Math and Life

Fibonacci Conversion

You, the not-to-be-seen-as-ignorant American, are having a drink with your newly acquired English friends while on vacation. You ask for directions. They, being amicable English, give them. And you, dear tourist, don't do kilometers.

There are several ways to approach this problem. The classic (read: boring) way is taught in middle school:

(# of miles) = 1.6(# of kilometers); plug and chug.

Fully at the command of perceived social expectations, you do not want to whip out your phone to solve this equation and mental math was never your forte. You're stuck.

Let's take a break from this harrowing tale of social anxiety and talk about some math1.

Start with one, and another one, then add the last two numbers. Two. Now add the last two numbers again. Three. Again. Five. Again. Eight.

After a few more steps, we get a nice list:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

This is called the Fibonacci sequence, and is defined mathematically like so:

F(n) is the expression for the nth term in the sequence, so F(7) is 13, the 7th number.

Now I’m going to ask a slightly weird question: As n approaches infinity, what is ratio of F(n) to F(n-1)?

In other words, as we go farther and farther down the sequence, what happens when I divide one number by the number before it? I’ll use x to represent this mystery ratio2:

Because we are going really far out into the sequence, F(n) divided by F(n-1) is essentially the same as F(n-1) divided by F(n-2)both are a term divided by the term before it.

After a bit of rearranging, we get

Going back to (1), we can replace F(n) with F(n-1) + F(n-2) using the definition of the Fibonacci sequence.

Break this fraction up:

The first term simplifies to 1. The second term, we notice, is the same as (2), so we can replace that with 1/x!

Now multiply everything by x, and we can solve for x using the quadratic equation (link).

There are two solutions here: one negative, one positive. If the ratio were negative, each successive term would switch signs. The Fibonacci sequence doesn't do that, so we’ll disregard that solution and go with the positive one:

At this point, you may not be entirely convinced. I’ve just done a full page of really abstract dark magic. I can understand if you aren’t entirely pursuaded after reading that. Hopefully I can alleviate your doubts with a list of the ratios between each successive Fibonacci pair:

The further you go, the closer it becomes.

This number, 1.618…, has a name: the golden ratio. It has many very interesting properties (some of which I hope to cover in future posts), but one of the most often overlooked fact about this special number is that it is almost exactly the ratio between a mile and a kilometer.

This means that consecutive Fibonacci numbers are also mile/kilometer pairs3:

How many miles is 34 kilometers? 21.

How many kilometers is 3 miles? 5.

So remember this the next time you’ve been given directions in a faraway land: the Fibonacci sequence is your friend.

1 In my experience, the solution to all social anxiety problems.

2 For the rigorous among us, yes, I did assume that the sequence converges.

3 Or at least the best integer approximation until F(12). Past this, the Fibonacci conversion diverges from the actual conversion because the golden ratio and the mile/kilometer ratio are ever so slightly off. (but still really close)

Image sources can be found here.