Thursday, 10 April 2008
Harmony, Record Floods, and Traffic Jams
I wrote recently about the study in Japan of traffic jams, and recently realized that the mathematics of traffic jams and record rainfalls (or any other kind of record) have some things in common. But to tie them together in a mathematical way, I'll give you a problem, but first a little information for those not rememering all the language you learned in Alg II.
The harmonic series is the sum of a sequence of unit fractions with the natural intergers as denominators, 1+ 1/2 + 1/3 + 1/4 + ..... and on and on "To Infinity and beyond" as my Buzz Lightyear toy calls out... Now it is well known that the function diverges. That means that there is no limit to the sum. If you go far enough you will pass any number you can name.
For some kids that seems perfectly natural. They kind of think that if you keep adding more and more onto it, you will just get bigger than any limit you could imagine, but in fact, high school math is full of series that DO have limits. Take the sum 1+ 1/2 + 1/4 + 1/8 + 1/16... Keep going as far as you want, you will never get past two. It is easy to see graphically that you can't. Draw a square and shade it (ok, that's one) and now draw another and shade 1/2 of it... now shade 1/4 of it (1/2 of what is unshaded) and continue forever. The second square will not get full in any finite time. Game over, you lose, do not pass 2, do not collect $200.
Ok, so now you know a little about the harmonic series. Each term gets smaller and smaller, but in the end it goes to infinity; so here is the problem. Can it get to infinity without ever landing on an integer (after one), or does the "getting smaller and smaller" part mean that it must land on one somewhere out there? If you take small enough steps you have to land on a crack in the sidewalk somewhere, or do you?
If you think you know, drop me a note and I will tell you if you have figured it out or not; but of course if you think it does, you should come up with a value, or at least an order of magnitude for what it will be. On the other hand, if you think it NEVER lands on an integer you ought to be able to come up with some explanation.
Ok, so what does that have to do with foods and traffic jams? Consider this problem for a moment; If we started keeping records today, how many years will have a record flood in the next ten years, or the next hundred years. Well, the first year MUST be a record, so there will be at least one. Now what is the probability that the second year beats the first? Sure, 1/2 is the obvious answer. Now what about the third year; how probable is it to beat both the previous years... 1/3... and you are beginning to see a pattern. So if we wait ten years, we would expect, on average, to have 1 + 1/2 + 1/3 +... + 1/10 records occur. My calculator picks 2.93 or not quite 3 record years. For 100 years, we would expect only about 5.18 record events.
Of course it doesn't have to be a flood. You could see how many heads you can juggle the family knife set (ok, maybe not a good ides). But how about a traffic jam. If we focus on a single lane of traffic there will be a bunch of cars all wanting to go their preferred speed. Some get stuck wanting to go faster than the one in front. How many bunches of cars will there be? Well, each front car in a bunch is a record breaker for slow speed. So with n cars, the expected number of bunches would be 1+ 1/2 + 1/3 + .... + 1/n....
Next time you have that pokey guy in front of you, instead of honking and screaming and having a brain seizure, just calmly think to yourself, "Well this record is not as bad as being in a 100 year flood."