At the word “infinity” each person has his own associations. Many people draw in their imagination a sea going beyond the horizon, while others have a picture of an endless starry sky in front of their eyes. In a completely different way, mathematicians who are accustomed to operating with numbers imagine an infinity. For many centuries they have been trying to find the largest of the physical quantities required for measuring. One of them is the Graham number. How many zeros are in it and what is it used for, this article will tell.
Infinitely large
In mathematics, this is the name of such a variable x n if for any given positive integer M we can specify a positive integer N such that for all numbers n greater than N, the inequality | x n | > M. However, no one, for example, an integer Z can be considered infinitely large, since it will always be less than the value (Z + 1).
A few words about the "giants"
The largest numbers that have physical meaning are considered:
- 10 80 . This number, which is usually called quinquavigintillion, is taken to indicate the approximate number of quarks and leptons (smallest particles) of the Universe.
- 1 Googol. Such a number in the decimal system is written as a unit with 100 zeros. According to some mathematical models, from the time of the big bang, before the explosion of a massive black hole, it takes 1 to 1.5 googol years, after which our universe will go to the last stage of its existence, i.e., we can assume that this number has some physical meaning.
- 8.5 x 10 185 . The Planck constant is 1.616199 x 10 -35 m, i.e., in decimal notation it looks like 0.00000000000000000000000000000616199 m. In 1 cubic meter. In an inch, there are about 1 goalkol of Planck lengths. It is estimated that in our entire Universe about 8.5 x 10 185 Planck lengths can fit.
- 2 77 232 917 - 1. This is the largest known prime number. If its binary notation has a rather compact form, then in order to depict it in decimal form, it will take no less - 13 million characters. It was found in 2017 as part of a project to search for Mersenne numbers. If enthusiasts continue to work in this direction, then at the current level of development of computer technology, in the near future they are unlikely to find the Mersenne number by an order of magnitude greater than 2 77 232 917 - 1, although such a lucky person will receive $ 150,000.
- Gugopleks. Then we just take 1 and add zeros after it in the amount of 1 googol. This number can be written as 10 ^ 10 ^ 100. It cannot be depicted in decimal form, since if the entire space of the Universe were filled with pieces of paper, on each of which would be written 0 with the size of the “Word” font 10, then in this case only half of all 0 after 1 for the number of googolplex would be obtained .
- 10 ^ 10 ^ 10 ^ 10 ^ 10 ^ 1.1. This is a number showing the number of years after which, according to the Poincare theorem, our Universe will return to a state close to today as a result of random quantum oscillations.
How did Graham's numbers appear?
In 1977, the well-known popularizer of science, Martin Gardner , published an article in Scientific American magazine regarding Graham’s proof of one of the problems of the Ramse theory. In it, he called the boundary established by the scientist the largest number ever used in serious mathematical reasoning.
Who is Ronald Lewis Graham
A scientist who is now over 80 was born in California. In 1962, he received his Ph.D. in mathematics from the University of Berkeley. For 37 years, he worked at Bell's laboratory, and later moved to AT&T Labs. The scientist actively collaborated with one of the greatest mathematicians of the 20th century, Pal Erdös, and is the winner of many prestigious awards. Graham's scientific bibliography contains more than 320 scientific papers.
In the mid-70s, the scientist became interested in the problem associated with the Ramsey theory. In its proof, the upper bound of the solution was determined, which is a very large number, subsequently named after Ronald Graham.
Hypercube problem
To understand the essence of the Graham number, you must first understand how it was obtained.
The scientist and his colleague Bruce Rothschild were engaged in solving the following problem:
- There is an n-dimensional hypercube. All pairs of its vertices are connected so that a complete graph with 2 n vertices is obtained. Each of its edges is painted either blue or red. It was necessary to find what the smallest number of vertices should be in a hypercube so that each such coloring contains a complete one-color subgraph with 4 vertices lying in one plane.
Decision
Graham and Rothschild proved that the problem has a solution N 'satisfying the condition 6 ⩽ N' ⩽ N where N is a well-defined, very large number.
The lower bound for N was subsequently refined by other scientists who proved that N must be greater than or equal to 13. Thus, the expression for the smallest number of vertices of a hypercube satisfying the conditions presented above took the form 13 ⩽ N'⩽ N.
Whip arrow notation
Before giving a definition of Graham's number, you should familiarize yourself with the method of its symbolic representation, since neither decimal nor binary notation is absolutely not suitable for this.
At present, it is customary to use Knuth's arrow notation to represent this quantity. According to her:
a b = a up arrow b.
For the operation of multiple exponentiation, the following entry was introduced:
a "up arrow" "up arrow" b = a b = "tower, consisting of a in the amount of b pieces."
And for pentation, that is, the symbolic designation of re-raising to the power of the previous operator, Knut has already used 3 arrows.
Using this writing option for the Graham number, we have “arrow” sequences nested in each other in the amount of 64 pieces.
Scale
His famous number, which excites the imagination and expands the boundaries of human consciousness, pushing it beyond the limits of the Universe, Graham and his colleagues received it as the upper limit for the number N in the proof of the hypercube problem presented above. It is extremely difficult to imagine how large its scale is for an ordinary person.
The question of the number of characters, or as it is sometimes mistakenly said, zeros in Graham's number is of interest to almost everyone who first hears about this value.
Suffice it to say that we are dealing with a rapidly growing sequence, which consists of 64 members. Even its first member is impossible to imagine, since it consists of n "towers" consisting of 3-k. Already its "lower floor" of 3 triples is 7 625 597 484 987, that is, it exceeds 7 billion, which is to say about the 64th floor (not a member!). Thus, to say exactly what the Graham number is equal to is currently impossible, since the combined capacities of all computers existing on Earth today are not enough to calculate it.
Is the record broken?
In the process of proving Kraskal’s theorem, Graham’s number was “dropped from the pedestal”. The scientist proposed the following task:
- There is an infinite sequence of leaf trees. Kraskal proved that there always exists a section of a graph, which is both part of a larger graph and its exact copy. This statement is not in doubt, since it is obvious that in infinity there will always be a precisely repeating combination.
Later, Harvey Friedman narrowed this task somewhat by considering only such acyclic graphs (trees) that for a particular one with coefficient i there are at most (i + k) vertices. He decided to find out what the number of acyclic graphs should be, so that with this method of their task one could always find a subtree that could be embedded in another tree.
As a result of research on this issue, it was found that N, depending on k, grows at an enormous rate. In particular, if k = 1 then N = 3. However, for k = 2, N already reaches 11. The most interesting thing starts when k = 3. In this case, N rapidly “takes off” and reaches a value that is many times greater than the Graham number. To imagine how large it is, it is enough to write down the number calculated by Ronald Graham in the form of G64 (3). Then the Friedmann-Kraskal value (vol. FinKraskal (3)) will have the order G (G (187196)). In other words, it turns out to be a mega-quantity that is infinitely larger than the unimaginably large Graham number. At the same time, even it will be a giant number of times less than infinity. It makes sense to talk about this concept in more detail.
Infinity
Now that we have explained what a Graham number is on our fingers, we need to understand the meaning that has been invested and is being invested in this philosophical concept. Indeed, “infinity” and “infinitely large number” in a certain context can be considered identical.
The greatest contribution to the study of this issue was made by Aristotle. The great thinker of antiquity divided infinity into potential and actual. By the latter, he meant the reality of the existence of infinite things.
According to Aristotle, the sources of ideas about this fundamental concept are:
- time;
- separation of values;
- the concept of the border and the existence of something beyond its borders;
- the inexhaustibility of creative nature;
- thinking that has no limits.
In the modern interpretation for infinity it is impossible to indicate a quantitative measure, so that the search for the largest number can continue forever.
Conclusion
Can the metaphor “Look into infinity” and the Graham number be considered synonymous in a sense? Rather, yes and no. Both that and another it is impossible to imagine, even having the strongest imagination. However, as already mentioned, it cannot be considered "the most, the most." Another thing is that at the moment the values greater than the Graham number do not have an established physical meaning.
In addition, it does not possess such properties of an infinite number ¥ as:
- ∞ + 1 = ∞;
- there is an infinite number of odd and even numbers;
- ∞ - 1 = ∞;
- the number of odd numbers is exactly half of all numbers;
- ∞ + ∞ = ∞;
- ∞ / 2 = ∞.
To summarize: the Graham number is the largest number in the practice of mathematical proof, according to the Guinness Book of Records. However, there are numbers that are many times greater than this value.
Most likely, in the future there will be a need for even larger “giants”, especially if a person goes beyond our solar system or invents something unimaginable at the current level of our consciousness.