# What does Graham's number mean?

## Portal for the student. Self preparation

In answering such a difficult question, what is the greatest number in the world, it should first be noted that there are two accepted ways of calling numbers today - English and American. Under the English system, the -billion or -million suffixes are added to each large number in turn, resulting in a number of millions, billions, trillions, trillions, and so on. If we start from the American system, then it is necessary to add the suffix million to each large number, forming the numbers trillions, quadrillions and larger. It should be noted here that the English number system is more widespread in the modern world and the numbers available in it are completely sufficient for the normal functioning of all systems in our world.

Of course, from a logical point of view, the answer to the question about the largest number may not be clear, because if you add just one to each subsequent digit, a new larger number will be obtained, so this process is unlimited. However, strangely enough, the largest number in the world still exists and it is inscribed on the Guinness Book of Records.

**Graham's number is the greatest number in the world**

It is this number that is recognized worldwide as the largest in the Book of Records, while it is very difficult to explain what it is and how big it is. Generally these are triples, multiplied by each other, forming a number 64 orders of magnitude higher than any person's point of understanding. As a result, we can only provide the last 50 digits of Graham's number **– ** 0322234872396701848518 64390591045756272 62464195387.

**Googol's number **

The history of how this number came about is not as complex as it was above. American mathematician Edward Kasner, who was talking to his nephews about large numbers, couldn't answer the question of how to call numbers with 100 zeros or more. The resourceful nephew suggested his name to such numbers - googol. It should be noted that this number doesn't have much practical value, but it is sometimes used in math to express infinity.

**Googlex **

This number was also invented by mathematician Edward Kasner and his nephew Milton Sirotta. It is generally a googol's tenth power. In answering the question of many curious people about how many zeros there are in Googleplex, it should be noted that even if you write down all of the paper on the planet with classic zeros, this number cannot be represented in the classic version.

**Skewes number **

Another contender for the highest number title is Skuse's number, proven by John Littlewood in 1914. Based on the evidence presented, that number is approximately 8.185 × 10370.

**Moser number **

This method of naming very large numbers was invented by Hugo Steinhaus, who suggested we call them polygons. As a result of three mathematical operations performed, the number 2 is born in a mega-gon (a polygon with mega-sides).

As you can see, a great number of mathematicians went to great lengths to find it - the greatest number in the world. Of course, we cannot judge to what extent these attempts were successful. It should be noted, however, that the actual applicability of such numbers is questionable as they are not even suitable for human understanding. Also, there is always a number that is bigger when you do a very simple math operation +1.

There are numbers so incredibly, incredibly big that even writing them down would take up the entire universe. But here's what really drives you crazy ... some of those unimaginably large numbers are extremely important to understanding the world.

When I say "the greatest number in the universe" I really mean the greatest *makes sense* number, the maximum possible number that is useful in any way. There are many contenders for this title, but I warn you right away: in fact, there is a risk that trying to understand all of this will blow you away. And besides, you don't have much fun with too much math.

**Googol and Googolplex**

Edward Kasner

We could start with two, probably the biggest numbers you've ever heard of, and these are indeed the two biggest numbers that have been widely accepted by definitions in English. (There is pretty precise nomenclature used to denote numbers that are as big as you'd like, but those two numbers aren't in dictionaries right now.) Google, as it became world famous (albeit with errors, note ... in fact it's googol) in the form of google, was born in 1920 to interest kids in big numbers.

To this end, Edward Kasner (pictured) took his two nephews Milton and Edwin Sirotte for a walk through the New Jersey Palisades. He invited them to come up with some ideas, and then nine-year-old Milton suggested "googol". It is not known where he got this word from, but Kasner decided that or a number with one hundred zeros after the unit is now called googol.

But young Milton didn't stop there, he suggested an even larger number, a googolplex. According to Milton, this is a number that has a 1 first and then as many zeros as you can write before you get tired. While this idea is intriguing, Kasner decided that a more formal definition was needed. As he explained in his 1940 book Mathematics and the Imagination, Milton's definition leaves open the risky possibility that the casual fool could become a mathematician superior to Albert Einstein simply because he has more stamina.

So Kasner decided that the googolplex would be equal to or 1 and then the googol of zeros. Otherwise, and in similar terms to other numbers, we'll say it's a googolplex. To show how fascinating this is, Carl Sagan once remarked that it is physically impossible to write down all the zeros on a googolplex because there is simply not enough room in the universe. If you fill the entire volume of the observable universe with fine dust particles about 1.5 microns in size, the number of different ways in which these particles are arranged corresponds roughly to a googolplex.

Linguistically, Googol and Googolplex are probably the two largest significant numbers (in English at least), but as we shall now see, there are infinite ways to define "meaning".

**Real world**

When we talk about the largest significant number there is a reasonable argument that it really means we need to find the largest number with real value in the world. We can start with the current human population, which is currently around 6,920 million. Global GDP was estimated at around $ 61.96 billion in 2010, but both numbers are insignificant when compared to the roughly 100 trillion cells that make up the human body. Of course, none of these numbers can be compared to the total number of particles in the universe, which is usually considered to be roughly the same, and this number is so large that our language has no word to match.

We can play a little with the measurement systems and make the numbers bigger and bigger. Thus the mass of the sun in tons is less than in pounds. An excellent way to do this is to use the Planck system of units, which are the smallest possible units for which the laws of physics still apply. For example, the age of the universe in Planck's time is approximate. When we return to the first unit of the Planck period after the Big Bang, we will see how dense the universe was back then. We're getting more and more, but we haven't even reached Googol yet.

The greatest number with a real world application - or in this case a real world application - is likely one of the most recent estimates of the number of universes in the multiverse. This number is so large that the human brain literally cannot perceive all these different universes because the brain can only approximate configurations. In fact, unless you take into account the idea of the multiverse as a whole, this number is likely the largest number with any practical significance. However, there are still much larger numbers hiding there. But to find them we have to venture into pure mathematics, and there is no better place to start than prime numbers.

**Mersenne primes**

Part of the difficulty is in finding a good definition of a "significant" number. One way is to think in terms of prime numbers and composite numbers. A prime number, as you probably remember from school mathematics, is a natural number (note, not equal to one) that is only divisible by itself. So and are prime numbers and and are composite numbers. This means that every composite number can ultimately be represented by its prime divisors. In a sense, a number is more important than, for example, because there is no way to express it as the product of smaller numbers.

Of course we can go a little further. For example, it's really simple, which means that in a hypothetical world where our knowledge of numbers is limited to a number, a mathematician can still express a number. But the next number is already prime, which means the only way to express it is to know directly about its existence. This means that the largest known prime numbers play an important role, and for example Googol - which is ultimately just a collection of numbers and is multiplied by one another - actually not. And since primes are largely random, there is no known way of predicting that an incredibly large number will actually be prime. To this day it is difficult to discover new prime numbers.

Ancient Greek mathematicians had at least 500 BC. An idea of prime numbers, and 2000 years later people still knew which numbers were prime only up to about 750. Euclidean thinkers saw the possibility of simplification, but until the Renaissance mathematicians could not really put it into practice. These numbers are known as the Mersenne numbers and are named after the 17th century French scientist Marina Mersenne. The idea is very simple: the Mersenne number is any number of this type. For example, if that number is prime, so is.

Identifying Mersenne primes is much faster and easier than any other type of prime, and computers have worked hard to find them for the past six decades. Until 1952, the largest known prime number was a number - a number with digits. That same year a computer calculated that the number was a prime, and that number is made up of numbers, which makes it much bigger than a googol.

Computers have been on the hunt since then, and Mersenne's i number is currently the largest prime number known to mankind. It was discovered in 2008 and is a number with almost a million digits. It is the largest known number that cannot be expressed in smaller numbers. If you'd like help finding an even larger Mersenne number, you (and your computer) can always do the search at http: //www.mersenne. org /.

**Skewes number**

Stanley Skewes

Let's go back to the prime numbers. As I said, they are fundamentally wrong, which means that there is no way of predicting what the next prime number will look like. Mathematicians have been forced to turn to some pretty fantastic measurements to find a way to predict future prime numbers, even in an obscure way. Probably the most successful of these attempts is the prime number function invented by the legendary mathematician Karl Friedrich Gauß in the late 18th century.

I'll save you the more complicated math - we have a long way to go in one way or another - but the essence of the function is this: for any whole number, you can guess how many prime numbers there are, fewer. For example, if the function predicts that there should be prime numbers, if - prime numbers, fewer, and if, then there are fewer numbers that are prime numbers.

The arrangement of the primes is in fact irregular and only an approximation of the actual number of primes. In fact, we know that there are prime numbers, fewer, fewer prime numbers, and prime numbers. While this is an excellent grade, it's always just a rating ... and a higher grade in particular.

In all previously known cases, the prime number function slightly exaggerates the actual number of a few prime numbers. Mathematicians once thought that this would always be ad infinitum, that it certainly holds true for some inconceivably large numbers, but in 1914 John Edenzor Littlewood proved that for an unknown, inconceivably large number, this function would begin to produce fewer prime numbers, and then will Changed infinitely often between the upper and lower limit.

The hunt was at the start of the races and this is where Stanley Skewes appeared (see photo). In 1933 he proved that if a function that approximates the number of prime numbers first gives a lower value, the upper bound is a number. It is difficult to really understand, even in the most abstract sense, what this number actually represents, and from that point of view it was the largest number ever used for serious mathematical proof. Since then, mathematicians have succeeded in reducing the upper limit to a relatively small number, but the original number is still known as the Skuse number.

What is the number that makes up even the mighty Googolplex dwarf? In the Penguin Dictionary of Curious and Interesting Numbers, David Wells describes one way Hardy's mathematician was able to grasp the size of Skuse's number:

"Hardy thought it was" the greatest number ever served any particular purpose in mathematics, "and suggested that if you were to play chess with all the particles in the universe, one step would be to swap two particles and the game would end if the same position were repeated a third time, then the number of all possible games would be roughly equal to the number of Skuse. ''

One last thing before we move on: we talked about the smaller of the two Skuse numbers. There is another Skuse number that a mathematician found in 1955. The first number is obtained on the basis that what is known as the Riemann Hypothesis is true - this is a particularly difficult hypothesis in mathematics that goes unproven and is very useful when it comes to prime numbers. However, if the Riemann hypothesis is false, Skuse determined that the starting point rises on.

**The size problem**

Before we get to the number that even Skuse's number looks tiny next to, we need to talk a little bit about scaling, otherwise we won't be able to estimate where we're going to go. Let's take a number first - it's a tiny number so small that people can actually intuitively understand what it means. There are very few numbers that fit this description, as numbers greater than six are no longer separate numbers and become "several," "many," and so on.

Now let's take, i.e. ... Although we really cannot intuit what it was like for a number, it is very easy to understand what it is, to imagine what it is. So far, so good. But what if we go? It's the same or. We are far from imagining this value like any other very large one - we are losing the ability to grasp individual parts somewhere around a million. (While it would take an insane amount of time to actually count to a million of everything, the point is, we can still sense that number.)

While we can't imagine that, we can at least generally understand what 7.6 billion is and maybe compare it to something like US GDP. We have moved from intuition to representation to simple understanding, but at least we still have a gap in understanding what a number is. This will soon change as we take a step up the ladder.

To do this, we need to go to the notation introduced by Donald Knuth, known as the arrow notation. In these terms it can be written as. Then when we go, the number we get is the same. This is the sum of the three. We have now clearly and truly surpassed all the other numbers we have already spoken of.After all, even the largest of them only had three or four members in the series of indicators. For example, even Skuse's super number is "only" - even taking into account the fact that both the base and indicators are much larger than, it is still absolutely nothing compared to the size of the number tower of one billion members.

Obviously there is no way to capture such large numbers ... and yet the process by which they are generated can still be understood. We couldn't understand the real number that a grade tower with billions of triples gives, but we can basically imagine such a tower with many members, and a really decent supercomputer can store such towers in memory, even if it knows their actual values can not calculate. ...

This is getting more and more abstract, but it only gets worse. You might think that a tower of forces whose exponent length is the same (also, I made this exact mistake in the previous version of this post), but it's simple. In other words, imagine if you had the opportunity to calculate the exact value of a triplet power tower made up of elements. Then you took that value and created a new tower with as many as it gives.

Repeat this process with each subsequent number ( *Note.* starting right) until you do it once and then you finally get it. This is a number that is just incredibly big, but at least the steps to get it seem understandable if everything is done very slowly. We can no longer understand the number or imagine how it will be obtained, but at least we can only understand the basic algorithm in a relatively long time.

Now let's prepare the mind to really blow it up.

**Graham's Number (Graham)**

Ronald Graham

This is how you get the Graham number, which is considered by the Guinness Book of Records to be the largest number ever used for mathematical proof. It is completely impossible to imagine how great it is, and just as difficult to explain exactly what it is. Basically, Graham's number appears when dealing with hypercubes, which are theoretical geometric shapes with more than three dimensions. The mathematician Ronald Graham (see photo) wanted to find out at what smallest number of dimensions certain properties of the hypercube remain stable. (Sorry for this vague explanation, but I'm sure we all need to get at least two degrees in math to be more specific.)

In either case, the Graham number is an upper limit to this minimum number of dimensions. How big is this upper limit? Let's go back to a number so large that we can only vaguely understand the algorithm for obtaining it. Instead of just jumping one step higher, we now count the number in which there are arrows between the first and last three. Now we are well beyond the slightest understanding of what that number is or what needs to be done to calculate it.

Now we repeat this process once ( *Note.* At each next step we write the number of arrows corresponding to the number obtained in the previous step.

This, ladies and gentlemen, is Graham's number, about an order of magnitude higher than the point of human understanding. So much larger than any number you can imagine - much more than any infinity you can ever imagine - this number, contradicts even the most abstract description.

But here's the strange thing. Since Graham's number is basically just three times multiplied, we know some of its properties without actually calculating them. We cannot represent Graham's number with a notation known to us, even after using the entire universe to write it down, but I can now tell you the last twelve digits of Graham's number: And that's not all: we at least know the last digits from Graham's number.

Of course, it should be remembered that this figure is only the upper limit of the original Graham problem. It is possible that the actual number of measurements required to perform the desired property is much, much less. In fact, according to most experts in the field since the 1980s, it has been assumed that the number of dimensions is actually only six - a number so small that we can intuitively understand it. Since then, the lower bound has been raised to, but there's still a very good chance that the solution to Graham's problem won't be next to a number as large as Graham's number.

**To infinity**

So there are numbers larger than Graham's number? To start with, there is of course the Graham number. As for the significant number ... well, there are some devilishly difficult areas of math (especially the area known as combinatorics) and computer science that have numbers even larger than Graham's number. But we have almost reached the limit of what I can hopefully ever reasonably explain. For those ruthless enough to go any further, further reading is offered at your own risk.

Well now an amazing quote attributed to Douglas Ray ( *Note.* to be honest it sounds pretty funny):

“I see collections of vague numbers hiding there in the darkness behind a small patch of light that the candle of the spirit gives. They whisper to one another; Conspiracy who knows what. Maybe they don't really like us because we caught their little brothers with our thoughts. Or maybe they just lead a distinct numerical lifestyle that is beyond our comprehension.

The question, "What is the largest number in the world?" Is wrong to say the least. There are different number systems - decimal, binary and hexadecimal, as well as different categories of numbers - semi-simple and simple, the latter being divided into legal and illegal. In addition, there are the Skewes "number", Steinhouse and other mathematicians who either jokingly or seriously invent exotic species such as "Megiston" or "Moser" and make them accessible to the public.

### What is the world's largest number in the decimal system?

Most "non-mathematicians" of the decimal system know the millions, billions and trillions. When the Russians associate a million with a dollar bribe that can be carried in a suitcase, where a billion (not to mention a trillion) North American banknotes are to be shoved, the majority don't have enough imagination. However, in large number theory there are concepts such as quadrillion (ten to fifteenth power - 1015), sextillion (1021), and octillion (1027).

In the English decimal system, the most widely used decimal system in the world, the decimal number is considered the maximum number - 1033.

In 1938, Edward Kasner, professor at Columbia University (USA) published his nine-year-old nephew's proposal, the decimal system of, on the pages of the journal "Scripta Mathematica" in connection with the development of applied mathematics and the expansion of the micro- and macrocosm to use a large number of "googol" ("googol") - ten to hundredths of a power (10100), which is expressed on paper as one with one hundred zeros. They did not stop there, however, and after a few years proposed that a new greatest number in the world be circulated - "Googolplex", ten, increased to the tenth power and again to the hundredth power - (1010) 100, expressed by one Unit to which a googol of zeros is assigned to the right. For the majority of even professional mathematicians, however, both "googol" and "googolplex" are of purely speculative interest and can hardly be applied to anything in everyday practice.

### Exotic numbers

What is the greatest number in the world among the prime numbers - those that are only divisible by themselves and by one? One of the first to establish the largest prime number, 2,147,483,647, was the great mathematician Leonard Euler. As of January 2016, this number is recorded as an expression that is calculated as 274 207 281 - 1.

It is impossible to answer this question correctly because the series of numbers has no upper limit. So to each number it is enough to add just one to get an even larger number. Although the numbers themselves are infinite, they don't have very many names of their own as most of them are content with names made up of smaller numbers. For example numbers and have their own names "one" and "one hundred", and the name of the number is already composed ("one hundred and one"). It is clear that in the finite set of numbers that humanity has given by its own name, there must be a greatest number. But what is it called and what is it the same? Let's try to find out and at the same time find out how many mathematicians invented.

"Short" and "Long" scale

The history of the modern system of naming large numbers goes back to the middle of the 15th century, when in Italy the words "million" (literally - large thousand) for thousand square, "bimillion" for one million square and "trillion" for a million cubes have been used. We know this system thanks to the French mathematician Nicolas Chuquet (approx. 1450 - approx. 1500): In his treatise "Science of Numbers" (Triparty en la science des nombres, 1484) he developed this idea and suggested a further use of Latin Cardinal numbers (see table), which are added "-million" at the end. Thus, Schukets "bimillion" became a billion, "trillion" became a trillion, and a million to the fourth power became "trillion".

In the Schücke system, the number between one million and one billion had no name of its own and was simply called "one thousand million", similar to "one thousand billion", "one thousand trillion" etc. It was not very practical, and in 1549 the French writer suggested and scientist Jacques Peletier du Mans (1517-1582) propose to name such "intermediate" numbers with the same Latin prefixes, but with the ending "-billion". So it was called "Billion" - "Billiards" - "Billion" and so on.

The Suke-Peletier system gradually became popular and was used across Europe. However, an unexpected problem arose in the 17th century. It found that some scientists got confused for some reason and called the number "billion" rather than "billion" or "thousand million". Soon this mistake quickly spread and a paradoxical situation arose - “billion” became synonymous with “billion” () and “million million” () at the same time.

This confusion lasted long enough and led the United States to create its own system for naming large numbers. According to the American system, the names of numbers are constructed in the same way as in the Schuke system - the Latin prefix and the ending "illion". However, the sizes of these numbers are different. If in the Shuke system names ending with "million" were given numbers that were degrees of one million, then in the American system the ending "-million" was given degrees of a thousand. That is, thousand million () were referred to as "billion", () - "trillion", () - "quadrillion", and so on.

The old system of naming large numbers continued to be used in conservative Britain and was referred to as "British" worldwide, although it was invented by the French Schuquet and Peletier. However, in the 1970s Britain officially switched to the "American system," which made it a bit strange to refer to one system as American and the other as British. As a result, the American system is now commonly referred to as the "short scale" and the British or Suquet-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 | ||

billiards | - | |

trillion | ||

trillion | - | |

Billiards | ||

Billiards | - | |

Trillion | ||

Quintilliard | - | |

Sextillion | ||

Sex billion | - | |

Septillion | ||

Septilliard | - | |

Octillion | ||

Octilliard | - | |

Trillion | ||

Not billions | - | |

Decillion | ||

Decilliard | - | |

Vigintillion | ||

Vigintilliard | - | |

Centillion | ||

Centilliard | - | |

million | ||

Billion | - |

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, with the exception that the number is referred to as a "billion" rather than a "billion". The long scale is still used in most other countries.

It is strange that in our country the final transition to the short scale did not take place until the second half of the 20th century. For example, Yakov Isidorovich Perelman (1882–1942) mentions in his Entertaining Arithmetic the parallel existence of two scales in the USSR. The short scale was used in everyday life and financial calculations, according to Perelman, 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 are big there too.

But back to the largest number. After the decillion, the names of the numbers are obtained by combining prefixes. In this way, numbers such as Undecillion, Duodecillion, Tredecillion, Quattordecillion, Quindecillion, Sexdecillion, Septemdecillion, Octodecillion, Novemdecillion, etc. are obtained. However, these names are no longer of interest to us as we have agreed to find the largest number with our own non-compound name.

If we turn to Latin grammar, we find that the Romans only had three non-compound names for numbers over ten: viginti - "twenty", centum - "one hundred" and mille - "thousand". The Romans did not have their own names for numbers greater than "a thousand". For example one million () The Romans called "decies centena milia", which means "ten times a hundred thousand". According to Schückes rule, these three remaining Latin digits give us names for numbers like "vigintillion", "centillion" and "milleillion".

So we found that on the "short scale" the maximum number that has its own name and is not made up of the smaller numbers is "million" (). If the "long scale" of name numbers were adopted in Russia, the largest number with its own name would be "billion" ().

However, there are names for even larger numbers.

Numbers outside the system

Some numbers have their own name without any association with the Latin prefixed naming system. And there are many such numbers. For example, you can remember the number e, the number "pi", a dozen, the number of the beast, and so on. Now that we are interested in large numbers, we will only look at those numbers with their own uncompound name that are more than a million.

Up until the 17th century, Russia used its own system of naming numbers. Tens of thousands have been called "Darkness", hundreds of thousands - "Legions", millions - "Leodra", tens of thousands - "Crows" and hundreds of millions - "Decks". This count of up to hundreds of millions has been called the "small count," and in some manuscripts the authors also considered the "large count," which used the same names for large numbers, but with a different meaning. So "darkness" didn't mean ten thousand, but a thousand thousand (), "Legion" - the darkness of those () ;; "Leodr" - legion of legions (), "Raven" - leodr leodrov (). For some reason the "deck" was not referred to as the "raven of ravens" in the great Slavic account. (), but only ten "ravens", that is (see table).

Number name | Meaning in "small number" | Value in the "Grand Score" | description |

dark | |||

legion | |||

Leodre | |||

Raven (vran) | |||

deck | |||

Obscurity of subjects |

The number also has its own name and was invented by a nine-year-old boy. And so it was. In 1938 the American mathematician Edward Kasner (1878-1955) walked through the park with his two nephews and discussed a large number with them. During the conversation we talked about a number with one hundred zeros that didn't have a name of its own. One of the nephews, nine-year-old Milton Sirott, suggested calling the number "googol". In 1940 Edward Kasner wrote the popular science book Mathematics and the Imagination together with James Newman, in which he told math lovers about the number of Googols. Google became even better known in the late 1990s thanks to the Google search engine named after it.

The name for an even larger number than Googol came about in 1950 thanks to the father of computer science, Claude Elwood Shannon (1916-2001).In his article "Programming a Computer to Play Chess" he tried to estimate the number of possible variations of a game of chess. According to him, each game takes turns on average, and with each move the player makes an average of a selection from the options that correspond (approximately equally) 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 from 100 BC The number of "Asankheya" is the same. This number is believed to be equal to the number of cosmic cycles required to attain nirvana.

Nine-year-old Milton Sirotta went down in the history of mathematics by not only inventing the number googol, but also suggesting another number at the same time - googolplex, which corresponds 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) to prove the Riemann hypothesis. The first number, later called "the first Skuse number", is the same degree for degree, that is. The "second Skuse number" is even bigger and is.

The more degrees there are in degrees, the more difficult it is of course to write numbers and understand their meaning while reading. In addition, it is possible to find such numbers (and, by the way, they have already been invented) if the degrees just don't fit on the page. Yes, what a side! They don't even fit in a book the size of the entire universe! In this case, the question arises of how to write such numbers. Fortunately, the problem is solvable, and mathematicians have developed various principles for writing such numbers. It is true that every mathematician who has dealt with this problem has invented his own notation, which led to the fact that there were several independent ways of writing large numbers - these are the notations of Knuth, Conway, Steinhaus, etc. We must now 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, Mathematisches Kaleidoskop, was published in Poland by Hugo Dionizy Steinhaus (1887-1972). 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 shapes - a triangle, a square, and a circle:

"In a triangle" means "",

"Square" means "in triangles"

"In a circle" means "in squares".

Steinhaus explains this spelling and finds the number "mega", which is the same in a circle, and shows that it is the same in a "square" or 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 pocket calculator in MS Windows cannot calculate in two triangles due to an overflow. This is roughly that huge number.

After Steinhaus has determined the number "Mega", he already invites readers to independently guess another number - "Mezons", which are the same in the circle. In another edition of the book, instead of Medzon, Steinhaus suggests guessing an even larger number - "Megiston", which is equal in a circle. After Steinhaus, I will also recommend readers to temporarily detach themselves from this text and try to write these numbers themselves with ordinary degrees in order to feel their gigantic size.

However, there are names for large numbers. Thus, the Canadian mathematician Leo Moser (Leo Moser, 1921-1970) modified the Steinhaus notation, which was limited by the fact that if the numbers had to be written down many large megistones, difficulties and inconveniences would arise as many circles inside would have to be drawn Another. Moser suggested not drawing circles, but pentagons after the squares, then hexagons, etc. He also suggested a formal notation for these polygons so that numbers can be written down without drawing complex drawings. Moser's notation looks like this:

"Triangle" \ u003d \ u003d;

"Square" \ u003d \ u003d "in triangles" \ u003d;

"In a pentagon" \ u003d \ u003d "in squares" \ u003d;

"Im -gon" \ u003d \ u003d "im -gons" \ u003d.

According to Moser's notation, the stone house "Mega" is written as "Mezon" as and "Megiston" as. In addition, Leo Moser suggested naming a polygon with a side number corresponding to a mega - "mega-gon". And suggested the number "in the Megagon", that is. This number became known as the Moser number or simply "Moser".

But Moser is not the largest number either. The largest number ever used in a mathematical proof is the "Graham number". This number was first used in 1977 by the American mathematician Ronald Graham to prove an estimate in Ramsey theory, namely in calculating the dimensions of certain -dimensional bichromatic hypercubes. But Graham's number only became famous after the story about him in Martin Gardner's 1989 book From Penrose Mosaics to Reliable Ciphers.

To explain how big the Graham number is, we need to explain another way of writing large numbers, introduced by Donald Knuth in 1976. The American professor Donald Knuth developed the concept of the supergrade, which he wanted to write down with arrows pointing upwards.

The usual arithmetic operations - addition, multiplication and exponentiation - can of course be extended to a sequence of hyperoperators as follows.

The multiplication of natural numbers can be defined by a repeated addition operation ("adding copies of a number"):

For example,

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

For example,

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

For example,

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

This already leads to pretty large numbers, but the notation doesn't 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 "-ICH arrow" is continued to the right in a sequential series of operators according to the correct associativity "arrow". Symbolically this can be written as follows:

For example:

The notation form is typically 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 (and also for descriptions with a variable number of arrows) or equivalent is preferred over hyperoperators. But some numbers are so big that even such a record is not enough. For example, Graham's number.

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

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

It can be written as where where the superscript y means to iterate 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 sextilons - and the number of atoms that make up the earth is on the order of Dodecalions), then the Googol is already "virtual", not to mention Graham's number. The scale of only the first term is so large that it is almost impossible to grasp, although the above entry is relatively easy to understand. Although this is just the number of towers in this formula, this number is already much larger than the number of Planck volumes (the smallest possible physical volume) contained (approximately) in the observable universe. After the first member, another member of the rapidly growing sequence awaits us.

17th June 2015

“I see collections of vague numbers hiding there in the darkness behind a small patch of light that the candle of the spirit gives. They whisper to one another; Conspiracy who knows what. Maybe they don't like us very much because they trapped their little brothers with our thoughts. Or maybe they just lead a clearly numerical way of life there that is beyond our understanding.

Douglas Ray

We continue ours. Today we have numbers ...

Sooner or later, everyone will be tormented by the question of what the greatest number is. A child's question can be answered in a million. What's next? Trillion. And further? In fact, the answer to the largest number question is simple. Just add one to the largest number as it won't be the largest anymore. This process can be continued indefinitely.

And if you ask the question: what is the greatest number in existence and what is its own name?

Now we're all going to find out ...

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

The American system is pretty simple. All names of large numbers are structured like this: They start with a Latin ordinal, and the suffix million is added at the end. 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 - trillions, quadrillion, trillion, sextillion, septillion, octillion, non-million and decillion. The American system is used in the United States, Canada, France, and Russia. You can find the number of zeros in a number written in the American system using the simple formula 3 x + 3 (where x is a Latin number).

The English naming system is the most widely used in the world. It is used, for example, in the UK and Spain, as well as most of the former English and Spanish colonies. The names of the numbers in this system are structured as follows: So: the suffix-million is added to the Latin digit, the next number (1000 times larger) is built according to the principle - the same Latin digit, but the suffix is -billion. That means that after the trillion in the English system there is a trillion and only then is there a quadrillion, followed by a quadrillion, and so on. A quadrillion in the English and American systems are completely different numbers! You can find the number of zeros in a number written in the English system that ends with the suffix-million by using the formula 6 x + 3 (where x is a Latin digit) and the number 6 x + 6 for numbers that end in -billion.

Only the number billions (10 9) went from the English system to the Russian language, which would be even more correct if you call it what the Americans call it - one billion since we adopted the American system. But who in our country does something according to the rules! ;-) By the way, the word trillion is sometimes used in Russian too (you can see for yourself by doing a search on Google or Yandex) and it apparently means 1000 trillion i.e. H. Billiards.

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 with their own names without Latin prefixes. There are several such numbers, but I'll talk more about them a little later.

Let's return to the notation with Latin numerals. It seems they can write numbers to infinity, but that's not entirely true. Let me explain why. First, let's see what the numbers from 1 to 10 are called 33:

The question now arises as to what comes next. What's behind the decillion? In principle, of course, it is possible to combine prefixes to create monsters like 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 the ones given above, you can only get three - Vigintillion (from lat.*viginti* - twenty), centillion (from lat.*centum* - one hundred) and one million (from lat.*mille* - one thousand). The Romans had no more than a thousand names of their own for numbers (all numbers over a thousand were compound). For example, one million (1,000,000) Romans called*decies centena milia*that is, "ten hundred thousand". And now the table indeed:

Thus, according to a similar system, the numbers are greater than 10 3003, which would have its own, non-compound name is impossible to get! Even so, numbers over a million million are known - these are the numbers outside the system. Let's finally tell you about it.

The smallest such number is innumerable (it's even in Dahl's dictionary), which means one hundred hundred, that is, 10,000. While this word is out of date and practically not used, it is curious that the word "innumerable" is widely used, meaning not a specific number at all, but an innumerable, innumerable amount of something. It is believed that the word innumerable came from ancient Egypt into European languages.

There are different opinions about the origin of this number. Some believe it came from Egypt while others believe it was only born in Ancient Greece. Be that as it may, the innumerable ones became famous 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 sandstone) Archimedes showed how one can systematically construct and name numbers of any size. In particular, if he lays 10,000 (innumerable) grains of sand in a poppy, he realizes that in the universe (a sphere with a diameter of innumerable earth diameters) there are no more than 1063 grains of sand. It is strange that modern calculations of the number of atoms in the visible universe lead to the number 1067 (just countless times more). Archimedes suggested the following names for numbers:

1 myriad \ u003d 10 4.

1 d myriad \ u003d myriad myriad \ u003d 108.

1 three-myriad \ u003d di-myriad di-myriad \ u003d 1016.

1 tetra-myriad \ u003d three-myriad three-myriad \ u003d 1032.

etc.

Googol (from the English googol) is the number ten to the hundredth power, that is, one with a hundred zeros. Googol was first described in the 1938 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 big number "googol". This number became known thanks to the search engine named after him. Google ... Note that "google" is a brand and googol is a number.

Edward Kasner.

You can often come across a mention on the internet - but it's not ...

In the famous Buddhist treatise Jaina Sutra from 100 BC Is the number asankheya (from chap. *asenci* - innumerable) equal to 10 140. This number is believed to be equal to the number of cosmic cycles required to attain nirvana.

Googolplex (Ger. *googolplex*) - a number that was also invented by Kasner with his nephew and means one with a googol of zeros, ie 10 10 100 ... This is how Kasner himself describes this "discovery":

Wisdom words are spoken at least as often by children as they are by scientists. The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to come up with a name for a very large number, namely 1 followed by a hundred zeros. He was very sure that this number was not infinite, and therefore just as certain that it had to have a name. At the same time that he proposed "googol", he gave a name to an even larger number: "Googolplex". A googolplex is much larger than a googol, but is still finite, as the inventor of the name quickly pointed out.

Math 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 Hypothesis about prime numbers. It means *e*so far *e*so far *e*to the 79th power, that is, ee e 79 ... Later Riele (te Riele, H.J.J. "About the sign of difference *P.*(x) -Li (x). *Mathematics. Comput.* 48, 323-328, 1987) reduced the Skewes number to ee 27/4, which corresponds to approximately 8.185 · 10 370. It is clear that the value of Skuse's number depends on the number *e*, then it's not a whole number, so we won't take it into account, otherwise we would have to remember other non-natural numbers - pi, e, etc.

It should be noted, however, that there is a second Skuse number, which in mathematics is called Sk2 and 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 is not valid. Sk2 is 1010 10103 1010 101000.

As you can see, the more degrees there are, the more difficult it is to understand which of the numbers is larger. For example, if you look at the Skuse numbers, it is almost impossible to find out which of these two numbers is larger without special calculations. Hence, it becomes impractical to use powers on very large numbers. In addition, you can think of such numbers (and they were already invented) when the degrees just don't fit on the page. Yes, what a side! You won't fit even a book the size of the entire universe! In this case, the question arises, how to write them down. As you can see, the problem is solvable, and mathematicians have developed various principles for writing such numbers. It is true that every mathematician who has dealt with this problem has invented his own notation, which led to the fact that there were several independent ways of writing numbers - these are the notations of Knuth, Conway, Steinhouse, etc.

Consider the notation by Hugo Steinhaus (H. Steinhaus. *Math snapshots*, 3rd ed. 1983), which is pretty straightforward. Stein House suggested writing large numbers in geometric shapes - a triangle, a square, and a circle:

Steinhaus came up with two new super big numbers. He called the number Mega and the number Megiston.

Mathematician Leo Moser refined Stenhouse's notation, which was limited by the fact that when numbers had to be written that were much larger than a megiston, difficulty and inconvenience would arise as many circles had to be drawn inside one another. Moser suggested not drawing circles, but pentagons after the squares, then hexagons, etc. He also suggested a formal notation for these polygons so that numbers can be written down without drawing complex drawings. Moser's notation looks like this:

According to Moser's notation, the stone house mega is written as 2 and the megiston as 10. In addition, Leo Moser suggested naming a polygon with the number of sides that corresponds to a mega-mega. And he suggested the number "2 in Megagon", which means 2. This number became known as the Moser number (Moser number) or simply Moser.

But Moser is not the largest number either. The largest number ever used for mathematical proofs is a limit quantity known as Graham's number, which was first used in 1977 to prove 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, the number written in Knuth's notation cannot be translated into the Moser system. So we have to explain this system too. In principle, there is nothing complicated about it. Donald Knuth (yes, yes, this is the same Knuth who wrote The Art of Programming and created the TeX editor) invented the concept of the supergrade, which he wanted to write down with arrows pointing upwards:

In general it looks like this:

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

- G1 \ u003d 3..3, where the number of supergrade arrows is 33.
- G2 \ u003d ..3, where the number of supergrade arrows is equal to G1.
- G3 \ u003d ..3, where the number of supergrade arrows is equal to G2.
- G63 \ u003d ..3, where the number of super degree arrows is equal to G62.

The G63 number came to be known as the Graham number (it is often referred to simply as the G). This number is the largest known number in the world and is even included in the Guinness Book of Records. But

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