Name the bigger number: 83! A tale of exponentials
Ever wanted a taste of the things Computer Scientist thinks about? This essay was so well written I actually read it twice. All you need is high school math to appreciate most of it.
Read Who Can Name the Bigger Number by Scott Aaronson (Asst. Prof, EECS, MIT)
In an old joke, two noblemen vie to name the bigger number. The first, after ruminating for hours, triumphantly announces “Eighty-three!” The second, mightily impressed, replies “You win.”
A biggest number contest is clearly pointless when the contestants take turns. But what if the contestants write down their numbers simultaneously, neither aware of the other’s? To introduce a talk on “Big Numbers,” I invite two audience volunteers to try exactly this. I tell them the rules:
You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number — not an infinity — on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.
So contestants can’t say “the number of sand grains in the Sahara,” because sand drifts in and out of the Sahara regularly. Nor can they say “my opponent’s number plus one,” or “the biggest number anyone’s ever thought of plus one” — again, these are ill-defined, given what our reasonable mathematician has available. Within the rules, the contestant who names the bigger number wins.
Are you ready? Get set. Go.
In about 30 minutes, the author brings you through Exponentials, Fermat’s Last Theorem, Halting Problem, NP Complete and even Busy Beavers 😀 But I thought to the laymen, the first section on exponentiation is the most important for their daily lives. As a result, we often fail to see the multiplicative effects of our actions.
In perhaps a worldly view, the author quotes examples like this:
But do people fear big numbers? Certainly they do. I’ve met people who don’t know the difference between a million and a billion, and don’t care. We play a lottery with ‘six ways to win!,’ overlooking the twenty million ways to lose. We yawn at six billion tons of carbon dioxide released into the atmosphere each year, and speak of ‘sustainable development’ in the jaws of exponential growth. Such cases, it seems to me, transcend arithmetical ignorance and represent a basic unwillingness to grapple with the immense.
As for me, besides hearing comments like “so what’s the difference between 1Mbps and 1Gbps, if the website is going to load slowly anyway”, I’m often amused by the smallest gestures like “let’s sell our product / idea by engaging each of our customer / stakeholder one by one, starting from the largest”, or “we have 100,000 companies to attend to, so we should start with the first 100 (and ignore 99,900 of them)”. This is plainly ignoring the large number, or unwilling to think out of the box, like writing an exponential number, or an Ackermann sequence. People are unwilling to open up to a number that they don’t have the “biological instinct” to manage, such as putting up something on the web for millions to see and comment versus conducting surveys manually. Those who grapple with such large numbers eventually win, because in the process the person would have managed to simplify the representation to an effective operation.
So the next time you’re scoping a problem, ask yourself is there a way to articulate your scope in an exponential manner. Instead of having, say, a branch office serve 1000 customer per day, ask yourself if you can virtualize your service and make them available in any shops, or resold by more distributors. Instead of saying that you will put up X number of posters for 100X eyeballs, ask yourself how to generate a campaign that yields a 1-to-10 person word-of-mouth effect. Instead of getting your boss for a remuneration amount (boss give me $1000 more please), perhaps ask for a numeration multiplier (boss increase my salary by 1% if I complete my weekly work on time) – who knows, he might be like the King who agreed to the Grand Vizier in Persia modest reward.
