Think of a big number. A really big one. How about a googol? That's 1 followed by 100 zeros, or 10^100. It's huge and finite, but it's nothing compared to Graham's number.
Graham's number was first used by mathematician Ronald Graham in 1977 as an upper bound for a problem he was working on. At the time, it was the largest number ever used in a mathematical proof and even made it into the Guinness Book of World Records. And no, you can't write it out in the traditional way.
It's not easy, let's be clear from the start. The very nature of the number makes it hard to explain, and even with all the facts on the table, the result is frustrating because we can never visualize it. The only alternatives to Graham's number are to know what it measures, have an idea of how to get to it, and know its last digits.
Our best ally for this titanic task is Graham himself. In July 2014, he spoke with the folks at Numberphile about his beast, and began by explaining its usefulness. What is the goal of Graham's number? What does it try to calculate or measure? Almost absurdly, the whole adventure starts with drawing a square…
Note: Subtitles are in English, but you can follow along.
Graham's Number
A square, four lines, four vertices, two dimensions. Now, there are two extra ways to connect the vertices. That's as simple as crossing two lines inside the square, joining opposite vertices. Your two-dimensional square now has six lines. You can paint those lines blue or red, in any combination except two (we'll get to that).
However, the exercise extends to other dimensions. The square becomes a cube, and you have to connect its vertices with lines painted red or blue. If we keep going up, the cube becomes a tesseract. The number of lines and red-blue patterns grows quickly, but there are two forbidden combinations you must avoid: a face with six lines and four vertices that is all blue, or all red. Your mission, should you choose to accept it, is to avoid those combinations as you go up in dimensions. The problem is you can't win. There is a point or number of dimensions where those blue or red faces are inevitable... and it's Graham's number.
How to Write Graham's Number
So... how do we write it? The simplest method is to say G = g₆₄, where g₁ equals 3↑↑↑↑3. The arrows are what we call "Knuth's up-arrow notation", created by Donald Knuth in 1976. Basic example: 3↑3 is 3 cubed, 27. If you write 3↑↑3, that is:
3↑(3↑3) =
3↑(3³) =
3↑(27) = 7,625,597,484,987A little over 7.62 trillion. g₁ demands continuing the process for four arrows. In very loose terms, each additional arrow makes the exponentiation of 3 ascend as determined by the number of the previous arrow:
3↑↑↑3 = 3↑↑(3↑↑3)
3↑↑↑3 = 3↑↑(7625597484987)
3↑↑↑3 = More than 3.6 trillion digits3↑↑↑↑3 equals 3↑↑↑(3↑↑↑3). If you think the number is already outrageously large, remember that we've barely reached g₁, so there are 63 more levels. The number of arrows for g₂ is g₁, and so on until we reach G = g₆₄.
It's impossible to calculate all of Graham's number, but what we can do is calculate the last digits. To say goodbye, here are the last 500:
02425950695064738395657479136519351798334535362521
43003540126026771622672160419810652263169355188780
38814483140652526168785095552646051071172000997092
91249544378887496062882911725063001303622934916080
25459461494578871427832350829242102091825896753560
43086993801689249889268099510169055919951195027887
17830837018340236474548882222161573228010132974509
27344594504343300901096928025352751833289884461508
94042482650181938515625357963996189939679054966380
03222348723967018485186439059104575627262464195387If you enjoyed this, you might also like this article about the golden ratio: https://old.neoteo.com/numero-aureo-belleza-matematica/
