The largest number in the world. The largest number in the world

The world of science is simply amazing with its knowledge. However, even the most brilliant person in the world will not be able to comprehend them all. But you need to strive for this. That is why in this article I want to figure out what it is, the most big number.

About systems

First of all, it is necessary to say that there are two systems for naming numbers in the world: American and English. Depending on this, the same number can be called differently, although it has the same meaning. And at the very beginning, you need to deal with these nuances in order to avoid uncertainty and confusion.

American system

It will be interesting that this system is used not only in America and Canada, but also in Russia. In addition, it also has its own scientific name: a system for naming numbers with a short scale. What are they called in this system? big numbers? So, the secret is quite simple. At the very beginning there will be a Latin ordinal number, after which the well-known suffix “-million” will simply be added. The following fact will be interesting: translated from Latin language the number "million" can be translated as "thousands". The following numbers belong to the American system: a trillion is 10 12, a quintillion is 10 18, an octillion is 10 27, etc. It will also be easy to figure out how many zeros are written in the number. To do this you need to know simple formula: 3*x + 3 (where “x” in the formula is a Latin numeral).

English system

However, despite the simplicity of the American system, the English system is still more widespread in the world, which is a system for naming numbers with a long scale. Since 1948, it has been used in countries such as France, Great Britain, Spain, as well as in countries - former colonies England and Spain. The construction of numbers here is also quite simple: to Latin designation add the suffix “-million”. Further, if the number is 1000 times larger, the suffix “-billion” is added. How can you find out the number of hidden zeros in a number?

1. If the number ends in “-million”, you will need the formula 6 * x + 3 (“x” is a Latin numeral).
2. If the number ends in “-billion”, you will need the formula 6 * x + 6 (where “x”, again, is a Latin numeral).

Examples

At this stage, as an example, we can consider how the same numbers will be called, but on a different scale.

You can easily see that the same name in different systems means different numbers. For example, a trillion. Therefore, when considering a number, you still first need to find out according to what system it is written.

Extra-system numbers

It is worth saying that, in addition to system ones, there are also non-system numbers. Perhaps the largest number was lost among them? It's worth looking into this.

1. Googol. This is the number ten to the hundredth power, that is, one followed by one hundred zeros (10,100). This number was first mentioned back in 1938 by scientist Edward Kasner. Very interesting fact: worldwide search system“Google” was named after a rather large number at that time - googol. And the name was invented by Kasner’s young nephew.
2. Asankheya. This is very interesting name, which is translated from Sanskrit as “innumerable.” Its numerical value is one with 140 zeros - 10 140. The following fact will be interesting: this was known to people back in 100 BC. e., as evidenced by the entry in the Jaina Sutra, a famous Buddhist treatise. This number was considered special, because it was believed that the same number of cosmic cycles was needed to achieve nirvana. Also at that time this number was considered the largest.
3. Googolplex. This number was invented by the same Edward Kasner and his aforementioned nephew. Its numerical designation is ten to the tenth power, which, in turn, consists of the hundredth power (i.e. ten to the googolplex power). The scientist also said that in this way you can get as large a number as you want: googoltetraplex, googolhexaplex, googoloctaplex, googoldecaplex, etc.
4. Graham's number is G. This is the largest number, recognized as such in the recent 1980 by the Guinness Book of Records. It is significantly larger than the googolplex and its derivatives. And scientists even said that the entire Universe is not able to contain the entire decimal notation of Graham’s number.
5. Moser number, Skewes number. These numbers are also considered one of the largest and they are used most often when solving various hypotheses and theorems. And since these numbers cannot be written down using generally accepted laws, each scientist does it in his own way.

Latest Developments

However, it is still worth saying that there is no limit to perfection. And many scientists believed and still believe that the largest number has not yet been found. And, of course, the honor of doing this will fall to them. On this project long time An American scientist from Missouri worked, his works were crowned with success. On January 25, 2012, he found the new largest number in the world, which consists of seventeen million digits (which is the 49th Mersenne number). Note: until this time, the largest number was considered to be the one found by the computer in 2008; it had 12 thousand digits and looked like this: 2 43112609 - 1.

Not for the first time

It is worth saying that this has been confirmed by scientific researchers. This number went through three levels of verification by three scientists on different computers, which took a full 39 days. However, this is not the first achievement in such a search by an American scientist. He had previously revealed the biggest numbers. This happened in 2005 and 2006. In 2008, the computer interrupted Curtis Cooper’s streak of victories, but in 2012 he still regained the palm and the well-deserved title of discoverer.

About the system

How does this all happen, how do scientists find the largest numbers? So, today the computer does most of the work for them. In this case, Cooper used distributed computing. What does it mean? These calculations are carried out by programs installed on the computers of Internet users who voluntarily decided to take part in the study. As part of this project, 14 Mersenne numbers were determined, named after French mathematician(these are prime numbers that are only divisible by themselves and one). In the form of a formula, it looks like this: M n = 2 n - 1 (“n” in this formula is a natural number).

About bonuses

A logical question may arise: what makes scientists work in this direction? So, this, of course, is passion and the desire to be a pioneer. However, there are bonuses here too: Curtis Cooper received a cash prize of $3,000 for his brainchild. But that's not all. The Electronic Frontier Foundation (EFF) encourages such searches and promises to immediately award cash prizes of $150,000 and $250,000 to those who submit prime numbers consisting of 100 million and a billion numbers. So there is no doubt that work is being done in this direction today great amount scientists around the world.

Simple conclusions

So what's the biggest number today? On this moment it was found by an American scientist from the University of Missouri, Curtis Cooper, which can be written as follows: 2 57885161 - 1. Moreover, it is also the 48th number of the French mathematician Mersenne. But it is worth saying that there can be no end to this search. And it is not surprising if through certain time Scientists will provide us with the next newly discovered largest number in the world for consideration. There is no doubt that this will happen in the very near future.

A child asked today: “What is the name of the largest number in the world?” Interesting question. I went online and found a detailed article in LiveJournal on the first line of Yandex. Everything is described there in detail. It turns out that there are two systems for naming numbers: English and American. And, for example, a quadrillion according to the English and American systems are completely different numbers! The largest non-composite number is Million = 10 to the 3003rd power.
As a result, the son came to a completely reasonable conclusion that it is possible to count endlessly.

Original taken from ctac in The largest number in the world

As a child, I was tormented by the question of what kind of
the biggest number, and I was tormented by this stupid
a question for almost everyone. Having learned the number
million, I asked if there was a higher number
million. Billion? How about more than a billion? Trillion?
How about more than a trillion? Finally, someone smart was found
who explained to me that the question is stupid, because
it is enough just to add to itself
a large number is one, and it turns out that it
has never been the biggest since there have been
the number is even greater.

And so, many years later, I decided to ask myself something else
question, namely: what is the most
a large number that has its own
Name? Fortunately, now there is an Internet and it’s puzzling
they can patient search engines that do not
they will call my questions idiotic ;-).
Actually, that's what I did, and this is the result
found out.

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

The American system is built quite
Just. 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 the numbers come out - trillion, quadrillion,
quintillion, sextillion, septillion, octillion,
nonillion and decillion. American system
used in the USA, Canada, France and Russia.
Find out the number of zeros in a number written by
American system, using a simple formula
3 x+3 (where x is a Latin numeral).

The English system of naming the most
widespread in the world. It is used, for example, in
Great Britain and Spain, as well as most
former English and Spanish colonies. Titles
numbers in this system are constructed like this: like this: to
a suffix is ​​added to the Latin numeral
-million, the next number (1000 times larger)
is built on the same principle
Latin numeral, but the suffix is ​​-billion.
That is, after a trillion in the English system
there is a trillion, and only then a quadrillion, after
followed by quadrillion, etc. So
Thus, quadrillion in English and
American systems are completely different
numbers! Find out the number of zeros in a number
written according to the English system and
ending with the suffix -illion, you can
formula 6 x+3 (where x is a Latin numeral) and
using the formula 6 x + 6 for numbers ending in
-billion.

From English system passed into Russian language
only the number billion (10 9), which is still
it would be more correct to call it what it is called
Americans - a billion, as we have adopted
exactly American system. But who is in our
the country is doing something according to the rules! ;-) By the way,
sometimes in Russian they use the word
trillion (you can see this for yourself,
by running a search in Google or Yandex) and it means, judging by
in total, 1000 trillion, i.e. quadrillion.

In addition to numbers written using Latin
prefixes according to the American or English system,
the so-called non-system numbers are also known,
those. numbers that have their own
names without any Latin prefixes. Such
There are several numbers, but I will tell you more about them
I'll tell you a little later.

Let's return to recording using Latin
numerals. It would seem that they can
write down numbers to infinity, but this is not
quite like that. Now I will explain why. Let's see for
beginning of what the numbers from 1 to 10 33 are called:

And now the question arises, what next. What
there behind a decillion? In principle, you can, of course,
by combining prefixes to generate such
monsters like: andecillion, duodecillion,
tredecillion, quattordecillion, quindecillion,
sexdecillion, septemdecillion, octodecillion and
newdecillion, but these will already be composite
names, but we were interested specifically
proper names for numbers. Therefore, own
names according to this system, in addition to those indicated above, more
you can only get three
- vigintillion (from lat. viginti —
twenty), centillion (from lat. centum- one hundred) and
million million (from lat. mille- thousand). More
thousands of proper names for numbers among the Romans
did not have (all numbers over a thousand they had
compound). For example, a million (1,000,000) Romans
called decies centena milia, that is, "ten hundred
thousand." And now, actually, the table:

Thus, according to a similar number system
greater than 10 3003 which would have
get your own, non-compound name
impossible! But still the numbers are higher
million are known - these are the same
non-system numbers. Let's finally talk about them.

The smallest such number is myriad
(it’s even in Dahl’s dictionary), which means
a hundred hundreds, that is, 10,000. This word, however,
outdated and practically not used, but
It's interesting that the word is widely used
"myriads", which does not mean at all
a certain number, and countless, uncountable
a lot of something. It is believed that the word myriad
(English: myriad) came to European languages from ancient
Egypt.

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

In the famous Buddhist treatise Jaina Sutra,
dating back to 100 BC, there is a number asankheya
(from China asenzi- uncountable), equal to 10 140.
It is believed that this number is equal to the number
cosmic cycles necessary to obtain
nirvana.

Googolplex(English) googolplex) - number also
invented by Kasner with his nephew and
meaning one followed by a googol of zeros, that is, 10 10 100.
This is how Kasner himself describes this “discovery”:

Words of wisdom are spoken by children at least as often as by scientists. The name
"googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was
asked to think up a name for a very big number, namely, 1 with a hundred zeros after it.
He was very certain that this number was not infinite, and therefore equally certain that
it had to have a name. At the same time that he suggested "googol" he gave a
name for a still larger number: "Googolplex." A googolplex is much larger than a
googol, but is still finite, as the inventor of the name was quick to point out.

Mathematics and the Imagination(1940) by Kasner and James R.
Newman.

An even larger number than a googolplex is a number
Skewes "number" was proposed by Skewes in 1933
year (Skewes. J. London Math. Soc. 8 , 277-283, 1933.) with
proof of hypothesis
Riemann concerning prime numbers. It
means e to a degree e to a degree e V
degrees 79, that is, e e e 79. Later,
Riele (te Riele, H. J. J. "On the Sign of the Difference P(x)-Li(x)."
Math. Comput. 48 , 323-328, 1987) reduced the Skuse number to e e 27/4,
which is approximately equal to 8.185 10 370. Understandable
the point is that since the value of the Skewes number depends on
numbers e, then it is not whole, therefore
we will not consider it, otherwise we would have to
remember other non-natural numbers - number
pi, number e, Avogadro's number, etc.

But it should be noted that there is a second number
Skuse, which in mathematics is denoted as Sk 2,
which is even greater than the first Skuse number (Sk 1).
Second Skewes number, was introduced by J.
Skuse in the same article to denote the number, up to
which the Riemann hypothesis is true. Sk 2
equals 10 10 10 10 3, that is, 10 10 10 1000
.

As you understand, the greater the number of degrees,
the more difficult it is to understand which number is greater.
For example, looking at the Skewes numbers, without
special calculations are almost impossible
understand which of these two numbers is greater. So
Thus, for super-large numbers use
degrees becomes uncomfortable. Moreover, you can
come up with such numbers (and they have already been invented) when
degrees of degrees just don't fit on the page.
Yes, that's on the page! They won't fit even in a book,
the size of the entire Universe! In this case it gets up
The question is how to write them down. The problem is how you
you understand, it is solvable, and mathematicians have developed
several principles for writing such numbers.
True, every mathematician who asked this question
problem I came up with my own way of recording that
led to the existence of several unrelated
with each other, ways to write numbers are
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 extra-large
numbers. He named the number - Mega, and the number is Megiston.

Mathematician Leo Moser refined the notation
Stenhouse, which was limited to what if
it was necessary to write down much larger numbers
megiston, difficulties and inconveniences arose, so
how I had to draw many circles alone
inside another. Moser suggested after squares
draw pentagons rather than circles, then
hexagons and so on. He also suggested
formal notation for these polygons,
so you can write numbers without drawing
complex drawings. Moser notation looks like this:

Thus, according to Moser's notation
Steinhouse's mega is written as 2, and
megiston as 10. In addition, Leo Moser suggested
call a polygon with the same number of sides
mega - megagon. And suggested the number "2 in
Megagone", that is, 2. This number became
known as Moser's number or simply
How Moser.

But Moser is not the largest number. The biggest
number ever used in
mathematical proof, is
limit value known as Graham number
(Graham's number), first used in 1977
proof of one estimate in Ramsey theory. It
related to bichromatic hypercubes and not
can be expressed without special 64-level
systems of special mathematical symbols,
introduced by Knuth in 1976.

Unfortunately, the number written in Knuth notation
cannot be converted into a Moser entry.
Therefore, we will have to explain this system too. IN
In principle, there is nothing complicated about it either. Donald
Knut (yes, yes, this is the same Knut who wrote
"The Art of Programming" and created
TeX editor) came up with the concept of superpower,
which he proposed to write down with arrows,
upward:

IN general view it looks like this:

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

The number G 63 became known as number
Graham(it is often designated simply as G).
This number is the largest known in
number in the world and is even included in the Book of Records
Guinness". Ah, that Graham number is greater than the number
Moser.

P.S. To bring great benefit
to all mankind and be glorified throughout the ages, I
I decided to come up with and name the biggest
number. This number will be called stasplex And
it is equal to the number G 100. Remember it and when
your children will ask what is the biggest
number in the world, tell them what this number is called stasplex.

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, maximum possible number, which is somewhat useful. There are many contenders for this title, but I'll warn you right away: there really is a risk that trying to understand it all 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 language. (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 is googol) in the form of Google, born in 1920 as a way to get children 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 charming, Kasner decided more was needed. formal definition. As he explained in his 1940 book Mathematics and the Imagination, Milton's definition leaves open the risky possibility that an accidental buffoon could become a mathematician superior to Albert Einstein simply because he has greater 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 can compare to the total number of particles in the Universe, which is generally considered to be approximately , and this number is so large that our language has no word for 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 in this case actual application in worlds is probably one of the latest estimates of the number of universes in the multiverse. This number is so large that 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 to find them we must go into the realm of pure mathematics, and there is no better place to start than prime numbers.

Mersenne primes

Part of the difficulty is coming up with good definition 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) that is divisible only by and itself. So, and are prime numbers, and and are composite numbers. This means that any composite number may ultimately be represented by its own simple divisors. In some ways, the number is more important than, say, , because there is no way to express it in terms of a product 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 important role, but, say, a googol - which is ultimately just a set of numbers and , multiplied together - actually 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 a concept of 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 it 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. legendary mathematician Carl Friedrich Gauss.

I'll spare you more complex mathematics- one way or another, we have a lot more to come - but the essence of the function is this: for any integer we can estimate how many prime numbers there are less 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 only 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 infinite number once.

The hunt was for the starting point of the races, and then Stanley Skewes appeared (see photo). In 1933 he proved that upper limit, when the function approximating the number of prime numbers first gives lower 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. Mathematicians have since 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 recounts 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 obtained on the basis that the so-called Riemann hypothesis is true - this is especially complex hypothesis mathematics that remains unproven is 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 a large number of 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 idea 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 the same as where in total threes. 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.

Obviously, there is no way to comprehend such 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 equal (indeed, in the previous version of this post I made exactly this mistake), but it is 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 hypercube will remain stable. (Sorry for such a vague explanation, but I'm sure we all need to get at least two academic degrees in mathematics 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 at each 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 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 last digits Graham numbers.

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. Concerning significant number...okay, there are some fiendishly complex areas of mathematics (specifically 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, here's some literature for additional reading at one's 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.

10 to the 3003rd power

Disputes about which one is the most big number in the world are ongoing. Different calculus systems offer different options and people don’t know what to believe and which number to consider as the largest.

This question has interested scientists since the times of the Roman Empire. The biggest problem lies in the definition of what a “number” is and what a “digit” is. At one time, people for a long time considered the largest number to be a decillion, that is, 10 to the 33rd power. But, after scientists began to actively study the American and English metric systems, it was discovered that the largest number in the world is 10 to the 3003rd power - a million. People in everyday life believe that the largest number is a trillion. Moreover, this is quite formal, since after a trillion, names are simply not given, because the counting begins to be too complex. However, purely theoretically, the number of zeros can be added indefinitely. Therefore, it is almost impossible to imagine even purely visually a trillion and what follows it.

In Roman numerals

On the other hand, the definition of “number” as understood by mathematicians is a little different. A number means a sign that is universally accepted and is used to indicate a quantity expressed in a numerical equivalent. The second concept of “number” means the expression of quantitative characteristics in convenient form through the use of numbers. It follows from this that numbers are made up of digits. It is also important that the number has symbolic properties. They are conditioned, recognizable, unchangeable. Numbers also have iconic properties, but they follow from the fact that numbers consist of digits. From this we can conclude that a trillion is not a figure at all, but a number. Then what is the largest number in the world if it is not a trillion, which is a number?

The important thing is that numbers are used as components of numbers, but not only that. A number, however, is the same number if we are talking about some things, counting them from zero to nine. This system of features applies not only to the familiar Arabic numerals, but also to Roman I, V, X, L, C, D, M. These are Roman numerals. On the other hand, V I I I is a Roman numeral. In Arabic calculus it corresponds to the number eight.

IN Arabic numerals

Thus, it turns out that counting units from zero to nine are considered numbers, and everything else is numbers. Hence the conclusion that the largest number in the world is nine. 9 is a sign, and a number is a simple quantitative abstraction. A trillion is a number, and not a number at all, and therefore cannot be the largest number in the world. A trillion can be called the largest number in the world, and that is purely nominally, since numbers can be counted ad infinitum. The number of digits is strictly limited - from 0 to 9.

It should also be remembered that the numerals and numbers of different number systems do not coincide, as we saw from the examples with Arabic and Roman numerals and numerals. This happens because numbers and numbers are simple concepts, which are invented by the person himself. Therefore, a number in one number system can easily be a number in another and vice versa.

Thus, the largest number is innumerable, because it can continue to be added indefinitely from digits. As for the numbers themselves, in the generally accepted system, 9 is considered the largest number.

June 17th, 2015

“I see clusters of vague numbers that are hidden there in the darkness, behind the small spot of light that the candle of reason gives. They whisper to each other; conspiring about who knows what. Perhaps they don't like us very much for capturing their little brothers in our minds. Or perhaps they simply lead a single-digit life, out there, beyond our understanding.
Douglas Ray

We continue ours. Today we have numbers...

Sooner or later, everyone is tormented by the question, what is the largest number. There are a million answers to a child's question. What's next? Trillion. And even further? In fact, the answer to the question of what are the largest numbers is simple. Just add one to the largest number, and it will no longer be the largest. This procedure can be continued indefinitely.

But if you ask the question: what is the largest number that exists, and what is its proper name?

Now we will find out everything...

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

The American system is built quite simply. All names of large numbers are constructed like this: at the beginning there is a Latin ordinal number, and at the end the suffix -million is added to it. An exception is the name "million" which is the name of the number thousand (lat. mille) and the magnifying suffix -illion (see table). This is how we get the numbers trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion and decillion. The American system is used in the USA, Canada, France and Russia. You can find out the number of zeros in a number written according to the American system using the simple formula 3 x + 3 (where x is a Latin numeral).

The English naming system is the most common in the world. It is used, for example, in Great Britain and Spain, as well as in most former English and Spanish colonies. The names of numbers in this system are built like this: like this: the suffix -million is added to the Latin numeral, the next number (1000 times larger) is built according to the principle - the same Latin numeral, but the suffix - billion. That is, after a trillion in the English system there is a trillion, and only then a quadrillion, followed by a quadrillion, etc. Thus, a quadrillion according to the English and American systems are completely different numbers! You can find out the number of zeros in a number written according to the English system and ending with the suffix -million, using the formula 6 x + 3 (where x is a Latin numeral) and using the formula 6 x + 6 for numbers ending in - billion.

Only the number billion (10 9) passed from the English system into the Russian language, which would still be more correct to be called as the Americans call it - billion, since we have adopted the American system. But who in our country does anything according to the rules! ;-) By the way, sometimes the word trillion is used in Russian (you can see this for yourself by running a search in Google or Yandex) and, apparently, it means 1000 trillion, i.e. quadrillion.

In addition to numbers written using Latin prefixes according to the American or English system, so-called non-system numbers are also known, i.e. numbers that have their own names without any Latin prefixes. There are several such numbers, but I will tell you more about them a little later.

Let's return to writing using Latin numerals. It would seem that they can write down numbers to infinity, but this is not entirely true. Now I will explain why. Let's first see what the numbers from 1 to 10 33 are called:

And now the question arises, what next. What's behind the decillion? In principle, it is, of course, possible, by combining prefixes, to generate such monsters as: andecillion, duodecillion, tredecillion, quattordecillion, quindecillion, sexdecillion, septemdecillion, octodecillion and novemdecillion, but these will already be compound names, and we were interested in our own names numbers. Therefore, according to this system, in addition to those indicated above, you can still get only three proper names - vigintillion (from Lat.viginti- twenty), centillion (from lat.centum- one hundred) and million (from lat.mille- thousand). The Romans did not have more than a thousand proper names for numbers (all numbers over a thousand were composite). For example, the Romans called a million (1,000,000)decies centena milia, that is, "ten hundred thousand." And now, actually, the table:

Thus, according to such a system, numbers are greater than 10 3003 , which would have its own, non-compound name is impossible to obtain! But nevertheless, numbers greater than a million are known - these are the same non-systemic numbers. Let's finally talk about them.

The smallest such number is a myriad (it is even in Dahl’s dictionary), which means a hundred hundreds, that is, 10,000. This word, however, is outdated and practically not used, but it is curious that the word “myriads” is widely used, does not mean a definite number at all, but an uncountable, uncountable multitude of something. It is believed that the word myriad came into European languages ​​from ancient Egypt.

There are different opinions about the origin of this number. Some believe that it originated in Egypt, while others believe that it was born only in Ancient Greece. Be that as it may in 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 visible universe lead to the number 10 67 (in total a myriad of times more). Archimedes suggested the following names for the numbers:
1 myriad = 10 4.
1 di-myriad = myriad of myriads = 10 8 .
1 tri-myriad = di-myriad di-myriad = 10 16 .
1 tetra-myriad = three-myriad three-myriad = 10 32 .
etc.

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

Edward Kasner.

On the Internet you can often find it mentioned that - but this is not true...

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

Googolplex (English) googolplex) - a number also invented by Kasner and his nephew and meaning one with a googol of zeros, that is, 10 10100 . This is how Kasner himself describes this “discovery”:

Words of wisdom are spoken by children at least as often as by scientists. The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "A googolplex is much larger than." a googol, but is still finite, as the inventor of the name was quick to point out.

Mathematics and the Imagination(1940) by Kasner and James R. Newman.

An even larger number than the googolplex, the Skewes number, was proposed by Skewes in 1933. J. London Math. Soc. 8, 277-283, 1933.) in proving the Riemann hypothesis concerning prime numbers. It means e to a degree e to a degree e to the power of 79, that is, ee e 79 . Later, te Riele, H. J. J. "On the Sign of the Difference P(x)-Li(x)." Math. Comput. 48, 323-328, 1987) reduced the Skuse number to ee 27/4 , which is approximately equal to 8.185·10 370. It is clear that since the value of the Skuse number depends on the number e, then it is not an integer, so we will not consider it, otherwise we would have to remember other non-natural numbers - the number pi, the number e, etc.

But it should be noted that there is a second Skuse number, which in mathematics is denoted as Sk2, which is even greater than the first Skuse number (Sk1). Second Skewes number, was introduced by J. Skuse in the same article to denote a number for which the Riemann hypothesis does not hold. Sk2 equals 1010 10103 , that is 1010 101000 .

As you understand, the more degrees there are, the more difficult it is to understand which number is greater. For example, looking at Skewes numbers, without special calculations, it is almost impossible to understand which of these two numbers is larger. Thus, for super-large numbers it becomes inconvenient to use powers. Moreover, you can come up with such numbers (and they have already been invented) when the degrees of degrees simply do not fit on the page. Yes, that's on the page! They won’t fit even into a book the size of the entire Universe! In this case, the question arises of how to write them down. The problem, as you understand, is solvable, and mathematicians have developed several principles for writing such numbers. True, every mathematician who asked about this problem came up with his own way of writing, which led to the existence of several, unrelated to each other, methods for writing numbers - these are the notations of Knuth, Conway, Steinhouse, etc.

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

Steinhouse came up with two new superlarge numbers. He named the number - Mega, and the number - Megiston.

Mathematician Leo Moser refined Stenhouse's notation, which was limited by the fact that if it was necessary to write down numbers much larger than a megiston, difficulties and inconveniences arose, since many circles had to be drawn one inside the other. Moser suggested that after the squares, draw not circles, but pentagons, then hexagons, and so on. He also proposed a formal notation for these polygons so that numbers could be written without drawing complicated pictures. Moser notation looks like this:

Thus, according to Moser's notation, Steinhouse's mega is written as 2, and megiston as 10. In addition, Leo Moser proposed calling a polygon with the number of sides equal to mega - megagon. And he proposed the number “2 in Megagon,” that is, 2. This number became known as Moser’s number or simply as Moser.

But Moser is not the largest number. The largest number ever used in a mathematical proof is the limiting quantity known as Graham's number, first used in 1977 in the proof of an estimate in Ramsey theory. It is associated with bichromatic hypercubes and cannot be expressed without the special 64-level system of special mathematical symbols introduced by Knuth in 1976.

Unfortunately, a number written in Knuth's notation cannot be converted into notation in the Moser system. Therefore, we will have to explain this system too. In principle, there is nothing complicated about it either. Donald Knuth (yes, yes, this is the same Knuth who wrote “The Art of Programming” and created the TeX editor) came up with the concept of superpower, which he proposed to write with arrows pointing upward:

In general it looks like this:

I think everything is clear, so let’s return to Graham’s number. Graham proposed so-called G-numbers:

1. G1 = 3..3, where the number of superpower arrows is 33.


2. G2 = ..3, where the number of superpower arrows is equal to G1.


3. G3 = ..3, where the number of superpower arrows is equal to G2.



4. G63 = ..3, where the number of superpower arrows is G62.

The G63 number came to be called the Graham number (it is often designated simply as G). This number is the largest known number in the world and is even listed in the Guinness Book of Records. And here