American mathematician Edward Kasner (1878 - 1955) in the first half of the 20th century proposed to callgoogol. In 1938, Kasner was walking through the park with his two nephews, Milton and Edwin Sirott, 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. Nine-year-old Milton suggested calling this numbergoogol (googol).
In 1940, Kasner, along with James Newman, published a book "Mathematics and Imagination" (Mathematics and the Imagination ), where this term was first used. According to other sources, he first wrote about googol in 1938 in the article " New Names in Mathematics" in the January issue of the magazine Scripta Mathematica.
Term googol has no serious theoretical and practical significance. Kasner proposed it to illustrate the difference between an unimaginably large number and infinity, and the term is sometimes used in mathematics teaching for this purpose.
Four decades after the death of Edward Kasner, the term googol used for self-name by the now world famous corporation Google .
Judge for yourself whether the googol is good and convenient as a unit of measurement for quantities that actually exist within the boundaries of our solar system:
- the average distance from the Earth to the Sun (1.49598 · 10 11 m) is taken as an astronomical unit (AU) - an insignificant tiny thing on the scale of a googol;
- Pluto, a dwarf planet in the Solar System, until recently the classical planet most distant from Earth, has an orbital diameter of 80 AU. (12 · 10 13 m);
- quantity elementary particles, from which the atoms of the entire Universe are composed, physicists estimate a number not exceeding 10 88 .
For the needs of the microcosm - the elementary particles of the atomic nucleus - the unit of length (non-systemic) is angstrom(Å = 10 -10 m). Introduced in 1868 by the Swedish physicist and astronomer Anders Angström. This unit of measurement is often used in physics because
10 -10 m = 0.000 000 000 1 m
This is the approximate diameter of the electron orbit in an unexcited hydrogen atom. The atomic lattice pitch in most crystals has the same order.
But even on this scale, the numbers expressing even interstellar distances are far from one googol. For example:
- The diameter of our Galaxy is considered to be 10 5 light years, i.e. equal to 10 5 times the distance traveled by light in one year; in angstroms it's just
10 31 Å;
- the distance to supposedly existing very distant Galaxies does not exceed
10 40 · Å.
Ancient thinkers called the universe the space bounded by the visible stellar sphere of finite radius. The ancients considered the Earth to be the center of this sphere, while Archimedes and Aristarchus of Samos gave way to the Sun as the center of the universe. So, if this universe is filled with grains of sand, then, as the calculations performed by Archimedes show in " Psammit" ("Calculus of grains of sand "), it would take about 10 63 grains of sand - a number that is
10 37 = 10 000 000 000 000 000 000 000 000 000 000 000 000
times smaller than a googol.
And yet the variety of phenomena even in earthly organic life is so great that there were physical quantities, surpassing one googol. Solving the problem of training robots to perceive voices and understand verbal commands, researchers found that variations in the characteristics of human voices reach a number
45 · 10 100 = 45 googol.
In mathematics itself there are many examples of giant numbers that have a specific affiliation.For example, positional notationthe largest known prime number as of September 2013, Mersenne numbers
2 57885161 - 1,
Would consist of more than 17 million digits.
By the way, Edward Kasner and his nephew Milton came up with a name for an even larger number than a googol - for a number equal to 10 to the power of a googol -
10 10 100 .
This number is called - googolplex. Let's smile - the number of zeros after one in decimal notation The googolplex exceeds the number of all elementary particles in our Universe.
“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 from 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 view 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
Countless different numbers surround us every day. Surely many people have at least once wondered what number is considered the largest. You can simply say to a child that this is a million, but adults understand perfectly well that other numbers follow a million. For example, all you have to do is add one to a number each time, and it will become larger and larger - this happens ad infinitum. But if you look at the numbers that have names, you can find out what the largest number in the world is called.
The appearance of number names: what methods are used?
Today there are 2 systems according to which names are given to numbers - American and English. The first is quite simple, and the second is the most common throughout the world. The American one allows you to give names to large numbers as follows: first, the ordinal number in Latin is indicated, and then the suffix “million” is added (the exception here is million, meaning a thousand). This system is used by Americans, French, Canadians, and it is also used in our country.
English is widely used in England and Spain. According to it, numbers are named as follows: the numeral in Latin is “plus” with the suffix “illion”, and the next (a thousand times larger) number is “plus” “billion”. For example, the trillion comes first, the trillion comes after it, the quadrillion comes after the quadrillion, etc.
So, the same number in various systems can mean different things, for example, an American billion in the English system is called a billion.
Extra-system numbers
In addition to the numbers that are written according to the known systems (given above), there are also non-systemic ones. They have their own names, which do not include Latin prefixes.
You can start considering them with a number called a myriad. It is defined as one hundred hundreds (10000). But according to its intended purpose, this word is not used, but is used as an indication of an innumerable multitude. Even Dahl's dictionary will kindly provide a definition of such a number.
Next after the myriad is a googol, denoting 10 to the power of 100. This name was first used in 1938 by the American mathematician E. Kasner, who noted that this name was invented by his nephew.
Google got its name in honor of googol ( search system). Then 1 with a googol of zeros (1010100) represents a googolplex - Kasner also came up with this name.
Even larger than the googolplex is the Skuse number (e to the power of e to the power of e79), proposed by Skuse in his proof of the Rimmann conjecture about prime numbers (1933). There is another Skuse number, but it is used when the Rimmann hypothesis is not valid. Which one is greater is quite difficult to say, especially when it comes to large degrees. However, this number, despite its “hugeness,” cannot be considered the very best of all those that have their own names.
And the leader among the largest numbers in the world is the Graham number (G64). It was used for the first time to carry out proofs in the field of mathematical science (1977).
When it comes to such a number, you need to know that you cannot do without a special 64-level system created by Knuth - the reason for this is the connection of the number G with bichromatic hypercubes. Knuth invented the superdegree, and in order to make it convenient to record it, he proposed the use of up arrows. So we found out what the largest number in the world is called. It is worth noting that this number G was included in the pages of the famous Book of Records.
The famous search engine, as well as the company that created this system and many other products, is named after the googol number - one of the largest numbers in the infinite set of natural numbers. However, the largest number is not even a googol, but a googolplex.
The googolplex number was first proposed by Edward Kasner in 1938; it represents a one followed by an incredible number of zeros. The name comes from another number - googol - one followed by a hundred zeros. Usually the number googol is written as 10 100, or 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.
Googolplex, in turn, is the number ten to the power of googol. It's usually written like this: 10 10 ^100, and that's a lot, a lot of zeros. There are so many of them that if you decided to count the number of zeros using individual particles in the universe, you would run out of particles before you ran out of zeros in the googolplex.
According to Carl Sagan, writing this number is impossible because writing it would require more space than exists in the visible universe.
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As a child, I was tormented by the question of what the largest number exists, and I tormented almost everyone with this stupid question. Having learned the number one million, I asked if there was a number greater than a million. Billion? How about more than a billion? Trillion? How about more than a trillion? Finally, there was someone smart who explained to me that the question was stupid, since it is enough just to add one to the largest number, and it turns out that it was never the largest, since there are even larger numbers.
And so, many years later, I decided to ask myself another question, namely: What is the largest number that has its own name? Fortunately, now there is the Internet and you can puzzle patient search engines with it, which will not call my questions idiotic ;-). Actually, that’s what I did, and this is what I found out as a result.
| 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 simply. All names of large numbers are constructed like this: at the beginning there is a Latin ordinal number, and at the end the suffix -million is added to it. An exception is the name "million" which is the name of the number thousand (lat. mille) and the magnifying suffix -illion (see table). This is how we get the numbers trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion and decillion. The American system is used in the USA, Canada, France and Russia. You can find out the number of zeros in a number written according to the American system using the simple formula 3 x + 3 (where x is a Latin numeral).
The English naming system is the most common in the world. It is used, for example, in Great Britain and Spain, as well as in most former English and Spanish colonies. The names of numbers in this system are built like this: like this: the suffix -million is added to the Latin numeral, the next number (1000 times larger) is built according to the principle - the same Latin numeral, but the suffix - billion. That is, after a trillion in the English system there is a trillion, and only then a quadrillion, followed by a quadrillion, etc. Thus, a quadrillion according to the English and American systems are completely different numbers! You can find out the number of zeros in a number written according to the English system and ending with the suffix -million, using the formula 6 x + 3 (where x is a Latin numeral) and using the formula 6 x + 6 for numbers ending in - billion.
Only the number billion (10 9) passed from the English system into the Russian language, which would still be more correct to be called as the Americans call it - billion, since we have adopted the American system. But who in our country does anything according to the rules! ;-) By the way, sometimes the word trillion is used in Russian (you can see this for yourself by running a search in Google or Yandex) and 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 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:
| 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's behind the decillion? In principle, it is, of course, possible, by combining prefixes, to generate such monsters as: andecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion and novemdecillion, but these will already be compound names, and we were interested in our own names numbers. Therefore, according to this system, in addition to those indicated above, you can still get only three proper names - vigintillion (from Lat. viginti- twenty), centillion (from lat. centum- one hundred) and million (from lat. mille- thousand). The Romans did not have more than a thousand proper names for numbers (all numbers over a thousand were composite). For example, the Romans called a million (1,000,000) decies centena milia, that is, "ten hundred thousand." And now, actually, the table:
Thus, according to such a system, it is impossible to obtain numbers greater than 10 3003, which would have its own, non-compound name! But nevertheless, numbers greater than a million are known - these are the same non-systemic numbers. Let's finally talk about them.
| Name | Number |
| Myriad | 10 4 |
| 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 is even in Dahl’s dictionary), which means a hundred hundreds, that is, 10,000. This word, however, is outdated and practically not used, but it is curious that the word “myriads” is widely used, which does not mean a specific number at all, but countless, uncountable multitudes of something. It is believed that the word myriad (English: myriad) came into European languages from ancient Egypt.
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.
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 10 100. This is how Kasner himself describes this “discovery”:
Words of wisdom are spoken by children at least as often as by scientists. The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex." A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out.
Mathematics and the Imagination(1940) by Kasner and James R. Newman.
An even larger number than 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, e e 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 e e 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 - pi, e, Avogadro's number, etc.
But it should be noted that there is a second Skuse number, 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 valid. Sk 2 is equal to 10 10 10 10 3, that is, 10 10 10 1000.
As you understand, the more degrees there are, the more difficult it is to understand which number is greater. For example, looking at Skewes numbers, without special calculations, it is almost impossible to understand which of these two numbers is larger. Thus, for super-large numbers it becomes inconvenient to use powers. Moreover, you can come up with such numbers (and they have already been invented) when the degrees of degrees simply do not fit on the page. Yes, that's on the page! They won’t fit even into a book the size of the entire Universe! In this case, the question arises of how to write them down. The problem, as you understand, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who wondered about this problem came up with his own way of writing, which led to the existence of several, unrelated to each other, methods for writing numbers - these are the notations of Knuth, Conway, Steinhouse, etc.
Consider the notation of Hugo Stenhouse (H. Steinhaus. Mathematical Snapshots, 3rd edn. 1983), which is quite simple. Stein House suggested writing large numbers inside geometric shapes - triangle, square and circle:
Steinhouse came up with two new superlarge numbers. He named the number - Mega, and the number is 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 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.
Unfortunately, a number written in Knuth's notation cannot be converted into notation in the Moser system. Therefore, we will have to explain this system too. In principle, there is nothing complicated about it either. Donald Knuth (yes, yes, this is the same Knuth who wrote “The Art of Programming” and created the TeX editor) came up with the concept of superpower, which he proposed to write with arrows pointing upward:
In general it looks like this:
I think everything is clear, so let’s return to Graham’s number. Graham proposed so-called G-numbers:
The number G 63 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 G 100. Remember it, and when your children ask what is the largest number in the world, tell them that this number is called stasplex.
Update (4.09.2003): Thank you all for the comments. It turned out that I made several mistakes when writing the text. I'll try to fix it now.
- I made several mistakes just by mentioning Avogadro's number. Firstly, several people pointed out to me that in fact 6.022 10 23 is the very best natural number. And secondly, there is an opinion, and it seems correct to me, that Avogadro’s number is not a number at all in the proper, mathematical sense of the word, since it depends on the system of units. Now it is expressed in “mol -1”, but if it is expressed, for example, in moles or something else, then it will be expressed as a completely different number, but this will not cease to be Avogadro’s number at all.
- drew my attention to the fact that the ancient Slavs also gave numbers their own names and it is not good to forget about them. So, here is a list of Old Russian names for numbers:
10,000 - darkness
100,000 - legion
1,000,000 - leodr
10,000,000 - raven or corvid
100,000,000 - deck
Interestingly, the ancient Slavs also loved large numbers and were able to count to a billion. Moreover, they called such an account a “small account.” In some manuscripts, the authors also considered the “great count”, reaching the number 10 50. About numbers greater than 10 50 it was said: “And more than this cannot be understood by the human mind.” The names used in the “small count” were transferred to the “great count”, but with a different meaning. So, darkness no longer meant 10,000, but a million, legion - the darkness of those (a million millions); leodre - legion of legions (10 to the 24th degree), then it was said - ten leodres, one hundred leodres, ..., and finally, one hundred thousand those legion of leodres (10 to 47); leodr leodrov (10 in 48) was called a raven and, finally, a deck (10 in 49). - The topic of national names of numbers can be expanded if we remember about the Japanese system of naming numbers that I had forgotten, which is very different from the English and American systems (I won’t draw hieroglyphs, if anyone is interested, they are):
10 0 - ichi
10 1 - jyuu
10 2 - hyaku
10 3 - sen
10 4 - man
10 8 - oku
10 12 - chou
10 16 - kei
10 20 - gai
10 24 - jyo
10 28 - jyou
10 32 - kou
10 36 - kan
10 40 - sei
10 44 - sai
10 48 - goku
10 52 - gougasya
10 56 - asougi
10 60 - nayuta
10 64 - fukashigi
10 68 - muryoutaisuu - Regarding the numbers of Hugo Steinhaus (in Russia for some reason his name was translated as Hugo Steinhaus). botev assures that the idea of writing superlarge numbers in the form of numbers in circles belongs not to Steinhouse, but to Daniil Kharms, who long before him published this idea in the article “Raising a Number.” I also want to thank Evgeniy Sklyarevsky, the author of the most interesting site on entertaining mathematics on the Russian-language Internet - Arbuza, for the information that Steinhouse came up with not only the numbers mega and megiston, but also suggested another number medical zone, equal (in his notation) to "3 in a circle".
- Now about the number myriad or mirioi. There are different opinions about the origin of this number. Some believe that it originated in Egypt, while others believe that it was born only in Ancient Greece. Be that as it may in fact, the myriad gained fame precisely thanks to the Greeks. Myriad was the name for 10,000, but there were no names for numbers greater than ten thousand. However, in his note “Psammit” (i.e., calculus of sand), Archimedes showed how to systematically construct and name arbitrarily large numbers. In particular, placing 10,000 (myriad) grains of sand in a poppy seed, he finds that in the Universe (a ball with a diameter of a myriad of the diameters of the Earth) no more than 10 63 grains of sand could fit (in our notation). It is curious that modern calculations of the number of atoms in the visible Universe lead to the number 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.
If you have any comments -