When a man was just learning to count, he had enough fingers to determine that the two mammoths walking around the cave were smaller than the herd behind the mountain. But as soon as he realized what positional numbering is (when a number has a specific place in a long row), he began to wonder: what next, what is the largest number?
Since then, the best minds have begun to look for how to calculate such quantities, and most importantly, in what sense to give them.
Ellipsis at the end of a row
When schoolchildren are introduced to the original concept of natural numbers, it is prudent to put an ellipsis around the edges of a number of numbers and explain that the largest and smallest number is a meaningless category. There is always the opportunity to add one to the largest number, and it will no longer be the largest. But progress would not be possible if there were not those who wanted to find meaning where it should not be.
The infinity of the number series, besides frightening and indefinite philosophical significance, created purely technical difficulties. I had to look for notation for very large numbers. At first, this was done separately for the main language groups, and with the development of globalization, words appeared that named the largest number generally accepted throughout the world.
Ten, a hundred, a thousand
Each language has its own name for numbers of practical importance.
In Russian, this is primarily a series from zero to ten. Up to a hundred further numbers are called either based on them, with a slight change of roots - “twenty” (two by ten), “thirty” (three by ten), etc., or are compound: “twenty one”, “fifty four ". An exception - instead of “fourteen” we have a more convenient “forty”.
The largest two-digit number, “ninety-nine,” has a composite name. Further from their own traditional names - “hundred” and “thousand”, the rest are formed from the necessary combinations. A similar situation in other common languages. It is logical to think that the established names were given to numbers and numbers with which most ordinary people dealt. Even what a thousand head of cattle is, an ordinary peasant could imagine. With a million it was harder, and confusion began.
Million, Quintillion, Decillion
In the middle of the 15th century, the Frenchman Nicolas Schouke, in order to designate the largest number, proposed a naming system based on the numerals from the generally accepted Latin scholars. In Russian, they underwent some modification for the convenience of pronunciation:
- 1 - Unus - oz
- 2 - Duo, Bi (double) - duo, bi.
- 3 - Tres - three.
- 4 - Quattuor - quad.
- 5 - Quinque - quinte.
- 6 - Sex - sex.
- 7 - Septem - septi.
- 8 - Octo - Octy.
- 9 - Novem - noni.
- 10 - Decem - dec.
The basis of the names was to be a million, from "million" - "big thousand" - that is, 1,000,000 - 1,000 ^ 2 - thousand squared. This word, to mention the largest number, was first used by the famous navigator and scientist Marco Polo. So, a thousand in the third degree became a trillion, 1000 ^ 4 - a quadrillion. Another Frenchman - Peletier - suggested for the numbers that Shuke called "thousand of millions" (10 ^ 9), "thousand of billions" (10 ^ 15) etc., use the ending "-billion". It turned out that 1,000,000,000 is a billion, 10 ^ 15 - billiard, unit with 21 zero - trillion and so on.
The terminology of French mathematicians began to be used in many countries. But it gradually became clear that 10 ^ 9 in some works they began to call not a billion, but a billion. And in the United States, they adopted a system according to which a million received degrees not of a million, as in the French, but in thousands. As a result, today there are two scales in the world: “long” and “short”. To understand what number is meant by a name, for example, a quadrillion, it is better to clarify to what degree the number 10 is raised. If in the 15th, this is a “short” scale adopted in the USA, Canada, Great Britain and some other countries, including in Russia (though we have 10 ^ 9– not a billion, but a billion), if in 24 it’s “long”, adopted in most regions of the world.
Tredecillion, vigintillion and milleillion
After the last numeral is used - dec, and a decillion is formed - the largest number without complex word formations - 10 ^ 33 on a short scale, combinations of the necessary prefixes are used for the next digits. Complex composite names of the type of tredecillion - 10 ^ 42, quindecillion - 10 ^ 48, etc. are obtained. The Romans were given non-composite, proper names: twenty - viginti, one hundred - centum and one thousand - mille. Following Shuke's rules, you can form monster names for an infinitely long time. For example, the number 10 ^ 308760 is called ducentuodomilianongendemic decillion.
But these constructions are of interest only to a limited number of people - they are not used in practice, and even these quantities themselves are not even tied to theoretical problems or theorems. It is precisely for purely theoretical constructions that giant numbers are intended, sometimes receiving very sonorous names or called by the surname of the author.
Darkness, Legion, Asankheya
The issue of huge numbers worried the “pre-computer” generation. The Slavs had several number systems, in some they reached tremendous heights: the largest number was 10 ^ 50. The names of numbers from the heights of our time seem to be poetry, and whether all of them had practical meaning, only historians and linguists know: 10 ^ 4 - "darkness", 10 ^ 5 - "legion", 10 ^ 6 - "leodr", 10 ^ 7 - lies, raven, 10 ^ 8 - "deck".
The equally beautiful number asaṃkhyeya is mentioned in Buddhist texts, in ancient Chinese and ancient Indian collections of sutras.
The quantitative value of the number of asankhey researchers give as 10 ^ 140. For those who understand it, it is full of divine meaning: it is precisely so many cosmic cycles that the soul must go through in order to be cleansed of everything bodily, accumulated over a long way of rebirth, and to achieve a blissful state of nirvana.
Googol, googolpleks
The mathematician from Columbia University (USA) Edward Kasner from the beginning of the 1920s began to think about large numbers. In particular, he was interested in a sonorous and expressive name for a beautiful number 10 ^ 100. Once he walked with his nephews and told them about this number. Nine-year-old Milton Sirotta suggested the word googol - googol. Uncle received a bonus from his nephews - a new number, which they explained as follows: a unit and as many zeros as you can write until you get tired. The name of this number was googolpleks. On reflection, Kashner decided that this would be the number 10 ^ googol.
Kashner saw the meaning in such numbers more pedagogically: science then did not know anything in such numbers, and he explained to future mathematicians by their example what the largest number could preserve the difference from infinity.
The chic idea of little naming geniuses was appreciated by the founders of the company to promote a new search engine. The googol domain turned out to be busy, and the letter o fell out, but a name appeared for which the ephemeral number may someday become real - how much will its shares cost.
Shannon's number, Skewse's number, Medzon, megiston
Unlike physicists who periodically stumble upon the restrictions imposed by nature, mathematicians continue their journey towards infinity. Chess game lover Claude Shannon (1916-2001) filled out the meaning of the number 10 ^ 118 - exactly so many options for positions can arise during 40 moves.
Stanley Squeeze from South Africa was one of seven tasks on the Millennium Challenge — the Riemann hypothesis. It concerns the search for patterns in the distribution of primes. In the course of reasoning, he first used the number 10 ^ 10 ^ 10 ^ 34, designated by him Sk 1 , and then 10 ^ 10 ^ 10 ^ 963 - the second Skuse number - Sk 2 .
Even the usual recording system is not suitable for handling such numbers. Hugo Steinghaus (1887-1972) proposed the use of geometric figures: n in a triangle is n to the power of n, n in a square is n in n triangles, n in a circle is n in n squares. He explained this system by the example of mega - 2 numbers in a circle, mezzon - 3 in a circle, megiston - 10 in a circle. It is so difficult to designate, for example, the largest two-digit number, but it has become easier to operate with colossal quantities.
Professor Donald Knuth proposed arrow notation, in which re- exponentiation was indicated by an arrow borrowed from the practice of programmers. Googol in this case looks like 10 ↑ 10 ↑ 2, and googolplex - 10 ↑ 10 ↑ 10 ↑ 2.
Graham Number
Ronald Graham (born 1935), an American mathematician, in the study of Ramsey's theory related to hypercubes - multidimensional geometric bodies - introduced singular numbers G 1 - G 64 , with the help of which he outlined the boundaries of the solution, where the highest multiple became the upper limit, which received his name. He calculated even the last 20 digits, and the following values served as the source data:
- G 1 = 3 ↑↑↑↑ 3 = 8.7 x 10 ^ 115.
- G 2 = 3 ↑ ... ↑ 3 (the number of arrows of the super degree = G 1 ).
- G 3 = 3 ↑ ... ↑ 3 (the number of arrows of superdegree = G 2 ).
...
- G 64 = 3 ↑ ... ↑ 3 (the number of arrows of the super degree = G 63 )
G 64 , denoted simply by G, is the largest number in the world used in mathematical calculations. It is listed in the book of records.
It is almost impossible to imagine its scale, given that the entire volume of the universe known to man, expressed in the smallest unit of volume (a cube with a face of Planck length (10
-35 m)), is expressed as 10 ^ 185.