The largest amount. The biggest numbers in mathematics

Once in childhood, we learned to count to ten, then to a hundred, then to a thousand. So what is the most big number you know? A thousand, a million, a billion, a trillion ... And then? Petallion, someone will say, will be wrong, because he confuses the prefix SI with a completely different concept.

In fact, the question is not as simple as it seems at first glance. First, we are talking about naming the names of the degrees of a thousand. And here, the first nuance that many know from American films - they call our billion a billion.

Further more, there are two types of scales - long and short. In our country, a short scale is used. On this scale, at each step, the mantisa increases by three orders of magnitude, i.e. multiply by a thousand - thousand 10 3, million 10 6, billion / billion 10 9, trillion (10 12). On a long scale, after a billion 10 9, there is a billion 10 12, and then the mantisa already increases by six orders of magnitude, and the next number, which is called a trillion, already denotes 10 18.

But back to our native scale. Want to know what's coming after the trillion? Please:

10 3 thousand
10 6 million
10 9 billion
10 12 trillion
10 15 quadrillion
10 18 quintillion
10 21 sextillion
10 24 septillion
10 27 octillion
10 30 nonillion
10 33 decillion
10 36 undecillion
10 39 dodecillion
10 42 tredecillion
10 45 quattuorddecillion
10 48 quindecillion
10 51 cedecillion
10 54 seventh decillion
10 57 duodevigintillion
10 60 undevigintillion
10 63 vigintillion
10 66 anvigintillion
10 69 duovigintillion
10 72 trevigintillion
10 75 quattorvigintillion
10 78 quinvigintillion
10 81 sexwigintillion
10 84 septemvigintillion
10 87 octovigintillion
10 90 novemvigintillion
10 93 trigintillion
10 96 antrigintillion

At this number, our short scale does not hold up, and in the future, the mantisa increases progressively.

10 100 googol
10 123 quadragintillion
10,153 quinquagintillion
10 183 sexagintillion
10,213 septuagintillion
10,243 octogintillion
10,273 nonagintillion
10,303 centillion
10,306 centunillion
10,309 centduollion
10 312 cent trillion
10,315 cents quadrillion
10 402 centretrigintillion
10 603 ducentillion
10,903 trecentillion
10 1203 quadringentillion
10 1503 quingentillion
10 1803 sescentillion
10 2103 septingentillion
10 2403 oxtingentillion
10 2703 nongentillion
10 3003 million
10 6003 duomillion
10 9003 tremillion
10 3000003 Million
10 6000003 duomiliamilillion
10 10 100 googolplex
10 3 × n + 3 zillion

Googol(from the English googol) - a number in decimal notation represented by one with 100 zeros:
10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
In 1938, the American mathematician Edward Kasner (1878-1955) walked in the park with his two nephews and discussed with them large numbers... 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 Sirotta, suggested calling the number "googol". In 1940, Edward Kasner, together with James Newman, wrote the popular science book "Mathematics and Imagination" ("New Names in Mathematics"), where he told the lovers of mathematics about the number of googols.
The term "googol" has no serious theoretical or practical meaning. Kasner proposed it in order to illustrate the difference between an unimaginably large number and infinity, and for this purpose the term is sometimes used in teaching mathematics.

Googolplex(from the English googolplex) - a number represented by one with a googol of zeros. Like googol, the term googolplex was coined by the American mathematician Edward Kasner and his nephew Milton Sirotta.
The number of googol is greater than the number of all particles in the known part of the universe, which ranges from 1079 to 1081. Thus, the number of googolplex, consisting of (googol + 1) digits, cannot be written in the classical "decimal" form, even if all matter in the known turn parts of the universe into paper and ink or into computer disk space.

Zillion(eng. zillion) is a common name for very large numbers.

This term does not have a strict mathematical definition. In 1996, Conway (eng. J. H. Conway) and Guy (eng. R. K. Guy) in their book eng. The Book of Numbers defined the nth power zillion as 10 3 × n + 3 for the short scale naming system.

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 with nine zeros (1,000,000,000) - what is the name of the number?

A short list of numbers and their quantitative designation

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

Grouping zeros

1,000,000,000 - what is the name of a number that has 9 zeros? This is a billion. For convenience, it is customary to group large numbers into three sets, separated from each other by a space or punctuation marks such as a comma or period.

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

Numbers with very many zeros

The most popular are Million and Billion (1,000,000,000). What is the name of a number with 100 zeros? This is the googol figure, also called 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 an infinite universe.

Is 1 billion a lot?

There are two scales of measurement - short and long. Worldwide in the field of 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 in France, and was previously used in Great Britain (until 1971), where a billion was 1 million million, that is, one and 12 zeros. This gradation is also called the long-term scale. The short scale is now dominant in financial and scientific matters.

Some European languages such as Swedish, Danish, Portuguese, Spanish, Italian, Dutch, Norwegian, Polish, German use a billion (or a billion) names 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 was called “lemon”.

The word “billion” is now used internationally. it natural number, which is displayed in decimal as 10 9 (one and 9 zeros). There is also another name - billion, which is not used in Russia and the CIS countries.

Billion = Billion?

Such a word as billion is used to designate a billion only in those states in which the "short scale" is taken as the basis. These are countries like Russian Federation, United Kingdom of Great Britain and Northern Ireland, USA, Canada, Greece and Turkey. In other countries, the term billion means the number 10 12, that is, one and 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, the billion had 12 zeros. However, everything changed after the appearance of the main textbook on arithmetic (by Tranchan) in 1558), where a billion is already a number with 9 zeros (one thousand million).

For the next several centuries, these two concepts were used on an equal basis with each other. In the middle of the 20th century, namely in 1948, France switched to a long-scale number 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 has used a long-term billion, but since 1974, UK official statistics have used a 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 persisted.

Countless different numbers surround us every day. Surely many people have asked at least once what number is considered the largest. You can simply tell a child that this is a million, but adults are well aware that other numbers follow a million. For example, it is only necessary to add one to the number each time, and it will become more and more - this happens ad infinitum. But if you take apart the numbers that have names, you can find out what the largest number in the world is called.

The emergence of the names of numbers: what methods are used?

Today there are 2 systems according to which numbers are given names - American and English. The first is fairly simple, while the second is the most common around the world. American allows you to give names to large numbers like this: first, the ordinal in Latin is indicated, and then the suffix "illion" is added (the exception here is a million, meaning a thousand). This system is used by Americans, French, Canadians, and it is also used in our country.

English is widely used in England and Spain. According to it, the numbers are named as follows: the numeral in Latin is "plus" with the suffix "illion", and the next (a thousand times larger) number is "plus" "illiard". For example, first comes a trillion, followed by a trillion, followed by a quadrillion, and so on.

So, the same number in different systems can mean different things, for example, the American billion in the English system is called a billion.

Off-system numbers

In addition to numbers that are written according to known systems (above), there are also non-systemic ones. They have their own names, which do not include Latin prefixes.

You can start considering them with a number called a myriad. It is defined as one hundred hundreds (10000). But for its intended purpose, this word is not used, but is used as an indication of the innumerable. Even Dahl's dictionary will kindly provide a definition of such a number.

The next after the myriad is googol, denoting 10 to the power of 100. This name was first used in 1938 - by a mathematician from America E. Kasner, who noted that this name was invented by his nephew.

Google (search engine) got its name in honor of googol. Then 1-tsa with a googol of zeros (1010100) is a googolplex - Kasner also invented this name.

Even larger in comparison with the googolplex is the Skuse number (e to the e to the e79 power), proposed by Skuse in the proof of the Rimmann conjecture about prime numbers(1933). There is one more Skuse number, but it is applied when the Rimmann hypothesis is not valid. Which of them is more difficult to say, especially when it comes to higher degrees. However, this number, despite its "enormity", cannot be considered the most-most of all those that have their own names.

And the leader among the largest numbers in the world is the Graham number (G64). It was he who was used for the first time to carry out proofs in the field of mathematical science (1977).

When it comes to such a number, you need to know that you cannot do without a special 64-level system created by Knut - the reason for this is the connection of the number G with bichromatic hypercubes. The whip invented a superdegree, and in order to make it convenient to take notes, he suggested using the up arrows. So we learned the name of the largest number in the world. It is worth noting that this G number got on the pages of the famous Book of Records.

Sooner or later, everyone is tormented by the question, what is the largest number. A child's question can be answered in a million. What's next? Trillion. And further? In fact, the answer to the question of what are the largest numbers is simple. You just need to add one to the largest number, as it will no longer be the largest. This procedure can be continued indefinitely. Those. is it not the largest number in the world? Is it infinity?

And if you ask the question: what is the largest number that exists, and what is its own name? Now we will all find out ...

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

The American system is pretty simple. All the names of large numbers are built 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 one thousand (lat. mille) and the increasing suffix-million (see table). This is how the numbers are obtained - trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion and decillion. The American system is used in the USA, Canada, France and Russia. You can find out the number of zeros in a number written in the American system using the simple formula 3 x + 3 (where x is a Latin numeral).

The English naming system is the most common in the world. It is used, for example, in Great Britain and Spain, as well as in most of the 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 is -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 in the English and American systems is completely different numbers! You can find out the number of zeros in a number written in the English system and ending with the suffix-million using the formula 6 x + 3 (where x is a Latin numeral) and by the formula 6 x + 6 for numbers ending in -billion.

Only the number billion (10 9) passed from the English system to the Russian language, which would still be more correct to call it as the Americans call it - a billion, since it is the American system that has been adopted in our country. But who in our country does something according to the rules! 😉 By the way, sometimes the word trillion is also used in Russian (you can see for yourself by running a search in Google or Yandex) and it means, apparently, 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, i.e. numbers that have their own names without any Latin prefixes. There are several such numbers, but I will talk about them in more detail a little later.

Let's go back to writing using Latin numerals. It would seem that they can write numbers to infinity, but this is not entirely true. Let me explain why. Let's see for a start how the numbers from 1 to 10 33 are called:

And so, now the question arises, what's next. What's behind the decillion? In principle, it is possible, of course, by combining prefixes to generate such monsters as: andecilion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion and novemdecillion, but these will already be compound names, but we were interested in numbers. Therefore, according to this system, in addition to those indicated above, you can still get only three - vigintillion (from lat. viginti- twenty), centillion (from lat. centum- one hundred) and a million (from lat. mille- thousand). The Romans did not have more than a thousand of their own names for numbers (all numbers over a thousand were 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 such a system, the number is greater than 10 3003, which would have its own, non-composite name, it is impossible to get! But nevertheless, numbers over a million million are known - these are the very off-system numbers. Let's finally tell you about them.

The smallest such number is a myriad (it is even in Dahl's dictionary), which means one hundred hundred, that is, 10,000 does not mean a definite number at all, but an uncountable, uncountable set 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 reality, but the myriad gained fame thanks to the Greeks. Myriad was the name for 10,000, but there were no names for numbers over ten thousand. However, in the note "Psammit" (ie the calculus of sand), Archimedes showed how one can 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 sphere with a diameter of a myriad of Earth's diameters) no more than 1063 grains of sand would fit (in our notation). It is curious that modern calculations of the number of atoms in the visible Universe lead to the number 1067 (just a myriad of times more). Archimedes suggested the following names for numbers:
1 myriad = 104.
1 d-myriad = myriad of myriads = 108.
1 three-myriad = di-myriad of di-myriads = 1016.
1 tetra-myriad = three-myriad three-myriad = 1032.
etc.

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

Edward Kasner.

On the Internet, you can often find a mention that Googol is the largest number in the world - but this is not so ...

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

Googolplex (eng. googolplex) - a number also invented by Kasner with his nephew and meaning one with a googol of zeros, that is, 10 10100. Here 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 (Skewes. J. London Math. Soc. 8, 277-283, 1933.) in proving the Riemann conjecture concerning prime numbers. It means e to the extent e to the extent e to the 79th power, that is, eee79. Later, Riele (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 ee27 / 4, which is approximately 8.18510370. It is clear that since the value of Skuse's number depends on the number e, then it is not an integer, therefore we will not consider it, otherwise we would have to remember other non-natural numbers - pi, e, etc.

But it should be noted that there is a second Skuse number, which in mathematics is denoted as Sk2, which is even greater than the first Skuse number (Sk1). The second Skuse number was introduced by J. Skuse in the same article to denote a number for which the Riemann hypothesis is not valid. Sk2 is equal to 101010103, which is 1010101000.

As you understand, the more there are in the number of degrees, the more difficult it is to understand which of the numbers is greater. For example, looking at the Skuse numbers, without special calculations, it is almost impossible to understand which of these two numbers is greater. Thus, it becomes inconvenient to use powers for very large numbers. Moreover, you can think of such numbers (and they have already been invented) when the degrees of degrees simply do not 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, the question arises 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 this problem came up with his own way of writing, which led to the existence of several unrelated ways to write numbers - these are the notations of Knuth, Conway, Steinhouse, etc.

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

Steinhaus came up with two new super-large numbers. He named the number Mega and the number Megiston.

The mathematician Leo Moser refined Stenhouse's notation, which was limited by the fact that if it was required to write numbers much larger than the megiston, difficulties and inconveniences arose, since it was necessary to draw many circles one inside the other. 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[k+1] = "n v n k-gons "= n[k]n.

Thus, according to Moser's notation, the Steinhaus mega is written as 2, and the megiston as 10. In addition, Leo Moser proposed to call a polygon with the number of sides equal to a mega - megaagon. And he suggested the number "2 in Megagon", that is 2. This number became known as the Moser's number (Moser's number) or simply as moser.

But the moser is not the largest number either. The largest number ever used in mathematical proof is a limiting quantity known as the Graham "s number, first used in 1977 to prove one 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, the number written in Knuth's notation cannot be translated into the Moser system. Therefore, we will have to explain this system as well. In principle, there is nothing complicated in 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 superdegree, which he proposed to write down with arrows pointing up:

V general view it looks like this:

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

The number G63 became known as the Graham number (it is often denoted simply as G). This number is the largest known number in the world and is even included in the Guinness Book of Records.

So there are numbers greater than Graham's number? There is, of course, the Graham number + 1 to start with. As for the significant number ... well, there are some devilishly complex areas of mathematics (in particular the area known as combinatorics) and computer science where numbers even larger than Graham's number occur. But we have almost reached the limit of what can be reasonably and intelligibly explained.

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

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