数学英语 25 What is the Fibonacci Sequence

by Jason Marshall

It’s not often someone suggests that knowing some math could make you the life of the party, but that’s exactly what I’m going to do. Yes, a properly timed delivery of a few fun facts about the famed Fibonacci sequence just might leave your friends clamoring for more—because it really is that cool. So, without further ado, let’s continue our exploration of sequences that we began a few articles ago by jumping right in and talking about Fibonacci’s famous sequence.

Review of Mathematical Sequences

As we’ve discussed, sequences in math are fairly simple things—they’re just lists of numbers arranged in some particular order. The number of sequences that can be written is infinite since any random 1 list of numbers will do. But some types of sequences are decidedly non-random—one of which being the geometric sequence. In such a sequence, each element is obtained from the previous one by multiplying it by the same fixed 2 number. For example: 2, 4, 8, 16, 32, is a geometric sequence where each successive element is obtained by multiplying the previous one by 2.

Exponential Population Growth

In the last article, I used this particular sequence to describe how populations might grow. Starting with a single pair of organisms that produce one additional pair of offspring each reproductive cycle, the number of organisms will grow as: 2, 4, 8, 16, 32, and so on. After a few more generations, this sequence predicts that the population will become very large, very quickly. But does this type of sequence actually describe nature?

Well, it depends. This type of growth—so-called geometric or exponential growth—can, in fact, occur in some situations. But even if geometric growth occurs for a while, it can’t last forever since the quickly multiplying organisms will eventually deplete 3 their resources—be it food or available living space—at which point their numbers must stop growing exponentially. But is that the only problem with describing population growth as a geometric sequence?

Fibonacci’s Rabbits

No, not really. To explain, let’s head way back to the year 1202, at which point we meet our hero: Fibonacci—a bright young Italian guy from Pisa in his 20s who’d done a lot of traveling. After returning to Italy, and inspired by what he learned about math during his travels, Fibonacci wrote a book. But this wasn’t just any book—this book turned out to be...well, important. For one thing, Fibonacci used it to introduce Europe to the 0 through 9 numeral system we still use today. Without that, who knows—we might all be stuck counting with Roman numerals!

Among other things, Fibonacci’s book also included a musing 4 about a math problem which turned out to have a far more interesting and lasting 5 solution than anyone could have imagined. Fibonacci’s math question seems simple enough: If two newborn rabbits are placed in a pen, how many rabbits will the pen contain after one year? To answer his question, Fibonacci wanted us to assume the following:

whenever a pair of rabbits reproduces, they always produce one male and one female offspring;

rabbits can reproduce once per month;

rabbits can start to reproduce when they are one month old; and

rabbits never die.

So that’s the question. What’s the solution?

The Fibonacci Sequence

To start answering the question, let’s think in terms of how many pairs of rabbits there are at the beginning of each month. Start with the 1 newborn pair that exists at the beginning of the first month. These first two newborns are too young to reproduce that month, so we begin the second month with 1 pair as well. So far the sequence is

1, 1.

Not very exciting, but let’s keep going. At the beginning of the second month, the original pair is mature enough to mate. As a result, one new pair of rabbits is born at the end of the second month. So at the beginning of the third moth 6, we have a total of 2 pairs. The original pair again mates at the beginning of that month, but the newborn pair is still immature 7. The original pair produces another pair of offspring, so at the beginning of the fourth month, we have a total of 3 pairs of rabbits. The sequence is now

1, 1, 2, 3,

which is a little more interesting, but still fairly mundane 8.

（见图）

However, now things start to get exciting...and potentially confusing too—so stick with me. In fact, if you’re finding this a little hard to follow, check out the Math Dude’s “Video Extra!” for episode 16 on YouTube for a more graphical explanation. But getting back to our story... At the beginning of the fourth month, two pairs mate (the original, and the first pair of offspring), and one pair is still immature. Those two pairs that mated each produce a new pair, giving us 5 pairs at the beginning of the fifth month. Let’s go through one more month. At the beginning of the fifth month, three pairs mate, but the newest two pairs that were just born are still immature. After the three new pairs of offspring are born, our total moves to 8 pairs.

At this point, the sequence is:

1, 1, 2, 3, 5, 8.

Do you see a pattern? Would it help if I said the next number is 13? And the next after that is 21?

1, 1, 2, 3, 5, 8, 13, 21, …

I’ll admit, the pattern isn’t totally obvious at first. But after you see it, it is. The trick is that each number in the Fibonacci sequence is obtained by adding together the previous two:

1 + 1 = 2 is the third number,

1 + 2 = 3 is the fourth number,

2 + 3 = 5 is the fifth number,

3 + 5 = 8 is the sixth number,

5 + 8 = 13 is the seventh number,

8 + 13 = 21 is the eighth number, and so on.

Fibonacci vs. Geometric Sequences

So how many rabbits are there after twelve months? Well, if you work it out, the thirteenth Fibonacci number is 233—so 233 pairs is 466 rabbits. (Note that we need to use the thirteenth, and not the twelfth, Fibonacci number because each represents the number of pairs at the beginning of the month. So, the thirteenth number corresponds to the beginning of the first day of the subsequent year—which is exactly what we want.) Clearly, taking into account the fact that organisms can’t reproduce immediately after they’re born has a dramatic effect on the rate of population growth. After 12 monthly doublings, exponential growth from the geometric sequence model we talked about before predicts 8192 rabbits—that’s more than 17 times the number predicted by the Fibonacci sequence! Of course, even the Fibonacci sequence is too simplistic—living beings eventually die, for example. But it’s a beautiful application of how a bit of simple math can model the very complex world. And there’s much, much more it can do too...

Math, Fun, and Fibonacci

You might be wondering: What’s practical about the Fibonacci numbers? For today, my answer may surprise you: nothing. Today’s quick and dirty tip is that you shouldn’t look at math as something that always has to be practical. In fact, at its core, math isn’t practical. It’s a puzzle. Mathematicians 9 don’t sit around doing tediously painful, although perhaps practical, long division problems all day; they make up problems and amuse themselves with them. And, as a result, they often discover really interesting things about the world—all because they allowed themselves to play. The Fibonacci sequence is a great example of that: it’s cool, it’s fun, it’s surprising, it’s beautiful, and if you play your hand right, it just might make you the life of the party.

Fibonacci Numbers in Nature

But what really makes this sequence so famous? Why was it in The Da Vinci Code? What about flowers and shells and the golden ratio? And, speaking of that: What’s the golden ratio? Stay tuned 10, because in the next article, we’re going to find out. And as for my whole “there’s nothing practical about the Fibonacci numbers”—well, that really was just for today. In truth, there actually are practical uses. We’ll be talking about those things too.

Wrap Up

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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!