What is the largest number you can imagine? I’m sure more than one reader will paraphrase Han Solo, thinking they can imagine a lot. But whatever the number they say, it surely doesn’t even come close to what the so-called Graham’s number represents; so much so that it is impossible to write it in conventional terms and one has to resort to scientific, exponential notation, and even that way it is quite complicated. Ronald Graham, its formulator, achieved with it something he probably did not think of: to enter the Guinness Book of Records.
All of us who watched Cosmos, Carl Sagan’s documentary series, in the distant 1980s, remember how fascinated we were by, among a thousand other things, the googol and the googolplex, which we had never heard of before.
The first is a number devised in 1938 by the mathematician Edward Kasner (although the peculiar name was not given by him but by his little nephew), conceived to show the difference between a colossally large number and infinity; it is equivalent to 10 to the power of 100; or, in other words: 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000.
Then he added the googolplex, which is a one followed by a googol of zeros, and the googolduplex, a one followed by a googolplex of zeros. These numbers are so huge that they have no practical application in mathematics.
To give you an idea, a normal calculator is below the googol in representation capacity and, as Sagan explained, there would be no paper large enough to write all the zeros of a googolplex in the entire Universe. Google did not choose this name for its company by chance.
With the googolplex, things get worse because, for example, the number of possible combinations for combining atoms is approximately 10 to 10 to 70, which means that in a universe with the dimensions of a googolplex, there would start to be repetitions; admittedly, many of us would find it fun to find our exact physical double.
If we are going to fantasize, we could continue with the googolduplex or even with the following ones in the sequence, googoltriplex, googolcuadruplex, etc; after all, there can be as many as we want. But it is not necessary because for that a baby came into this world in 1935 who was baptized as Ronald Graham. To be exact, he did so in the American town of Taft, a small Californian town of just over nine thousand inhabitants, even fewer at that time.
Graham studied mathematics at the famous Berkeley University, where he obtained his PhD in 1962. He dedicated his professional life to research in technology, first at Bell Labs (now owned by Nokia) and then at AT&T Research Labs, where he remained for thirty-seven years until his retirement in 1999, becoming director of information sciences. However, it was not exactly his work that made him famous, but a paper he published in 1977 on Ramsey Theory.
It takes its name from the English mathematician and philosopher who enunciated it, Frank Plumpton Ramsey, one of those privileged and precocious brains, capable of learning German in a week with a dictionary and a grammar just to be able to read Wittgenstein’s Tractatus logico-philosophicus in its original language or to be appointed professor at the prestigious King’s College of Cambridge University when he was barely twenty-one years old. A kidney disease ended his life as prematurely as his intellectual development, at twenty-six, but he had time to leave us the theory in question.
He formulated it in an article entitled On a problem of formal logic, which he published in 1928 in the journal of the London Mathematical Society and is based, in synthesis, on considering that complete disorder is impossible. Within a sufficiently large system, in spite of disorder there must be some order; therefore, it is necessary to study under what conditions such order is present.
Ramsey’s theory thus falls within the field of combinatorics, a branch of discrete mathematics (the study of discrete sets, i.e. finite or infinite numberable sets) that focuses on the properties of arrangements or groupings of a certain number of elements under certain established conditions. One of the issues it raised was the need to determine a large number that would serve as a maximum limit to facilitate the solution.
Obviously, he was not referring to large everyday numbers but to much larger numbers requiring scientific notation. When Ramsey wrote his article, Edward Kasner had not yet invented the googol; but in 1977 Ronald Graham turned the whole thing upside down with his proposal.
Perhaps the matter would not have been so far-reaching if Martin Gardner, a famous philosopher and popularizer of science in the style of Isaac Asimov, had not intervened, one of those who defended the need to give science a popular character, bringing it closer to the general public.
Gardner set an example by practicing illusionism (curiously, Graham juggled) and was a militant in recreational mathematics, which is applied for recreational purposes (for example, the Rubik’s Cube, Sudoku or chess itself). But, above all, he published a monthly column of that tone in Scientific American magazine, under the heading Mathematical Games. In the November 1977 issue he wrote a reference to something he had been working on with Graham for some time:
“In an unpublished proof, Graham has recently established … a bound so vast that it holds the record for the largest number ever used in a serious mathematical proof.”
That dimension was so extraordinary that it caught people’s attention because it surpassed all the large numbers formulated until then, such as the one devised by the South African mathematician Stanley Skewes or the Steinhaus-Moser notation, which were already larger than the googolplex. All with a number of digits impossible to represent graphically even if each one occupied one of the Planck units, which are the smallest measurable spaces, sometimes nicknamed God units because they are beyond human measurements.
In fact, Graham’s number cannot even be represented as a power (i.e., x to the power of …) or even to towers of exponents or tetrations (itinerant exponentiations, i.e., a number powered to itself several times), so it is necessary to resort to formulas of a type of notation called Knuth’s arrow. With them it has been possible to calculate the last digits of Graham’s number, since they must have certain properties common to towers of that type. In its entirety it is impossible to write it down, because the observable universe is so small that it would not fit in it, but, in case anyone is curious, the final five hundred digits are:
…02425950695064738395657479136519351798334535362521430035401260267716226721604198106522631693551887803881448314065252616878509555264605107117200099709291249544378887496062882911725063001303622934916080254594614945788714278323508292421020918258967535604308699380168924988926809951016905591995119502788717830837018340236474548882222161573228010132974509273445945043433009010969280253527518332898844615089404248265018193851562535796399618993967905496638003222348723967018485186439059104575627262464195387
We said at the beginning that in 1980 the Guinness Book of Records registered Graham’s number as the largest, although, as usually happens, it was soon surpassed by others; this is the case of those related to the various finite forms made by the American mathematician William Friedman of the TREE (also known as Kruskal’s theorem tree, after the mathematician who proved it, also American Joseph Kruskal, based on an idea of the Hungarian Andrew Vázsonyi).
This article was first published on our Spanish Edition on January 28, 2019. Puedes leer la versión en español en El número de Graham, tan grande que su representación ordinaria no cabría en el universo observable
Sources
Mathematical games (Martin Gardner en Scientific American)/Cosmos (Carl Sagan)/Wonders of numbers. Adventures in mathematics, mind, and meaning (Clifford A. Pickover)/How big is big and how small is small. The sizes of everything and why (Timothy Paul Smith)/On a problem of formal logic (Frank P. Ramsey)/Wikipedia
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