When you see the word googleplex, odds are good that you have one of three reactions:
- What’s a googleplex?
- Hey, that’s Google’s headquarters.
- Somebody doesn’t know how to spell that number.
Yes, as many of you undoubtedly are aware, Google, the search engine-powered media giant, is a clever misspelling of googol, a number represented by a one with 100 zeroes behind it (10100). Google, Inc. extended the quirky pun by naming its major company offices Googleplexes after another number, the googolplex, which is a one with a googol zeroes behind it. (That is, 10^10100 or, in what passes for mathematic humor, 10googol.)
Okay, so where did the word googol come from? After all, it doesn’t fit into the more familiar etymology of large numbers laid out with Greco-Latin roots such as quad- for quadrillion (1015), oct- for octillion (1027), or dec- for decillion (1033).
These terms conform to the short scale of long numbers, which is the primary English language system for naming numbers, wherein each power of 1,000 has its own [root] + -illion designation. (This is opposed to the large scale, which names every power of one million.)
A googol doesn’t fit into this system because it’s not a recognized number in orthodox mathematics. In the short scale, 10100 is actually 10 duotrigintillion.
Mathematician Edward Kasner coined the term googol in his 1940 book Mathematics and the Imagination. More properly, Kasner’s nephew coined the term when asked to make up a memorable name for a one with 100 zeroes behind it. That same nephew also coined the term googolplex, with Kasner supplying the ten to the googol definition. Both terms have since slipped in the popular consciousness, with one or the other often named as the “biggest number there is.”
In truth, the largest designated number in the short scale is the centillion (10303). The mathematics world considers any number larger than a one with 303 zeroes behind it to be infinite, at least for all reasonable purposes. That’s not to say that mathematicians don’t use numbers larger than the centillion, just that these numbers usually defy description — and sometimes even scientific notation.
Such is the case for the largest number ever put to practical mathematical use — one so large it dwarfs the number of atoms in the universe.
WHAT IS THE LARGEST NUMBER EVER PUT TO PRACTICAL USE IN MATHEMATICS?
What’s the largest number ever put to practical use in mathematics — a figure so significant that it dwarfs a centillion, a googolplex, and even the total number of atoms in the known universe?(For the record, rough calculations suggest that there are 1080 atoms in the known universe or, in the short scale notation, 100 quinvigintillion atoms. There aren’t even a googol’s worth of atoms out there.)
We’re talking about Graham’s Number, G, a mathematic concept so esoteric — and so massive — that there’s no simple way to even describe the number of digits that comprise Graham’s Number, let alone the number itself. Graham’s Number is just part of the answer to the following word problem:
“Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on 2n vertices. Then color each of the edges of this graph using only the colors red and black. What is the smallest value of n for which every possible such coloring must necessarily contain a single-colored complete sub-graph with four vertices that lie in a plane?”
Now, this Trivia Geek will be the first to admit that he has no idea what the above problem means, but it sounds impressive. As noted, Graham’s Number isn’t even the answer to this problem — which no one has yet solved.
Instead, G is merely the smallest known upper bound of all the possible numbers that could answer the problem, based on existing mathematic proofs. The established lower bound is six, which is yet another example of mathematic humor.
Put another way, mathematicians have proven that the answer to the above problem is a number between six and G, where G is a concept so large we can’t conventionally describe using scientific notation — not exactly narrowing it down.
To write out Graham’s Number using a concise expression still requires using some rather oblique terminology, the hyper operator. In this context, G = f64(4) where f(n) = hyper(3,n + 2,3). Good luck explaining that in a bar bet, though it does explain why the Guinness Book of World Records crowned G as the Champion Largest Number.
All that adds up to a serious computational conundrum — and some numerically notorious Geek Trivia.
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