Not included in the collection of essays. The largest numbers in mathematics

Many people are interested in questions about what they are called big numbers and what is the largest number 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 the 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 the Russian language uses the American system for naming 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

• 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 for numbers were then composite).

Compound names of large numbers

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

Compound names of large numbers

• 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 trcentillion, 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 — octientillion
• 10 2703 — nongentillion
• 10 3003 - million
• 10 6003 - duo-million
• 10 9003 - three million
• 10 15003 — quinquemilliallion
• 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 well known thanks to the Google search engine named after him.

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 proposed the number “2 in Megagon” - 2. The last number is 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 proposed to write with arrows pointing up:

In general

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.

Sometimes people who are not involved in mathematics wonder: what is the largest number? On the one hand, the answer is obvious - infinity. Bores will even clarify that “plus infinity” or “+∞” is used by mathematicians. But this answer will not convince the most corrosive, especially since it is not natural number, but a mathematical abstraction. But having understood the issue well, they can discover a very interesting problem.

Indeed, the size limit is in this case does not exist, but there is a limit to human imagination. Each number has a name: ten, one hundred, billion, sextillion, and so on. But where does people's imagination end?

Not to be confused with a trademark of Google Corporation, although they have a common origin. This number is written as 10100, that is, one followed by a hundred zeros. It is difficult to imagine, but it was actively used in mathematics.

It's funny that it was invented by a child - the nephew of the mathematician Edward Kasner. In 1938, my uncle entertained his younger relatives with discussions about very large numbers. To the child’s indignation, it turned out that such a wonderful number had no name, and he gave his own version. Later, my uncle inserted it into one of his books, and the term stuck.

Theoretically, a googol is a natural number, because it can be used for counting. But it’s unlikely that anyone will have the patience to count to the end. Therefore, only theoretically.

As for the name of the company Google, a common mistake has crept in here. The first investor and one of the co-founders was in a hurry when he wrote out the check and missed the letter “O”, but in order to cash it, the company had to be registered with this particular spelling.

Googolplex

This number is a derivative of googol, but is significantly larger than it. The prefix “plex” means raising ten to a power equal to the base number, so guloplex is 10 to the power of 10 to the power of 100 or 101000.

The resulting number exceeds the number of particles in the observable Universe, which is estimated to be about 1080 degrees. But this did not stop scientists from increasing the number by simply adding the prefix “plex” to it: googolplexplex, googolplexplexplex and so on. And for particularly perverted mathematicians, they invented a variant of magnification without the endless repetition of the prefix “plex” - they simply put Greek numbers in front of it: tetra (four), penta (five) and so on, up to deca (ten). The last option sounds like a googoldecaplex and means a tenfold cumulative repetition of the procedure of raising the number 10 to the power of its base. The main thing is not to imagine the result. You still won’t be able to realize it, but it’s easy to get mentally injured.

48th Mersen number

Main characters: Cooper, his computer and a new prime number

Relatively recently, about a year ago, we managed to discover the next, 48th Mersen number. On this moment it is the largest prime number in the world. Let us recall that prime numbers are those that are divisible without a remainder only by one and themselves. The simplest examples are 3, 5, 7, 11, 13, 17 and so on. The problem is that the further into the wilds, the less common such numbers are. But the more valuable is the discovery of each next one. For example, the new prime number consists of 17,425,170 digits if represented in the form of the decimal number system familiar to us. The previous one had about 12 million characters.

It was discovered by the American mathematician Curtis Cooper, who delighted the mathematical community with a similar record for the third time. It took 39 days of running his personal computer just to check his result and prove that this number was indeed prime.

This is what the Graham number looks like in Knuth arrow notation. It’s difficult to say how to decipher this without having a complete higher education in theoretical mathematics. It is also impossible to write it down in our usual decimal form: the observable Universe is simply not able to accommodate it. Building one degree at a time, as is the case with googolplexes, is also not a solution.

Good formula, just incomprehensible

So why do we need this seemingly useless number? Firstly, for the curious, it was placed in the Guinness Book of Records, and this is already a lot. Secondly, it was used to solve a problem included in the Ramsey problem, which is also unclear, but sounds serious. Thirdly, this number is recognized as the largest ever used in mathematics, and not in comic proofs or intellectual games, but to solve a very specific mathematical problem.

Attention! the following information dangerous for your mental health! By reading it, you accept responsibility for all consequences!

For those who want to test their mind and meditate on the Graham number, we can try to explain it (but only try).

Imagine 33. It's pretty easy - it turns out 3*3*3=27. What if we now raise three to this number? The result is 3 3 to the 3rd power, or 3 27. In decimal notation, this is equal to 7,625,597,484,987. A lot, but for now it can be realized.

In Knuth's arrow notation, this number can be displayed somewhat more simply - 33. But if you add only one arrow, it becomes more complicated: 33, which means 33 to the power of 33 or in power notation. If we expand to decimal notation, we get 7,625,597,484,987 7,625,597,484,987. Are you still able to follow your thoughts?

Next stage: 33= 33 33 . That is, you need to calculate it wild number from the previous action and raise it to the same power.

And 33 is only the first of 64 terms of Graham's number. To get the second one, you need to calculate the result of this mind-blowing formula and substitute the corresponding number of arrows into diagram 3(...)3. And so on, another 63 times.

I wonder if anyone other than him and a dozen other supermathematicians will be able to get to at least the middle of the sequence without going crazy?

Did you understand something? We are not. But what a thrill!

Why do we need the largest numbers? This is difficult for the average person to understand and comprehend. But with their help, a few specialists are able to introduce new technological toys to ordinary people: phones, computers, tablets. Ordinary people are also unable to understand how they work, but they are happy to use them for their entertainment. And everyone is happy: ordinary people get their toys, “supernerds” have the opportunity to continue playing their mind games.

The question “What is the largest number in the world?” is, to say the least, incorrect. There are different number systems - decimal, binary and hexadecimal, as well as various categories of numbers - semi-prime and simple, the latter being divided into legal and illegal. In addition, there are Skewes numbers, Steinhouse and other mathematicians who, either as a joke or seriously, invent and present to the public such exotics as “Megiston” or “Moser”.

What is the largest number in the world in decimal system

Of the decimal system, most “non-mathematicians” are familiar with million, billion and trillion. Moreover, if Russians generally associate a million with a dollar bribe that can be carried away in a suitcase, then where to stuff a billion (not to mention a trillion) North American banknotes - most people lack imagination. However, in the theory of large numbers there are such concepts as quadrillion (ten to the fifteenth power - 1015), sextillion (1021) and octillion (1027).

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

In 1938, due to the development applied mathematics and the expansion of the micro- and macrocosm, professor at Columbia University (USA), Edward Kasner published in the pages of the journal “Scripta Mathematica” his nine-year-old nephew’s proposal to use “googol” as the largest number in the decimal system. – representing ten to the hundredth power (10100), which on paper is expressed as one followed by one hundred zeros. However, they did not stop there and a few years later proposed introducing a new largest number in the world - the “googolplex”, which represents ten raised to the tenth power and again raised to the hundredth power - (1010)100, expressed by a unit, to which a googol of zeros is assigned to the right. However, for the majority of even professional mathematicians, both “googol” and “googolplex” are of purely speculative interest, and it is unlikely that they can be applied to anything in everyday practice.

Exotic numbers

What is the largest number in the world among prime numbers - those that can only be divided by themselves and one. One of the first to record the largest prime number, equal to 2,147,483,647, was the great mathematician Leonhard Euler. As of January 2016, this number is recognized as the expression calculated as 274,207,281 – 1.

Countless different numbers surround us every day. Surely many people have at least once wondered what number is considered the largest. You can simply say to a child that this is a million, but adults understand perfectly well that other numbers follow a million. For example, all you have to do is add one to a number each time, and it will become larger and larger - this happens ad infinitum. But if you look at the numbers that have names, you can find out what the largest number in the world is called.

The appearance of number names: what methods are used?

Today there are 2 systems according to which names are given to numbers - American and English. The first is quite simple, and the second is the most common throughout the world. The American one allows you to give names to large numbers as follows: first, the ordinal number in Latin is indicated, and then the suffix “million” is added (the exception here is million, meaning a thousand). This system is used by Americans, French, Canadians, and it is also used in our country.

English is widely used in England and Spain. According to it, numbers are named as follows: the numeral in Latin is “plus” with the suffix “illion”, and the next (a thousand times larger) number is “plus” “billion”. For example, the trillion comes first, the trillion comes after it, the quadrillion comes after the quadrillion, etc.

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

Extra-system numbers

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

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

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

Google got its name in honor of googol ( search system). Then 1 with a googol of zeros (1010100) represents a googolplex - Kasner also came up with this name.

Even larger than the googolplex is the Skuse number (e to the power of e to the power of e79), proposed by Skuse in his proof of the Rimmann conjecture about prime numbers (1933). There is another Skuse number, but it is used when the Rimmann hypothesis is not 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.

Answering such a difficult question as to what it is, the largest number in the world, it should first be noted that today there are 2 accepted ways of naming numbers - English and American. According to the English system, the suffixes -billion or -million are added to each large number in order, resulting in the numbers million, billion, trillion, trillion, and so on. If we proceed from the American system, then according to it, the suffix -million must be added to each large number, resulting in the formation of the numbers trillion, quadrillion and large ones. It should also be noted here that the English number system is more common in modern world, and the numbers in it are quite sufficient for normal functioning all systems of our world.

Of course, the answer to the question about the largest number from a logical point of view cannot be unambiguous, because if you just add one to each subsequent digit, you get a new larger number, therefore, this process has no limit. However, oddly enough, there is still the largest number in the world and it is listed in the Guinness Book of Records.

Graham's number is the largest number in the world

It is this number that is recognized in the world as the largest in the Book of Records, but it is very difficult to explain what it is and how large it is. In a general sense, these are triplets multiplied together, resulting in a number that is 64 orders of magnitude higher than the point of understanding of each person. As a result, we can only give the final 50 digits of Graham's number – 0322234872396701848518 64390591045756272 62464195387.

Googol number

The history of this number is not as complex as the one mentioned above. Thus, the American mathematician Edward Kasner, talking with his nephews about large numbers, could not answer the question of how to name numbers that have 100 zeros or more. A resourceful nephew suggested his own name for such numbers - googol. It should be noted that this number does not have much practical significance, however, it is sometimes used in mathematics to express infinity.

Googleplex

This number was also invented by mathematician Edward Kasner and his nephew Milton Sirotta. In a general sense, it represents a number to the tenth power of a googol. Answering the question of many inquisitive people, how many zeros are in the Googleplex, it is worth noting that in classic version There is no way to imagine this number, even if you cover all the paper on the planet with classical zeros.

Skewes number

Another contender for the title of largest number is the Skewes number, proven by John Littwood in 1914. According to the evidence given, this number is approximately 8.185 10370.

Moser number

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

As you can already see, a huge number of mathematicians have made efforts to find it - greatest number in the world. The extent to which these attempts were successful, of course, is not for us to judge, however, it must be noted that the real applicability of such numbers is doubtful, because they are not even amenable to human understanding. In addition, there will always be a number that will be greater if you perform a very simple mathematical operation +1.