Tuesday, October 23, 2012

The Fibonacci Sequence

1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots.

Looks like a bunch of random numbers, right?

Look closer.  Each number is the sum of the two that came before it. This set of numbers is called the Fibonacci sequence, after Leonardo of Pisa (not DaVinci) who was also known as Fibonacci.  He first discovered these numbers after traveling widely in the Arabic world as a boy, where he was exposed to the numbers we know now, Arabic numerals.  These proved to be much easier to do arithmetic with than Roman numerals, and he chose to study with Arabic masters.

Upon his return to Europe, he began work on his book, Liber Abaci or Book of Calculations.  Published in 1202, it detailed the efficiency and simplicity of Arabic-Hindu numbers, as well as introducing 0-9 and place values. He demonstrated the new system by using it in bookkeeping and interest problems, rendering it incredibly useful to the merchants of the day. It was widely read by educated Europeans.

But what about the Fibonacci sequence?  In Liber Abaci, Fibonacci presented a problem involving the growth of rabbit populations. In it, he assumed that two idealistic rabbits in a field can mate one month after birth and produce another pair of rabbits once a month thereafter.  The question is, if they mate once a month, how many rabbits will there be in one year?  

We start off with one pair. In a month, they mate, but there's still only one pair.  This is the 1, 1 of the sequence. At 2 months, there are now 2 pairs of rabbits (1,1,2). At the end of 3 months, the original pair produces another pair, which makes 3 (1,1,2,3).  At the end of the fourth month, the original female and her daughter each produce a pair, which makes 5 pairs total (1,1,2,3,5).  This continues until the 12th month, when you have 144 pairs of rabbits.

Okay, so there's lots of story problems in modern math books too. What makes the Fibonacci sequence cool?

Maybe the fact that Fibonacci numbers occur in nature, and that this is probably where the sequence came from? Interestingly enough, Fibonacci number and their cousins, Lucas numbers, appear quite often in things like flowers and pinecones and cauliflower. Luckily, Vi Hart from Khan Academy explains it in this 3 part video that's fun and informative.

Isn't math neat?

Next week: Felix Baumgartner's jump from the "edge" of space.