The largest power of a number. The largest numbers in mathematics

Many people are interested in questions about what large numbers are called and what number is the largest in the world. With these interesting questions and we will look into this in this article.

Story

The southern and eastern Slavic peoples used alphabetical numbering to record numbers, and only those letters that are in greek alphabet. A special “title” icon was placed above the letter that designated the number. The numerical values of the letters increased in the same order as the letters in the Greek alphabet (in the Slavic alphabet the order of the letters was slightly different). In Russia, Slavic numbering was preserved until the end of the 17th century, and under Peter I they switched to “Arabic numbering,” which we still use today.

The names of the numbers also changed. Thus, until the 15th century, the number “twenty” was designated as “two tens” (two tens), and then it was shortened for faster pronunciation. The number 40 was called “fourty” until the 15th century, then it was replaced by the word “forty,” which originally meant a bag containing 40 squirrel or sable skins. The name “million” appeared in Italy in 1500. It was formed by adding an augmentative suffix to the number “mille” (thousand). Later this name came to the Russian language.

In the ancient (18th century) “Arithmetic” of Magnitsky, a table of the names of numbers is given, brought to the “quadrillion” (10^24, according to the system through 6 digits). Perelman Ya.I. the book “Entertaining Arithmetic” gives the names of large numbers of that time, slightly different from today: septillion (10^42), octalion (10^48), nonalion (10^54), decalion (10^60), endecalion (10^ 66), dodecalion (10^72) and it is written that “there are no further names.”

Ways to construct names for large numbers

There are 2 main ways to name large numbers:

American system, which is used in the USA, Russia, France, Canada, Italy, Turkey, Greece, Brazil. The names of large numbers are constructed quite simply: the Latin ordinal number comes first, and the suffix “-million” is added to it at the end. An exception is the number “million,” which is the name of the number thousand (mille) and the augmentative suffix “-million.” The number of zeros in a number, which is written according to the American system, can be found out by the formula: 3x+3, where x is the Latin ordinal number
English system most common in the world, it is used in Germany, Spain, Hungary, Poland, Czech Republic, Denmark, Sweden, Finland, Portugal. The names of numbers according to this system are constructed as follows: the suffix “-million” is added to the Latin numeral, the next number (1000 times larger) is the same Latin numeral, but the suffix “-billion” is added. The number of zeros in a number, which is written according to the English system and ends with the suffix “-million,” can be found out by the formula: 6x+3, where x is the Latin ordinal number. The number of zeros in numbers ending with the suffix “-billion” can be found using the formula: 6x+6, where x is the Latin ordinal number.

Only the word billion passed from the English system into the Russian language, which is still more correctly called as the Americans call it - billion (since in Russian it is used American system names of numbers).

In addition to numbers that are written according to the American or English system using Latin prefixes, non-system numbers are known that have their own names without Latin prefixes.

Proper names for large numbers

Number		Latin numeral	Name	Practical significance
10 1	10		ten	Number of fingers on 2 hands
10 2	100		one hundred	About half the number of all states on Earth
10 3	1000		thousand	Approximate number of days in 3 years
10 6	1000 000	unus (I)	million	5 times more than the number of drops per 10 liter. bucket of water
10 9	1000 000 000	duo (II)	billion (billion)	Estimated Population of India
10 12	1000 000 000 000	tres (III)	trillion
10 15	1000 000 000 000 000	quattor (IV)	quadrillion	1/30 of the length of a parsec in meters
10 18		quinque (V)	quintillion	1/18th of the number of grains from the legendary award to the inventor of chess
10 21		sex (VI)	sextillion	1/6 of the mass of planet Earth in tons
10 24		septem (VII)	septillion	Number of molecules in 37.2 liters of air
10 27		octo (VIII)	octillion	Half of Jupiter's mass in kilograms
10 30		novem (IX)	quintillion	1/5 of all microorganisms on the planet
10 33		decem (X)	decillion	Half the mass of the Sun in grams

Vigintillion (from Latin viginti - twenty) - 10 63
Centillion (from Latin centum - one hundred) - 10,303
Million (from Latin mille - thousand) - 10 3003

For numbers greater than a thousand, the Romans did not have their own names (all names of numbers were then composite).

Compound names of large numbers

In addition to proper names, for numbers greater than 10 33 you can get compound names by combining prefixes.

Compound names of large numbers

Number	Latin numeral	Name	Practical significance
10 36	undecim (XI)	andecillion
10 39	duodecim (XII)	duodecillion
10 42	tredecim (XIII)	thredecillion	1/100 of the number of air molecules on Earth
10 45	quattuordecim (XIV)	quattordecillion
10 48	quindecim (XV)	quindecillion
10 51	sedecim (XVI)	sexdecillion
10 54	septendecim (XVII)	septemdecillion
10 57		octodecillion	So many elementary particles in the sun
10 60		novemdecillion
10 63	viginti (XX)	vigintillion
10 66	unus et viginti (XXI)	anvigintillion
10 69	duo et viginti (XXII)	duovigintillion
10 72	tres et viginti (XXIII)	trevigintillion
10 75		quattorvigintillion
10 78		quinvigintillion
10 81		sexvigintillion	So many elementary particles in the universe
10 84		septemvigintillion
10 87		octovigintillion
10 90		novemvigintillion
10 93	triginta (XXX)	trigintillion
10 96		antigintillion

10 123 - quadragintillion
10 153 — quinquagintillion
10 183 — sexagintillion
10,213 - septuagintillion
10,243 — octogintillion
10,273 — nonagintillion
10 303 - centillion

Further names can be obtained by direct or reverse order of Latin numerals (which is correct is not known):

10 306 - ancentillion or centunillion
10 309 - duocentillion or centullion
10 312 - trcentillion or centtrillion
10 315 - quattorcentillion or centquadrillion
10 402 - tretrigyntacentillion or centretrigintillion

The second spelling is more consistent with the construction of numerals in Latin and avoids ambiguities (for example, in the number trecentillion, which according to the first spelling is both 10,903 and 10,312).

10 603 - decentillion
10,903 - trcentillion
10 1203 - quadringentillion
10 1503 — quingentillion
10 1803 - sescentillion
10 2103 - septingentillion
10 2403 — octingentillion
10 2703 — nongentillion
10 3003 - million
10 6003 - duo-million
10 9003 - three million
10 15003 — quinquemillillion
10 308760 -ion
10 3000003 — mimiliaillion
10 6000003 — duomimiliaillion

Myriad– 10,000. The name is outdated and practically not used. However, the word “myriads” is widely used, which does not mean a specific number, but an innumerable, uncountable number of something.

Googol ( English . googol) — 10 100. The American mathematician Edward Kasner first wrote about this number in 1938 in the journal Scripta Mathematica in the article “New Names in Mathematics.” According to him, his 9-year-old nephew Milton Sirotta suggested calling the number this way. This number became publicly known thanks to the Google search engine named after it.

Asankheya(from Chinese asentsi - uncountable) - 10 1 4 0 . This number is found in the famous Buddhist treatise Jaina Sutra (100 BC). It is believed that this number is equal to the number of cosmic cycles required to achieve nirvana.

Googolplex ( English . Googolplex) — 10^10^100. This number was also invented by Edward Kasner and his nephew; it means one followed by a googol of zeros.

Skewes number (Skewes' number, Sk 1) means e to the power of e to the power of e to the power of 79, that is, e^e^e^79. This number was proposed by Skewes in 1933 (Skewes. J. London Math. Soc. 8, 277-283, 1933.) when proving the Riemann hypothesis concerning prime numbers. Later, Riele (te Riele, H. J. J. “On the Sign of the Difference П(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. However, this number is not an integer, so it is not included in the table of large numbers.

Second Skewes number (Sk2) equals 10^10^10^10^3, that is, 10^10^10^1000. This number was introduced by J. Skuse in the same article to indicate the number up to which the Riemann hypothesis is valid.

For super-large numbers it is inconvenient to use powers, so there are several ways to write numbers - Knuth, Conway, Steinhouse notations, etc.

Hugo Steinhouse suggested writing large numbers inside geometric shapes(triangle, square and circle).

Mathematician Leo Moser refined Steinhouse's notation, proposing to draw pentagons, then hexagons, etc. after squares rather than circles. Moser also proposed a formal notation for these polygons so that the numbers could be written without drawing complex pictures.

Steinhouse came up with two new super-large numbers: Mega and Megiston. In Moser notation they are written as follows: Mega – 2, Megiston– 10. Leo Moser also proposed to call a polygon with the number of sides equal to mega – megagon, and also suggested the number “2 in Megagon” - 2. Last number known as Moser's number or just like Moser.

There are numbers larger than Moser. The largest number that has been used in a mathematical proof is number Graham(Graham's number). It was first used in 1977 to prove an estimate in Ramsey theory. This number is associated with bichromatic hypercubes and cannot be expressed without a special 64-level system of special mathematical symbols introduced by Knuth in 1976. Donald Knuth (who wrote “The Art of Programming” and created the TeX editor) came up with the concept of superpower, which he suggested writing with arrows pointing up:

IN general view

Graham proposed G-numbers:

The number G 63 is called Graham's number, often denoted simply G. This number is the largest known number in the world and is listed in the Guinness Book of Records.

John Sommer

Place zeros after any number or multiply with tens raised to an arbitrary power. It won't seem enough. It will seem like a lot. But the bare records are still not very impressive. The piling up of zeros in the humanities causes not so much surprise as a slight yawn. In any case, to any largest number in the world that you can imagine, you can always add another one... And the number will come out even larger.

And yet, are there words in Russian or any other language to denote very large numbers? Those that are more than a million, a billion, a trillion, a billion? And in general, how much is a billion?

It turns out that there are two systems for naming numbers. But not Arab, Egyptian, or any other ancient civilizations, but American and English.

In the American system numbers are called like this: take the Latin numeral + - illion (suffix). This gives the numbers:

Trillion - 1,000,000,000,000 (12 zeros)

Quadrillion - 1,000,000,000,000,000 (15 zeros)

Quintillion - 1 followed by 18 zeros

Sextillion - 1 and 21 zeros

Septillion - 1 and 24 zeros

octillion - 1 followed by 27 zeros

Nonillion - 1 and 30 zeros

Decillion - 1 and 33 zeros

The formula is simple: 3 x+3 (x is a Latin numeral)

In theory, there should also be the numbers anilion (unus in Latin - one) and duolion (duo - two), but, in my opinion, such names are not used at all.

English number naming system more widespread.

Here, too, the Latin numeral is taken and the suffix -million is added to it. However, the name of the next number, which is 1,000 times greater than the previous one, is formed using the same Latin number and the suffix - illiard. That is:

Trillion - 1 and 21 zeros (in the American system - sextillion!)

Trillion - 1 and 24 zeros (in the American system - septillion)

Quadrillion - 1 and 27 zeros

Quadrillion - 1 followed by 30 zeros

Quintillion - 1 and 33 zeros

Quinilliard - 1 and 36 zeros

Sextillion - 1 and 39 zeros

Sextillion - 1 and 42 zeros

The formulas for counting the number of zeros are:

For numbers ending in - illion - 6 x+3

For numbers ending in - billion - 6 x+6

As you can see, confusion is possible. But let us not be afraid!

In Russia, the American system of naming numbers has been adopted. We borrowed the name of the number “billion” from the English system - 1,000,000,000 = 10 9

Where is the “cherished” billion? - But a billion is a billion! American style. And although we use the American system, we took “billion” from the English one.

Using the Latin names of numbers and the American system, we name the numbers:

- vigintillion- 1 and 63 zeros

- centillion- 1 and 303 zeros

- million- one and 3003 zeros! Oh-ho-ho...

But this, it turns out, is not all. There are also non-system numbers.

And the first of them is probably myriad- one hundred hundreds = 10,000

Google(it is in his honor that the famous search engine) - one and one hundred zeros

In one of the Buddhist treatises the number is named asankheya- one and one hundred and forty zeros!

Number name googolplex(like googol) was invented by the English mathematician Edward Kasner and his nine-year-old nephew - unit c - dear mother! - googol zeros!!!

But that's not all...

The mathematician Skuse named the Skuse number after himself. It means e to a degree e to a degree e to the power of 79, that is e e e 79

And then a big difficulty arose. You can come up with names for numbers. But how to write them down? The number of degrees of degrees of degrees is already such that it simply cannot be removed onto the page! :)

And then some mathematicians began to write numbers in geometric figures. And they say he was the first to come up with this method of recording outstanding writer and thinker Daniil Ivanovich Kharms.

And yet, what is the BIGGEST NUMBER IN THE WORLD? - It’s called STASPLEX and is equal to G 100,

where G is the Graham number, the most large number, ever used in mathematical proofs.

This number - stasplex - was invented by a wonderful person, our compatriot Stas Kozlovsky, LJ to which I direct you :) - ctac

There are numbers that are so incredibly, incredibly large that it would take the entire universe to even write them down. But here's what's really crazy... some of these unfathomably large numbers are crucial to understanding the world.

When I say “the largest number in the universe,” I really mean the largest significant number, the maximum possible number that is useful in some way. There are many contenders for this title, but I'll warn you right away: there really is a risk that trying to figure it all out will blow your mind. And besides, with too much math, you won't have much fun.

Googol and googolplex

Edward Kasner

We could start with what are quite possibly the two biggest numbers you've ever heard of, and these are indeed the two biggest numbers that have generally accepted definitions in English. (There is a fairly precise nomenclature used to denote numbers as large as you would like, but these two numbers you will not find in dictionaries nowadays.) Googol, since it became world famous (albeit with errors, note. in fact it's googol) Google view, was born in 1920 as a way to get kids interested in big numbers.

To this end, Edward Kasner (pictured) took his two nephews, Milton and Edwin Sirott, for a walk through the New Jersey Palisades. He invited them to come up with any ideas, and then nine-year-old Milton suggested “googol.” Where he got this word from is unknown, but Kasner decided that or a number in which one hundred zeros follow the unit will henceforth be called a googol.

But young Milton did not stop there; he proposed an even larger number, the googolplex. This is a number, according to Milton, in which the first place is 1, and then as many zeros as you could write before you got tired. While the idea is fascinating, Kasner decided 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 a random buffoon could become a mathematician superior to Albert Einstein simply because he has more stamina.

So Kasner decided that a googolplex would be , or 1, and then a googol of zeros. Otherwise, and in notation similar to that which we will deal with for other numbers, we will say that a googolplex is . To show how fascinating this is, Carl Sagan once noted that it is physically impossible to write down all the zeros of a googolplex because there simply isn't enough space in the universe. If we fill the entire volume of the observable Universe small particles dust approximately 1.5 microns in size, then the number in various ways the location of these particles will be approximately equal to one googolplex.

Linguistically speaking, googol and googolplex are probably the two largest significant numbers (at least in the English language), but, as we will now establish, there are infinitely many ways to define “significance.”

Real world

If we talk about the largest significant number, there is a reasonable argument that this really means that we need to find the largest number with a value that actually exists in the world. We can start with the current human population, which is currently around 6920 million. World GDP in 2010 was estimated to be around $61,960 billion, but both of these numbers are insignificant compared to the approximately 100 trillion cells that make up the human body. Of course, none of these numbers compare to full number particles in the Universe, which is generally considered to be approximately , and this number is so large that our language does not have a word corresponding to it.

We can play a little with the systems of measures, making the numbers larger and larger. Thus, the mass of the Sun in tons will be less than in pounds. A great way to do this is to use the Planck system of units, which are the smallest possible measures for which the laws of physics still apply. For example, the age of the Universe in Planck time is about . If we return to the first unit of Planck time after Big Bang, then we will see that the density of the Universe was then . We're getting more and more, but we haven't even reached googol yet.

The largest number with any real world application - or in this case real world application - is probably one of the latest estimates of the number of universes in the multiverse. This number is so large that the human brain will literally not be able to perceive all these different universes, since the brain is only capable of approximately configurations. In fact, this number is probably the largest number that makes any practical sense unless you take into account the idea of the multiverse as a whole. However, there are still much larger numbers lurking there. But in order to find them we must go into the realm of pure mathematics, and no better start than prime numbers.

Mersenne primes

Part of the challenge is coming up with a good definition of what a “significant” number is. One way is to think in terms of prime and composite numbers. A prime number, as you probably remember from school mathematics, is any natural number(note not equal to one), which is divisible only by and itself. So, and are prime numbers, and and are composite numbers. This means that any composite number can ultimately be represented by its prime factors. In some ways, the number is more important than, say, , because there is no way to express it in terms of the product of smaller numbers.

Obviously we can go a little further. , for example, is actually just , which means that in a hypothetical world where our knowledge of numbers is limited to , a mathematician can still express the number . But the next number is prime, which means that the only way to express it is to directly know about its existence. This means that the largest known prime numbers play an important role, but, say, a googol - which is ultimately just a collection of numbers and , multiplied together - actually does not. And since prime numbers are basically random, there is no known way to predict that an incredibly large number will actually be prime. To this day, discovering new prime numbers is a difficult undertaking.

Mathematicians Ancient Greece had an idea about prime numbers, at least as early as 500 BC, and 2000 years later people still knew which numbers were prime only up to about 750. Thinkers in Euclid's time saw the possibility of simplification, but until the Renaissance mathematicians could not really put this into practice . These numbers are known as Mersenne numbers, named after the 17th century French scientist Marin Mersenne. The idea is quite simple: a Mersenne number is any number of the form . So, for example, , and this number is prime, the same is true for .

It is much faster and easier to determine Mersenne primes than any other kind of prime number, and computers have been hard at work searching for them for the past six decades. Until 1952, the largest known prime number was a number—a number with digits. In the same year, the computer calculated that the number is prime, and this number consists of digits, which makes it much larger than a googol.

Computers have been on the hunt ever since, and currently the -th Mersenne number is the largest prime number. known to mankind. Discovered in 2008, it amounts to a number with almost millions of digits. This is the biggest known number, which cannot be expressed in terms of any smaller numbers, and if you want help finding an even larger Mersenne number, you (and your computer) can always join the search at http://www.mersenne.org/.

Skewes number

Stanley Skews

Let's look at prime numbers again. As I said, they behave fundamentally wrong, meaning that there is no way to predict what the next prime number will be. Mathematicians have been forced to resort to some pretty fantastic measurements to come up with some way to predict future prime numbers, even in some nebulous way. The most successful of these attempts is probably the prime number counting function, which was invented in the late 18th century by the legendary mathematician Carl Friedrich Gauss.

I'll spare you the more complicated math - we have a lot more to come anyway - but the gist of the function is this: for any integer, you can estimate how many prime numbers there are that are smaller than . For example, if , the function predicts that there should be prime numbers, if there should be prime numbers smaller than , and if , then there should be smaller numbers that are prime.

The arrangement of the prime numbers is indeed irregular and is merely an approximation of the actual number of prime numbers. In fact, we know that there are prime numbers less than , prime numbers less than , and prime numbers less than . This is an excellent estimate, to be sure, but it is always only an estimate... and, more specifically, an estimate from above.

In all known cases to , the function that finds the number of primes slightly overestimates the actual number of primes smaller than . Mathematicians once thought that this would always be the case, ad infinitum, and that this would certainly apply to some unimaginably huge numbers, but in 1914 John Edensor Littlewood proved that for some unknown, unimaginably huge number, this function would begin to produce fewer primes, and then it will switch between the top estimate and the bottom estimate an infinite number of times.

The hunt was for the starting point of the races, and then Stanley Skewes appeared (see photo). In 1933, he proved that the upper limit when a function approximating the number of prime numbers first produces a smaller value is the number . It is difficult to truly understand even in the most abstract sense what this number actually represents, and from this point of view it was the largest number ever used in a serious mathematical proof. Since then, mathematicians have been able to reduce the upper bound to a relatively small number, but the original number remains known as the Skewes number.

So how big is the number that dwarfs even the mighty googolplex? In The Penguin Dictionary of Curious and Interesting Numbers, David Wells talks about one way in which the mathematician Hardy was able to conceptualize the size of the Skuse number:

“Hardy thought it was “the largest number ever served for any particular purpose in mathematics,” and suggested that if a game of chess were played with all the particles of the universe as pieces, one move would consist of swapping two particles, and the game would stop when the same position was repeated a third time, then the number of all possible games would be approximately equal to Skuse's number.'

One last thing before we move on: we talked about the smaller of the two Skewes numbers. There is another Skuse number, which the mathematician discovered in 1955. The first number is derived from the fact that the so-called Riemann hypothesis is true - this is a particularly difficult hypothesis in mathematics that remains unproven, very useful when we're talking about about prime numbers. However, if the Riemann hypothesis is false, Skuse found that the starting point of the jumps increases to .

Problem of magnitude

Before we get to the number that makes even the Skewes number look tiny, we need to talk a little about scale, because otherwise we have no way of assessing where we're going to go. First let's take a number - it's a tiny number, so small that people can actually have an intuitive understanding of what it means. There are very few numbers that fit this description, since numbers greater than six cease to be separate numbers and become “several”, “many”, etc.

Now let's take , i.e. . Although we actually cannot intuitively, as we did for the number, understand what it is, it is very easy to imagine what it is. So far so good. But what happens if we move to ? This is equal to , or . We are very far from being able to imagine this quantity, like any other very large one - we lose the ability to comprehend individual parts somewhere around a million. (Really, it's crazy large number It would take a while to actually count to a million of anything, but the fact is that we are still capable of perceiving that number.)

However, although we cannot imagine, we are at least able to understand general outline, what is 7600 billion, perhaps comparing it to something like US GDP. We have moved from intuition to representation to simple understanding, but at least we still have some gap in our understanding of what a number is. That's about to change as we move another rung up the ladder.

To do this, we need to move to a notation introduced by Donald Knuth, known as arrow notation. This notation can be written as . When we then go to , the number we get will be . This is equal to where the total of threes is. We have now far and truly surpassed all the other numbers we have already talked about. After all, even the largest of them had only three or four terms in the indicator series. For example, even the super-Skuse number is “only” - even with the allowance for the fact that both the base and the exponents are much larger than , it is still absolutely nothing compared to the size of a number tower with a billion members.

It is obvious that there is no way to comprehend so much huge numbers...and yet the process by which they are created can still be understood. We couldn't understand the real quantity that is given by a tower of powers with a billion triplets, but we can basically imagine such a tower with many terms, and a really decent supercomputer would be able to store such towers in memory even if it couldn't calculate their actual values .

This is becoming more and more abstract, but it will only get worse. You might think that a tower of degrees whose exponent length is (moreover, in previous version this post I made exactly this mistake), but it's simple. In other words, imagine that you have the ability to calculate exact value power tower of triplets, which is made up of elements, and then you took that value and created a new tower with as many in it... as gives .

Repeat this process with each subsequent number ( note starting from the right) until you do it times, and then finally you get . This is a number that is simply incredibly large, but at least the steps to get it seem understandable if you do everything very slowly. We can no longer understand the numbers or imagine the procedure by which they are obtained, but at least we can understand the basic algorithm, only in a long enough time.

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

Graham number (Graham)

Ronald Graham

This is how you get Graham's number, which holds a place in the Guinness Book of World Records as the largest number ever used in a mathematical proof. It is absolutely impossible to imagine how big it is, and equally 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. Mathematician Ronald Graham (see photo) wanted to find out at what least number measurements, certain properties of the hypercube will remain stable. (Sorry for such a vague explanation, but I'm sure we all need to get at least two degrees in math to make it more accurate.)

In any case, the Graham number is an upper estimate of this minimum number of dimensions. So how big is this upper bound? Let's return to the number, so large that we can only vaguely understand the algorithm for obtaining it. Now, instead of just jumping up one more level to , we will count the number that has arrows between the first and last three. We are now far beyond even the slightest understanding of what this number is or even what we need to do to calculate it.

Now let's repeat this process once ( note on every next step we write the number of arrows, equal to the number obtained in the previous step).

This, ladies and gentlemen, is Graham's number, which is about an order of magnitude higher than the point of human understanding. It is a number that is so much greater than any number you can imagine—it is so much greater than any infinity you could ever hope to imagine—it simply defies even the most abstract description.

But here's a strange thing. Since the Graham number is basically just triplets multiplied together, we know some of its properties without actually calculating it. We can't represent the Graham number using any familiar notation, even if we used the entire universe to write it down, but I can tell you the last twelve digits of the Graham number right now: . And that's not all: we know at least the last digits of Graham's number.

Of course, it's worth remembering that this number is only an upper bound in Graham's original problem. It is possible that the actual number of measurements required to perform the desired property much, much less. In fact, it has been believed since the 1980s, according to most experts in the field, that there are actually only six dimensions - a number so small that we can understand it intuitively. Since then the lower limit has been increased to , but there is still a very big chance that the solution to Graham's problem does not lie anywhere near a number as large as Graham's number.

Towards infinity

So are there numbers greater than Graham's number? There is, of course, for starters there is the Graham number. As for the significant number... well, there are some fiendishly complex areas of mathematics (particularly the area known as combinatorics) and computer science in which numbers even larger than Graham's number occur. But we have almost reached the limit of what I can hope will ever be rationally explained. For those foolhardy enough to go even further, further reading is suggested at your own risk.

Well, now an amazing quote that is attributed to Douglas Ray ( note Honestly, it sounds pretty funny:

“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.

June 17th, 2015

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. The exception is the name "million" which is the name of the number thousand (lat. mille) and the magnifying suffix -illion (see table). This is how 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 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 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 possible, of course, 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 trademark, 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 is the Skewes number, which 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 Skewes 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 himself 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 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 limit value, known as Graham's number, first used in 1977 to prove an estimate in Ramsey theory. It is related to 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 the 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. Oh, here you go

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, a trillion comes first, followed by a trillion, followed by a quadrillion and so on.

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 (search engine) got its name in honor of googol. 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 true. 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.