Quick: What’s the biggest number you can think of? Okay, time’s up. Mine was 85. I’m proud of it. But as it turns out, there are even larger numbers. In a recent post at the explainer blog Wait But Why, Tim Urban teaches readers about Graham’s Number, one of the largest figures commonly referenced in mathematics.
Urban approaches Graham’s Number by going beyond multiplication (e.g., 3 times 3) and exponentiation (3^3) into stranger realms of tetration, pentation, and hexation. With care and enthusiasm, Urban details these mathematical processes and how they eventually lead to Graham’s Number, which once held the world record for the largest number cited in a “serious mathematical proof” (until it was superseded, presumably, by Graham’s Number + 1).
Graham’s Number is so big that it’s impossible to explicitly compute, let alone write down. We actually can figure out the last few digits, but that’s it—nobody knows how it begins, so you couldn’t write it down in decimal form. We can talk about it, though, and express it in more abstract terms. As you might imagine, getting at Graham’s Number involves some mind-bending mathematics. Urban’s explanation includes images like this…
…but if you take it slow, you should be able to follow along. Once the outlandish vastness of Graham’s Number is fomenting a deep existential terror within you, then you’re getting the idea.
If Urban’s article doesn’t do it for you, here is mathematician Ron Graham himself explaining the origin of the number with his name on it:
And here’s another video in which Graham talks about how the number is calculated:
I still think 85 is pretty big.