How do you like your chances? Sure, that giant “9” probably won’t fall on this digit… nor on this digit… but how long do you think your luck will last? Eventually you’ll get unlucky, and that “9” will come plummeting from the sky. The chances are in your favor.īut now you begin to type a 100-digit number. For short numbers, with just 2 or 3 digits, you’re not very worried. It’s as if, every time you type a digit, there’s a 10% chance that a giant numeral “9” falls from the sky and crushes you.
That might not feel like a big danger-after all, those 9’s will pop up only 10% of the time. Every time you add a digit, you add a new opportunity for a 9. You see, the longer a number gets, the harder it is to avoid a 9. Like 9 million, or 47 billion, or 228 trillion.” When you think of “numbers,” you’re only picturing little numbers. “Why should throwing out the numbers with 9’s make such a difference? We’ve still got all the other numbers!”Īgain, I say: not so fast. “Nonsense, you gullible old toad!” you are perhaps shouting to your screen. If you throw out the numbers with 9’s in them, the series is small enough to converge. But now we get to the truly strange part of the whole ordeal, the truth that prompted the inimitable Zach Weinersmith of SMBC to build a punchline around it: In mathematical language: the series diverges. This sum eventually exceeds a million, then a billion, on its way to the stars. That line you’re in? It moves like ball lightning compared with the growth of this series.Īnd yet… this series never settles down. Instead, imagine that you’re waiting in line at the DMV.Īnd that employee is one of those talking trees from Lord of the Rings.Īnd the line includes all 7 billion people on earth. In fact, the word “slowly” fails to evoke its agonizingly incremental pace. To be clear, this total grows incredibly slowly. Well… let’s look at some running totals as we go along. Now, back to our original sum, the harmonic series. Sure, you’ll get close-achingly, painfully, infinitesimally close-but you’ll never surpass it. But it’s finite in the sense that no matter how many numbers you add, you’ll never exceed 2. This sum is “infinite” in the sense that it goes on forever, with no final term. The next term brings us halfway AGAIN to 2. The next term brings us halfway again to 2. Perhaps your first thought is this: “You silly man you’ve already answered your own question.
Or does it just keep rising, forever and ever, eventually exceeding a million, then a billion, then a trillion, and so on, surpassing any ceiling or limit we might imagine? That is to say: Does it level out at some value? Lovely, yes, but does it-in any meaningful sense-“equal” anything? Now, what is the harmonic series? It’s this: (Obviously it’s a mistake to post an actual cartoonist’s work alongside my own second-grade-quality scrawl, but hey, maybe I’ll benefit from a math humor cheerleader effect.) A few weeks ago, the webcomic Saturday Morning Breakfast Cereal posted a cartoon about the harmonic series.
0 kommentar(er)
