What is the biggest number you know. The largest number in the world

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 largest numbers you've ever heard of, and these are indeed the two largest numbers that have generally accepted definitions in the 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 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 with small dust particles approximately 1.5 microns in size, then the number of different ways these particles can be arranged 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. Global 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 go back to the first Planck time unit after the Big Bang, 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 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 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) that 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 of 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 from Euclid's time saw the possibility of simplification, but it wasn't until the Renaissance mathematicians could not really use it in 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, 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 Mersenne number is the largest prime number known to mankind. Discovered in 2008, it amounts to a number with almost millions of digits. It is the largest known number that 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 Skewes

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 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 up 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. 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 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 it comes to 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. (Admittedly, it would take an insanely long time to actually count to a million of anything, but the point is that we are still capable of perceiving that number.)

However, although we cannot imagine, we are at least able to understand in general terms what 7600 billion is, perhaps by 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.

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 being able to calculate the exact value of a power tower of triplets that is made up of elements, and then you took that value and created a new tower with as many in it as... that 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 smallest number of dimensions certain properties of a hypercube would 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 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'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 quite possible that the actual number of measurements required to satisfy the desired property is 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. The lower bound has since been raised to , but there is still a very good 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.

Have you ever thought how many zeros there are in one million? This is a pretty simple question. What about a billion or a trillion? One followed by nine zeros (1000000000) - what is the name of the number?

A short list of numbers and their quantitative designation

• Ten (1 zero).
• One hundred (2 zeros).
• One thousand (3 zeros).
• Ten thousand (4 zeros).
• One hundred thousand (5 zeros).
• Million (6 zeros).
• Billion (9 zeros).
• Trillion (12 zeros).
• Quadrillion (15 zeros).
• Quintilion (18 zeros).
• Sextillion (21 zeros).
• Septillion (24 zeros).
• Octalion (27 zeros).
• Nonalion (30 zeros).
• Decalion (33 zeros).

Grouping of zeros

1000000000 - what is the name of a number that has 9 zeros? This is a billion. For convenience, large numbers are usually grouped into sets of three, separated from each other by a space or punctuation marks such as a comma or period.

This is done to make the quantitative value easier to read and understand. For example, what is the name of the number 1000000000? In this form, it’s worth straining a little and doing the math. And if you write 1,000,000,000, then the task immediately becomes visually easier, since you need to count not zeros, but triples of zeros.

Numbers with a lot of zeros

The most popular are million and billion (1000000000). What is the name of a number that has 100 zeros? This is a Googol number, so called by Milton Sirotta. This is a wildly huge amount. Do you think this number is large? Then what about a googolplex, a one followed by a googol of zeros? This figure is so large that it is difficult to come up with a meaning for it. In fact, there is no need for such giants, except to count the number of atoms in the infinite Universe.

Is 1 billion a lot?

There are two measurement scales - short and long. Around the world in science and finance, 1 billion is 1,000 million. This is on a short scale. According to it, this is a number with 9 zeros.

There is also a long scale that is used in some European countries, including France, and was formerly used in the UK (until 1971), where a billion was 1 million million, that is, a one followed by 12 zeros. This gradation is also called the long-term scale. The short scale is now predominant in financial and scientific matters.

Some European languages, such as Swedish, Danish, Portuguese, Spanish, Italian, Dutch, Norwegian, Polish, German, use billion (or billion) in this system. In Russian, a number with 9 zeros is also described for the short scale of a thousand million, and a trillion is a million million. This avoids unnecessary confusion.

Conversational options

In Russian colloquial speech after the events of 1917 - the Great October Revolution - and the period of hyperinflation in the early 1920s. 1 billion rubles was called “limard”. And in the dashing 1990s, a new slang expression “watermelon” appeared for a billion; a million were called “lemon.”

The word "billion" is now used internationally. This is a natural number, which is represented in the decimal system as 10 9 (one followed by 9 zeros). There is also another name - billion, which is not used in Russia and the CIS countries.

Billion = billion?

A word such as billion is used to designate a billion only in those states in which the “short scale” is adopted as a basis. These are countries such as the Russian Federation, the United Kingdom of Great Britain and Northern Ireland, the USA, Canada, Greece and Türkiye. In other countries, the concept of a billion means the number 10 12, that is, one followed by 12 zeros. In countries with a “short scale”, including Russia, this figure corresponds to 1 trillion.

Such confusion appeared in France at a time when the formation of such a science as algebra was taking place. Initially, a billion had 12 zeros. However, everything changed after the appearance of the main manual on arithmetic (author Tranchan) in 1558), where a billion is already a number with 9 zeros (a thousand millions).

For several subsequent centuries, these two concepts were used on an equal basis with each other. In the mid-20th century, namely in 1948, France switched to a long scale numerical naming system. In this regard, the short scale, once borrowed from the French, is still different from the one they use today.

Historically, the United Kingdom used the long-term billion, but since 1974 official UK statistics have used the short-term scale. Since the 1950s, the short-term scale has been increasingly used in the fields of technical writing and journalism, although the long-term scale still persists.

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. Based on the American system, 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. Here it should be noted that the English number system is more common in the modern world, and the numbers it contains are quite sufficient for the normal functioning of 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 the classical version there is no way to represent 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 - the largest 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.

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.

Thus, the same number in different 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.

In the names of Arabic numbers, each digit belongs to its own category, and every three digits form a class. Thus, the last digit in a number indicates the number of units in it and is called, accordingly, the ones place. The next, second from the end, digit indicates the tens (tens place), and the third from the end digit indicates the number of hundreds in the number - the hundreds place. Further, the digits are also repeated in turn in each class, denoting units, tens and hundreds in the classes of thousands, millions, and so on. If the number is small and does not have a tens or hundreds digit, it is customary to take them as zero. Classes group digits in numbers of three, often placing a period or space between classes in computing devices or records to visually separate them. This is done to make large numbers easier to read. Each class has its own name: the first three digits are the class of units, then the class of thousands, then millions, billions (or billions) and so on.

Since we use the decimal system, the basic unit of quantity is ten, or 10 1. Accordingly, as the number of digits in a number increases, the number of tens also increases: 10 2, 10 3, 10 4, etc. Knowing the number of tens, you can easily determine the class and rank of the number, for example, 10 16 is tens of quadrillions, and 3 × 10 16 is three tens of quadrillions. The decomposition of numbers into decimal components occurs in the following way - each digit is displayed in a separate term, multiplied by the required coefficient 10 n, where n is the position of the digit from left to right.
For example: 253 981=2×10 6 +5×10 5 +3×10 4 +9×10 3 +8×10 2 +1×10 1

The power of 10 is also used in writing decimal fractions: 10 (-1) is 0.1 or one tenth. In a similar way to the previous paragraph, you can also expand a decimal number, n in this case will indicate the position of the digit from the decimal point from right to left, for example: 0.347629= 3×10 (-1) +4×10 (-2) +7×10 (-3) +6×10 (-4) +2×10 (-5) +9×10 (-6 )

Names of decimal numbers. Decimal numbers are read by the last digit after the decimal point, for example 0.325 - three hundred twenty-five thousandths, where the thousandth is the place of the last digit 5.

Table of names of large numbers, digits and classes