in my opinion, such names are not used at all.
English number naming system more widespread.
Here, too, a Latin numeral is taken and the suffix-million is added to it. However, the name of the next number, which is 1000 times larger than the previous one, is formed using the same Latin number and the suffix - illiard. I mean:
Trillion - 1 and 21 zero (in the American system - sextillion!)
Trillion - 1 and 24 zeros (in the American system - septillion)
Quadrillion - 1 and 27 zeros
Quadrillion - 1 and 30 zeros
Quintillion - 1 and 33 zeros
Queenilliard - 1 and 36 zeros
Sextillion - 1 and 39 zeros
Sexbillion - 1 and 42 zeros
The formulas for counting the number of zeros are as follows:
For numbers ending in - illion - 6 x + 3
For numbers ending in - illiard - 6 x + 6
As you can see, confusion is possible. But let's not be afraid!
In Russia, the American system of naming numbers is adopted. From the English system, we borrowed the name of the number "billion" - 1,000,000,000 = 10 9
And where is the "cherished" billion? - Why, a billion is a billion! American style. And we, although we use the American system, took the "billion" from the English one.
Using the Latin names of numbers and the American system, we will call the numbers:
- vigintillion- 1 and 63 zeros
- centillion- 1 and 303 zeros
- million- one and 3003 zeros! Whoa ...
But this, it turns out, is not all. There are also non-systemic numbers.
And the first one is probably myriad- one hundred hundred = 10,000
Googol(it is after him that the famous search engine is named) - one and one hundred zeros
In one of the Buddhist treatises the number asankheya- one and one hundred forty zeros!
Number name googolplex(like googol) was invented by the English mathematician Edward Kasner and his nine-year-old nephew - the unit s - dear mom! - googol zeros !!!
But that's not all ...
The mathematician Skuse named Skuse's number after himself. It means e to the extent e to the extent e to the 79th power, that is, e e e 79
And then a great difficulty arose. You can come up with names for numbers. But how to write them down? The number of degrees of degrees of degrees is already such that it simply does not disappear on the page! :)
And then some mathematicians began to write numbers in geometric shapes. And the first, they say, this method of recording was invented by the outstanding writer and thinker Daniil Ivanovich Kharms.
And yet, what is the BIGGEST NUMBER IN THE WORLD? - It is called STASPLEX and is equal to G 100,
where G is the Graham number, the most big number ever used in mathematical proofs.
This number - stasplex - came up with wonderful person, our compatriot Stas Kozlovsky, to LJ which I am addressing you :) - ctac
It is impossible to correctly answer this question, since the number series has no upper limit. So, to any number it is enough just to add one to get an even larger number. Although the numbers themselves are infinite, they do not have many names of their own, since most of them are content with names made up of smaller numbers. So, for example, numbers and have their own names "one" and "one hundred", and the name of the number is already composite ("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 is it equal to? Let's try to figure it out and at the same time find out how large the numbers were invented by mathematicians.
"Short" and "long" scale
The history of the modern naming system for large numbers dates back to the middle of the 15th century, when in Italy they began to use the words "million" (literally - a large thousand) for a thousand squared, "bimillion" for a million squared and "trillion" for a million cubed. We know about this system thanks to the French mathematician Nicolas Chuquet (c. 1450 - c. 1500): in his treatise "Science of numbers" (Triparty en la science des nombres, 1484), he developed this idea, suggesting further use of Latin cardinal numbers (see table), adding them to the ending "-million". Thus, Schuquet's “bimillion” became a billion, “trillion” into a trillion, and a million to the fourth power became “quadrillion”.
In the Schücke system, the number between a million and a billion did not have its own name and was simply called “one thousand million”, similarly it was called “one thousand billion,” “one thousand trillion,” and so on. It was not very convenient, and in 1549 french writer and the scientist Jacques Peletier du Mans (1517-1582) proposed to name such "intermediate" numbers using the same Latin prefixes, but the ending "-billion". So, it began to be called "billion" - "billiard" - "trillion", etc.
The Suke-Peletier system gradually became popular and began to be used throughout Europe. However, in the 17th century, an unexpected problem arose. It turned out that some scientists for some reason began to get confused and call the number not “billion” or “thousand million”, but “billion”. Soon this error quickly spread, and a paradoxical situation arose - “billion” became simultaneously synonymous with “billion” () and “million million” ().
This confusion lasted long enough and led to the fact that the United States created its own system of naming large numbers. According to the American system, the names of numbers are constructed in the same way as in the Schuke system - the Latin prefix and the ending “illion”. However, the magnitudes of these numbers are different. If in the Shuke system names with the ending "million" received numbers that were degrees of a million, then in the American system the ending "-million" received degrees of a thousand. That is, one thousand million () began to be called “billion”, () - “trillion”, () - “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 Schuquet and Peletier. However, in the 1970s, the UK officially switched to “ American system", Which led to the fact that calling one system American and the other British became somehow strange. As a result, the American system is now commonly referred to as the "short scale", and the British system, or the Schuke-Peletier system, as the "long scale."
In order not to get confused, let's summarize the intermediate result:
| Number name | Short scale value | Long Scale Value |
| Million | ||
| Billion | ||
| Billion | ||
| Billiard | - | |
| Trillion | ||
| Trillion | - | |
| Quadrillion | ||
| Quadrillion | - | |
| Quintillion | ||
| Quintilliard | - | |
| Sextillion | ||
| Sexbillion | - | |
| Septillion | ||
| Septilliard | - | |
| Octillion | ||
| Octilliard | - | |
| Quintillion | ||
| Nonbillion | - | |
| Decillion | ||
| Decilliard | - | |
| Vigintillion | ||
| Vigintilliard | - | |
| Centillion | ||
| Centilliard | - | |
| Million | ||
| Milliard | - |
The short naming scale is now used in the United States, United Kingdom, Canada, Ireland, Australia, Brazil and Puerto Rico. Russia, Denmark, Turkey and Bulgaria also use a short scale, except that the number is not called “billion”, but “billion”. The long scale, however, continues to be used in most other countries at the present time.
It is curious that in our country the final transition to the short scale took place 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 the long scale in Russia, although the numbers there turn out to be large.
But back to looking for the largest number. After decillion, the names of numbers are obtained by combining prefixes. This is how numbers such as undecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion, novemdecillion, etc. are obtained. However, these names are no longer interesting to us, since we agreed to find the largest number with our own non-composite name.
If we turn to Latin grammar, we find that the Romans had only three non-compound names for numbers more than ten: viginti - "twenty", centum - "one hundred" and mille - "thousand". For numbers greater than "a thousand", the Romans did not have their own names. For example, a million () the Romans called it "decies centena milia", that is, "ten times a hundred thousand." According to Schücke's rule, these three remaining Latin numerals give us names for numbers like "vigintillion", "centillion" and "milleillion".
So, we found out that on the “short scale” the maximum number that has its own name and is not a composite of the smaller numbers is “million” (). If the "long scale" of naming numbers were adopted in Russia, then the largest number with its own name would be "milliard" ().
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", a 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, which are more than a million.
Until the 17th century, Russia used its own system of naming numbers. Tens of thousands were called "darkness", hundreds of thousands - "legions", millions - "leodra", tens of millions - "crows", and hundreds of millions - "decks". This counting up to hundreds of millions was called the "little count", and in some manuscripts the authors also considered the "great count", in which the same names were used for large numbers, but with a different meaning. So, "darkness" did not mean ten thousand, but a thousand thousand () , "Legion" - the darkness of those () ; "Leodr" - legion of legions () , "Raven" - leodr leodrov (). For some reason, the "deck" in the great Slavic account was not called the "raven of ravens" () , but only ten "ravens", that is (see table).
| Number name | Meaning in "small count" | Value in the "grand score" | Designation |
| Darkness | |||
| Legion | |||
| Leodre | |||
| Raven (vran) | |||
| Deck | |||
| Darkness of themes |  |
The number also has its own name and was invented by a nine-year-old boy. And it was like this. In 1938, the American mathematician Edward Kasner (1878-1955) walked in the park with his two nephews and discussed large numbers with them. During the conversation, they talked about a number with one hundred zeros, which did not have its own name. One of the nephews, nine-year-old Milton Sirott, suggested calling the number "googol". In 1940, Edward Kasner, together with James Newman, wrote the popular science book "Mathematics and the Imagination", where he told math lovers about the number of googols. Google gained even more prominence 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 for Playing Chess," he tried to estimate the number of possible variants of a chess game. According to him, each game lasts an average of moves and on each move the player makes a choice on the average of the options, which corresponds (approximately equal) to the options of the game. This work became widely known, and this number became known as the "Shannon number".
In the famous Buddhist treatise Jaina Sutra, dating back to 100 BC, the number "asankheya" is found equal. It is believed that this number is equal to the number of cosmic cycles required to attain 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 - "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) when proving the Riemann hypothesis. The first number, which later came to be called "the first Skuse number", is equal in degree to degree in degree, that is. However, the "second Skewes number" is even larger and is.
Obviously, the more degrees there are in degrees, the more difficult it is to write numbers and understand their meaning when reading. Moreover, it is possible to come up with such numbers (and they, by the way, have already been invented), when the degrees of degrees simply do not fit on the page. Yes, what a page! They won't even fit into a book the size of the entire Universe! In this case, the question arises how to write such numbers. The problem, fortunately, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who asked this problem came up with his own way of writing, which led to the existence of several unrelated ways to write large numbers - these are the notations of Knuth, Conway, Steinhaus, etc. We now have to deal with some of them.
Other notations
In 1938, the same year that nine-year-old Milton Sirotta invented the numbers googol and googolplex, a book about entertaining mathematics, Mathematical Kaleidoscope, written by Hugo Dionizy Steinhaus (1887-1972) was published in Poland. This book has become very popular, has gone through many editions and has been translated into many languages, including English and Russian. In it, Steinhaus, discussing large numbers, offers an easy 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 way of writing, Steinhaus comes up with the number "mega" 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 a power, raise the resulting number to a power, then raise the resulting number to the power of the resulting number, and so on, raise everything to a power of times. For example, a calculator in MS Windows cannot calculate due to overflow even in two triangles. Approximately this huge number is.
Having determined the number "mega", Steinhaus invites the readers to independently estimate another number - "mezons", equal in the circle. In another edition of the book, Steinhaus, instead of Medzon, suggests evaluating an even larger number - "megiston", equal in a circle. Following Steinhaus, I will also recommend readers to temporarily break away from this text and try to write these numbers themselves using ordinary degrees in order to feel their gigantic magnitude.
However, there are names for large numbers. For example, the Canadian mathematician Leo Moser (1921-1970) refined the Steinhaus notation, which was limited by the fact that if it was required to write down the numbers many large megistones, then difficulties and inconveniences would arise, since many circles would have to be drawn one inside another. Moser suggested drawing not circles, but pentagons after the squares, then hexagons, and so on. He also proposed a formal notation for these polygons so that numbers could be written down without drawing complex drawings. Moser's notation looks like this:
"Triangle" = =;
"Squared" = = "in triangles" =;
"In a pentagon" = = "in squares" =;
"In the -gon" = = "in the -gons" =.
Thus, according to Moser's notation, the Steinhaus "mega" is written as, "mezon" as, and "megiston" as. In addition, Leo Moser proposed to call a polygon with the number of sides equal to mega - "mega-gon". And suggested the number « in the megagon ", that is. This number became known as the Moser number, or simply as "Moser".
But even the Moser is not the largest number. So, the largest number ever used in a 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 dimensions of certain -dimensional bichromatic hypercubes. But Graham's number gained fame only after the story about him in Martin Gardner's book "From Penrose Mosaics to Reliable Ciphers", published in 1989.
To explain how large the Graham 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 superdegree, which he proposed to write down with arrows pointing up.
The usual arithmetic operations - addition, multiplication, and exponentiation - can naturally be extended into a sequence of hyperoperators as follows.
Multiplication of 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 repetitive multiplication operation (“multiply 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 a degree icon in the Algol programming language.
For example,
Hereinafter, the expression is always evaluated from right to left, and Knuth's arrow operators (like the exponentiation operation), by definition, have right associativity (order from right to left). According to this definition,
This already leads to quite large numbers, but the notation 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 operator "quadruple arrow":
Etc. General rule operator "-I am arrow ", in accordance with the 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 writing 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 preferred (and also for descriptions with a variable number of arrows), or equivalently, to hyperoperators. But some numbers are so huge that even such a record is not sufficient. For example, Graham's number.
When using Knuth Arrow notation, Graham's 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 shows the total number of arrows. In other words, it is calculated in steps: in the first step, we calculate with four arrows between the triplets, in the second - with arrows between the triplets, in the third - with arrows between the triplets, and so on; at the end we calculate from the arrows between the triplets.
It can be written as, where, where the superscript y means iterating over the functions.
If other numbers with "names" can be matched with the corresponding number of objects (for example, the number of stars in the visible part of the Universe is estimated in sextillons -, and the number of atoms that make up the globe is of the order of dodecalions), then the googol is already "virtual", not to mention about Graham's number. The scale of only the first term is so great that it is almost impossible to grasp it, although the entry above is relatively easy to understand. Although this is just the number of towers in this formula for, this number is already a lot more quantity Planck volumes (the smallest possible physical volume) that are contained in the observable universe (approximately). After the first member, another member of the rapidly growing sequence awaits us.
A child today asked: "What is the name of the largest number in the world?" An interesting question. I went online and on the first line of Yandex I found a detailed article in LiveJournal. Everything is detailed there. It turns out that there are two systems for naming numbers: English and American. And, for example, a quadrillion in the English and American systems are completely different numbers! The largest non-composite number is
Million = 10 to the 3003 power.
As a result, the son came to a completely reasonable input that you can count infinitely.
Original taken from ctac c The largest number in the world
As a child, I was tormented by the question of what kind of
the largest number, and I've been plagued with these stupid
the question of almost everyone. Finding out the number
million, I asked, is there a number greater
million. Billion? And more than a billion? Trillion?
More than a trillion? Finally, someone smart was found
who explained to me that the question is stupid, since
just add to yourself
large number one, and it turns out that it
has never been the biggest since exist
the number is even greater.
And so, after many years, I decided to ask myself another
a question, namely: what is the most
a large number that has its own
title? Fortunately, now there is an Internet and to puzzle
they can have patient search engines that do not
will call my questions idiotic ;-).
Actually, this is what I did, and this is what the result is
found out.
| Number | Latin name | Russian prefix |
| 1 | unus | an- |
| 2 | duo | duo- |
| 3 | tres | three- |
| 4 | quattuor | quadri- |
| 5 | quinque | quinti- |
| 6 | sex | sex- |
| 7 | septem | septi- |
| 8 | octo | oct- |
| 9 | novem | non- |
| 10 | decem | deci- |
There are two systems for naming numbers -
American and English.
The American system is pretty
simply. All names for large numbers are constructed like this:
at the beginning there is a Latin ordinal number,
and the suffix-million is added to it at the end.
The exception is the name "million"
which is the name of the number thousand (lat. mille)
and the increasing suffix-million (see table).
This is how the numbers turn 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 the number written by
the American system, you can use a simple formula
3 x + 3 (where x is a Latin numeral).
The English naming system is most
widespread in the world. It is used, for example, in
Great Britain and Spain, as well as in most
former English and Spanish colonies. Names
numbers in this system are constructed like this: so: to
suffix is added to the Latin numeral
-million, next number (1000 times more)
is built on the principle - the same
Latin numeral, but the suffix is -billion.
That is, after trillion in the English system
there is a trillion, and only then a quadrillion, for
followed by a quadrillion, etc. So
way, a quadrillion in English and
American systems are completely different
numbers! Find out the number of zeros in a number,
written in the English system and
ending with the suffix-million, you can use
the formula 6 x + 3 (where x is a Latin numeral) and
by the formula 6 x + 6 for numbers ending in
-billion.
From the English system to the Russian language passed
only the number billion (10 9), which is still
it would be more correct to call it as it is called
Americans - by a billion, since we have adopted
it is the American system. But who do we have in
country does something according to the rules! ;-) By the way,
sometimes in Russian they also use the word
trillion (you can see for yourself,
by running a search in Google or Yandex) and means it, judging by
everything, 1000 trillion, i.e. quadrillion.
In addition to numbers written using Latin
prefixes according to the American or English system,
so-called off-system numbers are also known,
those. numbers that have their own
names without any Latin prefixes. Of such
there are several numbers, but more about them I
I'll tell you a little later.
Let's go back to recording using Latin
numerals. It would seem that they can
write down numbers ad infinitum, but this is not
quite so. Let me explain why. Let's see for
beginnings as the numbers from 1 to 10 33 are called:
| Name | Number |
| Unit | 10 0 |
| Ten | 10 1 |
| 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 so, now the question arises, what's next. What
there behind the decillion? In principle, you can, of course,
by combining prefixes, generate such
monsters like: andecillion, duodecillion,
tredecillion, quattordecillion, quindecillion,
sexdecillion, septemdecillion, octodecillion and
novemdecillion, but these will already be composite
names, but we were interested in exactly
own names of numbers. Therefore, their own
names on this system, in addition to the above, more
you can get only three
- vigintillion (from lat. viginti —
twenty), centillion (from lat. centum- one hundred) and
million (from lat. mille- thousand). More
thousands of proper names for numbers among the Romans
was not available (all numbers over a thousand they had
composite). For example, a million (1,000,000) Romans
called decies centena milia, that is, "ten hundred
thousand ". And now, in fact, the table:
Thus, according to a similar system, the numbers
more than 10 3003, which would have
get your own, non-compound name
impossible! But nevertheless the numbers are greater
million are known - these are the same
off-system numbers. Let's finally tell you about them.
| Name | Number |
| Myriad | 10 4 |
| Googol | 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's number | G 63 (in Graham notation) |
| Stasplex | G 100 (in Graham notation) |
The smallest such number is myriad
(it is even in Dahl's dictionary), which means
a hundred hundred, that is - 10,000. This word, really,
deprecated and practically not used, but
curious that the word is widely used
"myriad", which means not at all
a certain number, but an uncountable, uncountable
a lot of something. It is believed that the word myriad
(English myriad) came to European languages from the ancient
Egypt.
Googol(from the English googol) is the number ten in
hundredth degree, that is, one followed by one hundred zeros. O
"googole" was first written in 1938 in the article
"New Names in Mathematics" in the January issue of the magazine
Scripta Mathematica American mathematician Edward Kasner
(Edward Kasner). According to him, to call it "googol"
a large number suggested his nine year old
nephew of Milton Sirotta.
This number became well-known thanks to,
named after him, the search engine Google... note that
Google is a trademark and googol is a number.
In the famous Buddhist treatise Jaina Sutras,
dating back to 100 BC, there is a number asankheya
(from whale. asenci- uncountable) equal to 10 140.
It is believed that this number is equal to the number
cosmic cycles necessary to gain
nirvana.
Googolplex(eng. googolplex) is a number also
invented by Kasner with his nephew and
meaning one with a googol of zeros, that is, 10 10 100.
This is how Kasner himself describes this "discovery":
Words of wisdom are spoken by children at least as often as by scientists. The name
"googol" was invented by a child (Dr. Kasner "s nine-year-old nephew) who was
asked to think up a name for a very big number, namely, 1 with a hundred zeros after it.
He was very certain that this number was not infinite, and therefore equally certain that
it had to have a name. At the same time that he suggested "googol" he gave a
name for a still larger number: "Googolplex." A googolplex is much larger than a
googol, but is still finite, as the inventor of the name was quick to point out.
Mathematics and the Imagination(1940) by Kasner and James R.
Newman.
Even more than a googolplex number is a number
Skewes "number was proposed by Skewes in 1933.
year (Skewes. J. London Math. Soc. 8
, 277-283, 1933.) at
proof of hypothesis
Riemann on prime numbers. It
means e to the extent e to the extent e v
degree 79, that is, e e e 79. Later,
Riel (te Riele, H. J. J. "On the Sign of the Difference NS(x) -Li (x). "
Math. Comput. 48
, 323-328, 1987) reduced the Skewes number to e e 27/4,
which is approximately equal to 8.185 · 10 370. Understandable
the point is that since the value of Skuse's number depends on
the numbers e, then it is not whole, therefore
we will not consider it, otherwise we would have to
remember other unnatural 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 a number, up to
which the Riemann hypothesis is valid. Sk 2
is equal to 10 10 10 10 3, i.e. 10 10 10 1000
.
As you understand, the more in the number of degrees,
the more difficult it is to understand which of the numbers is larger.
For example, looking at the Skuse numbers, without
special calculations are almost impossible
understand which of these two numbers is greater. So
way, for very large numbers, use
degrees it 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, what a page! They will not fit, even in a book,
the size of the entire universe! In this case, it rises
the question is how to write them down. The problem is how you
you understand is solvable, and mathematicians have developed
a few guidelines for writing such numbers.
True, every mathematician who asked this
problem 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 by Knuth, Conway, Steinhouse, etc.
Consider the notation of Hugo Steinhaus (H. Steinhaus. Mathematical
Snapshots, 3rd edn. 1983), which is pretty simple. Stein
hauz suggested recording large numbers inside
geometric shapes - triangle, square and
circle:
Steinhaus came up with two new extra-large
numbers. He called the number - Mega and the number is Megiston.
Mathematician Leo Moser refined the notation
Stenhouse, which was limited by the fact that if
it was required to write many more numbers
megiston, difficulties and inconveniences arose, so
how I had to draw many circles one
inside the other. Moser suggested after squares
draw pentagons instead of circles, then
hexagons and so on. He also suggested
a formal notation for these polygons,
so that you can write numbers without drawing
complex drawings. Moser's notation looks like this:
Thus, according to Moser's notation
Steinhouse mega is written as 2, and
megiston as 10. In addition, Leo Moser suggested
call a polygon with the number of sides equal
mege - mega-gon. And he suggested the number "2 in
Megagon ", that is 2. This number became
known as the Moser "s number" or simply
how moser.
But Moser is not the largest number either. The biggest
number ever used in
mathematical proof is
limiting value known as Graham's number
(Graham "s number), first used in 1977 in
proof of one estimate in Ramsey theory. It
associated with bichromatic hypercubes and not
can be expressed without much 64-level
systems of special mathematical symbols,
introduced by Knuth in 1976.
Unfortunately, the number written in Knuth notation
cannot be translated into a Moser record.
Therefore, we will have to explain this system as well. V
in principle, there is nothing complicated in 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 superdegree,
which he suggested to write down with arrows,
upward:
V general view it looks like this:
I think everything is clear, so back to the number
Graham. Graham proposed the so-called G-numbers:
The number G 63 became known as number
Graham(it is often denoted simply as G).
This number is the largest known in
the world in number and is entered even in the "Book of Records
Guinness. "Oh, here's that Graham's number is greater than the number
Moser.
P.S. To be of great benefit
to all mankind and become famous for centuries, I
decided to come up with and name the biggest one
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
world number, tell them this number is called stasplex.
Once I read a tragic story, which tells about the Chukchi, whom polar explorers taught to count and write numbers. The magic of numbers amazed him so much that he decided to write down absolutely all the numbers in the world in a row, starting with one, in the notebook donated by the polar explorers. The Chukchi abandons all his affairs, stops communicating even with his own wife, no longer hunts for seals and seals, but writes everything and writes numbers in a notebook .... So a year goes by. In the end, the notebook ends and the Chukchi understands that he was able to write down only a small part of all the numbers. He cries bitterly and, in despair, burns down his scribbled notebook in order to start living the simple life of a fisherman again, no longer thinking about the mysterious infinity of numbers ...
We will not repeat the feat of this Chukchi and try to find the largest number, since any number only needs to add one to get an even larger number. Let us ask ourselves, albeit similar, but a different question: which of the numbers that have their own name is the largest?
Obviously, although the numbers themselves are infinite, they do not have so many proper names, since most of them are content with names made up of smaller numbers. So, for example, the numbers 1 and 100 have their own names "one" and "one hundred", and the name of the number 101 is already compound ("one hundred and one"). It is clear that in the finite set of numbers that humanity has awarded with its own name, there must be some largest number. But what is it called and what is it equal to? Let's try to figure it out and find, in the end, this is the largest number!
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"Short" and "long" scale
The history of the modern naming system for large numbers dates back to the middle of the 15th century, when in Italy they began to use the words "million" (literally - a large thousand) for a thousand squared, "bimillion" for a million squared and "trillion" for a million cubed. We know about this system thanks to the French mathematician Nicolas Chuquet (c. 1450 - c. 1500): in his treatise "Science of numbers" (Triparty en la science des nombres, 1484), he developed this idea, suggesting further use of Latin cardinal numbers (see table), adding them to the ending "-million". Thus, Schuquet's “bimillion” became a billion, “trillion” into a trillion, and a million to the fourth power became “quadrillion”.
In the Schücke system, the number 10 9, which was between a million and a billion, did not have its own name and was simply called “one thousand million”, similarly, 10 15 was called “one thousand billion,” 10 21, “one thousand trillion,” and so on. It was not very convenient, and in 1549 the French writer and scientist Jacques Peletier du Mans (1517-1582) proposed to name such “intermediate” numbers using the same Latin prefixes, but the ending “-billion”. So, 10 9 began to be called “billion”, 10 15 - “billiard”, 10 21 - “trillion”, etc.
The Suke-Peletier system gradually became popular and began to be used throughout Europe. However, in the 17th century, an unexpected problem arose. It turned out that some scientists for some reason began to get confused and call the number 10 9 not “a billion” or “a thousand million”, but “a billion”. Soon, this mistake quickly spread, and a paradoxical situation arose - “billion” became simultaneously synonymous with “billion” (10 9) and “million million” (10 18).
This confusion lasted long enough and led to the fact that the United States created its own system of naming large numbers. According to the American system, the names of numbers are constructed in the same way as in the Schuke system - the Latin prefix and the ending “illion”. However, the magnitudes of these numbers are different. If in the Shuke system names with the ending "million" received numbers that were degrees of a million, then in the American system the ending "-million" received degrees of a thousand. That is, one thousand million (1000 3 = 10 9) began to be called “billion”, 1000 4 (10 12) - “trillion”, 1000 5 (10 15) - “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 Schuquet and Peletier. However, in the 1970s, Great Britain officially switched to the "American system", which led to the fact that it became somewhat strange to call one system American and the other British. As a result, the American system is now commonly referred to as the "short scale", and the British system, or the Schuke-Peletier system, as the "long scale."
In order not to get confused, let's summarize the intermediate result:
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The short naming scale is now used in the United States, United Kingdom, Canada, Ireland, Australia, Brazil and Puerto Rico. Russia, Denmark, Turkey and Bulgaria also use a short scale, except that the number 10 9 is not called “billion”, but “billion”. The long scale, however, continues to be used in most other countries at the present time.
It is curious that in our country the final transition to the short scale took place only in the second half of the 20th century. For example, even 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 the long scale in Russia, although the numbers there turn out to be large.
But back to looking for the largest number. After decillion, the names of numbers are obtained by combining prefixes. This is how numbers such as undecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion, novemdecillion, etc. are obtained. However, these names are no longer interesting to us, since we agreed to find the largest number with our own non-composite name.
If we turn to Latin grammar, we find that the Romans had only three non-compound names for numbers more than ten: viginti - "twenty", centum - "one hundred" and mille - "thousand". For numbers greater than "a thousand", the Romans did not have their own names. For example, the Romans called a million (1,000,000) "decies centena milia", that is, "ten times a hundred thousand." According to Schücke's rule, these three remaining Latin numerals give us names for numbers like "vigintillion", "centillion" and "milleillion".
So, we found out that on the “short scale” the maximum number that has its own name and is not a composite of the smaller numbers is “million” (10 3003). If the "long scale" of naming numbers was adopted in Russia, then the largest number with its own name would be "milliard" (10 6003).
However, there are names for even larger numbers.
Numbers outside the system
Some numbers have their own name, without any connection with the naming system using Latin prefixes. And there are many such numbers. You can, for example, remember the number e, the number "pi", a 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, which are more than a million.
Until the 17th century, Russia used its own system of naming numbers. Tens of thousands were called "darkness", hundreds of thousands - "legions", millions - "leodra", tens of millions - "crows", and hundreds of millions - "decks". This counting up to hundreds of millions was called the "little count", and in some manuscripts the authors also considered the "great count", in which the same names were used for large numbers, but with a different meaning. So, "darkness" meant not ten thousand, but a thousand thousand (10 6), "legion" - the darkness of those (10 12); "Leodr" - legion of legions (10 24), "raven" - leodr leodr (10 48). For some reason, the “deck” in the great Slavic account was called not “ravens of ravens” (10 96), but only ten “ravens”, that is, 10 49 (see table).
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The number 10 100 also has its own name and was invented by a nine-year-old boy. And it was like this. In 1938, the American mathematician Edward Kasner (1878-1955) walked in the park with his two nephews and discussed large numbers with them. During the conversation, they talked about a number with one hundred zeros, which did not have its own name. One of the nephews, nine-year-old Milton Sirott, suggested calling the number "googol". In 1940, Edward Kasner, together with James Newman, wrote the popular science book "Mathematics and the Imagination", where he told math lovers about the number of googols. Google gained even more prominence 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 for Playing Chess," he tried to estimate the number of possible variants of a chess game. According to him, each game lasts on average 40 moves and on each move the player makes a choice on average out of 30 options, which corresponds to 900 40 (approximately equal to 10 118) options for the game. This work became widely known, and this number became known as the "Shannon number".
In the famous Buddhist treatise Jaina Sutra, dating back to 100 BC, the number "asankheya" is found equal to 10 140. It is believed that this number is equal to the number of cosmic cycles required to attain nirvana.
Nine-year-old Milton Sirotta went down in the history of mathematics not only for inventing the number of googol, but also for the fact that at the same time he proposed another number - googolplex, which is equal to 10 to the power of googol, that is, one with googol of zeros.
Two more numbers, larger than the googolplex, were proposed by the South African mathematician Stanley Skewes (1899-1988) when proving the Riemann hypothesis. The first number, which later became known as the "first Skuse number", is e to the extent e to the extent e to the 79th power, that is e e e 79 = 10 10 8.85.10 33. However, the "second Skewes number" is even larger and amounts to 10 10 10 1000.
Obviously, the more degrees there are in degrees, the more difficult it is to write numbers and understand their meaning when reading. Moreover, it is possible to come up with such numbers (and they, by the way, have already been invented), when the degrees of degrees simply do not fit on the page. Yes, what a page! They won't even fit into a book the size of the entire Universe! In this case, the question arises how to write such numbers. The problem, fortunately, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who asked this problem came up with his own way of writing, which led to the existence of several unrelated ways to write large numbers - these are the notations of Knuth, Conway, Steinhaus, etc. We now have to deal with some of them.
Other notations
In 1938, the same year that nine-year-old Milton Sirotta invented the numbers googol and googolplex, a book about entertaining mathematics, Mathematical Kaleidoscope, written by Hugo Dionizy Steinhaus (1887-1972) was published in Poland. This book has become very popular, has gone through many editions and has been translated into many languages, including English and Russian. In it, Steinhaus, discussing large numbers, offers a simple way to write them, using three geometric shapes - a triangle, a square and a circle:
"N in a triangle "means" n n»,
« n squared "means" n v n triangles ",
« n in a circle "means" n v n squares ".
Explaining this way of writing, Steinhaus comes up with the number "mega" equal to 2 in a circle and shows that it is equal to 256 in a "square" or 256 in 256 triangles. To calculate it, you need to raise 256 to the power of 256, raise the resulting number 3.2.10 616 to the power of 3.2.10 616, then raise the resulting number to the power of the resulting number, and so on, raise the total to the power of 256 times. For example, a calculator in MS Windows cannot calculate because of overflow 256 even in two triangles. Approximately this huge number is 10 10 2.10 619.
Having determined the number "mega", Steinhaus invites readers to independently estimate another number - "mezons", equal to 3 in a circle. In another edition of the book, Steinhaus, instead of Medzon, proposes to estimate an even higher number - "megiston", equal to 10 in a circle. Following Steinhaus, I will also recommend readers to temporarily break away from this text and try to write these numbers themselves using ordinary degrees in order to feel their gigantic magnitude.
However, there are names for b O higher numbers. So, the Canadian mathematician Leo Moser (Leo Moser, 1921-1970) modified the Steinhaus notation, which was limited by the fact that if it was required to write down the numbers many large megistones, then difficulties and inconveniences would arise, since many circles would have to be drawn one inside another. Moser suggested drawing not circles, but pentagons after the squares, then hexagons, and so on. He also proposed a formal notation for these polygons so that numbers could be written down without drawing complex drawings. Moser's notation looks like this:
« n triangle "= n n = n;
« n squared "= n = « n v n triangles "= nn;
« n in a pentagon "= n = « n v n squares "= nn;
« n v k + 1-gon "= n[k+1] = " n v n k-gons "= n[k]n.
Thus, according to Moser's notation, the Steinhaus “mega” is written as 2, the “mezon” as 3, and the “megiston” as 10. In addition, Leo Moser suggested calling a polygon with the number of sides equal to mega - “mega-gon”. And he proposed the number "2 in mega", that is, 2. This number became known as Moser's number or simply as "Moser".
But even the Moser is not the largest number. So, the largest number ever used in a 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 dimensions of certain n-dimensional bichromatic hypercubes. But Graham's number gained fame only after the story about him in Martin Gardner's book "From Penrose Mosaics to Reliable Ciphers", published in 1989.
To explain how large the Graham 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 superdegree, which he proposed to write down with arrows pointing up:
I think everything is clear, so let's go back to Graham's number. Ronald Graham proposed the so-called G-numbers:
Here is the number G 64 and is called the Graham number (it is often denoted simply as G). This number is the largest known number in the world used in mathematical proof, and even entered the Guinness Book of Records.
And finally
Having written this article, I can't help but be tempted to come up with my own number. Let this number be called " stasplex"And will be equal to the number G 100. Memorize it, and when your children ask what is the largest number in the world, tell them that this number is called stasplex.
Partners news
The question "What is the largest number in the world?" Is, to say the least, incorrect. Exist as various systems calculus - decimal, binary and hexadecimal, and various categories of numbers - semi-simple and simple, and the latter are divided into legal and illegal. In addition, there are the Skewes "number", Steinhouse and other mathematicians who, either jokingly or seriously, invent and publish such exotics as "megiston" or "moser" to the public.
What is the largest decimal number in the world
Of the decimal system, most "non-mathematicians" are well aware of the million, billion and trillion. Moreover, if Russians associate a million with a dollar bribe that can be carried away in a suitcase, then where to shove a billion (not to mention a trillion) North American banknotes - the majority do not have enough imagination. However, in the theory of large numbers, there are concepts such as quadrillion (ten to the fifteenth power - 1015), sextillion (1021) and octillion (1027).
In the English decimal system, the most widely used decimal system in the world, the decimal is considered the maximum number - 1033.
In 1938, in connection with the development of applied mathematics and the expansion of the micro- and macrocosm, a professor at Columbia University (USA), Edward Kasner, published on the pages of the journal "Scripta Mathematica" the proposal of his nine-year-old nephew to use the decimal system of a large number of "googol" ("googol") - representing ten to the hundredth power (10100), which on paper is expressed as one with one hundred zeros. However, they did not stop there and after a few years proposed to introduce into circulation a new largest number in the world - "googolplex", which is ten, raised to the tenth power and once again raised to the hundredth power - (1010) 100, expressed by a unit to which a googol of zeros is assigned to the right. However, for the majority of even professional mathematicians, both "googol" and "googolplex" are of purely speculative interest, and they can hardly be applied to anything in everyday practice.
Exotic numbers
What is the largest number in the world among prime numbers - those that can only be divisible by themselves and by one. One of the first to fix the largest prime number, 2,147,483,647, was the great mathematician Leonard Euler. As of January 2016, this number is recognized as an expression calculated as 274 207 281 - 1.
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