Miles to kilometers

The number of kilometers in a mile is k = 1.609344 which is close to the golden ratio φ = 1.6180334.

The ratio of consecutive Fibonacci numbers converges to φ, and so you can approximately convert miles to kilometers by multiplying by a Fibonacci number and dividing by the previous Fibonacci number. For example, you could multiply by 8 and divide by 5, or you could multiply by 13 and divide by 8.

As you start going down the Fibonacci sequence, consecutive ratios get closer to k and closer to φ. But since the ratios converge to φ, at some point the ratios get closer to φ and further from k. That means there’s an optimal Fibonacci ratio for converting miles to kilometers.

I was curious what this optimal ratio is, and it turns out to be 21/13. There we have

|k − 21/13| = 0.0060406

and so the error in the approximation is 0.375%. The error is about a third smaller than using φ as the conversion factor.

The Lucas numbers satisfy the same recurrence relation as the Fibonacci numbers, but start with L0 = 2 and L1 = 1. The ratio of consecutive Lucas numbers also converges to φ, and so you could also use Lucas numbers to convert miles to kilometers.

There is an optimal Lucas ratio for converting miles to kilometers for the same reasons there is an optimal Fibonacci ratio. That ratio turns out to be 29/18, and

|k − 29/18| = 0.001767

which is about 4 times more accurate than the best Fibonacci ratio.

3 thoughts on “Miles to kilometers”

1. Evan Berkowitz

If you need something you can do in your head while driving, it may be more practical to simply use the Fibonacci sequence itself. If you’re going 30 mph, that’s about 50 kph, since (5 follows 3 in the sequence)*10. If you’re going 34 mph you can of course estimate 55 kph, but you can also estimate (30+3+1 –> 50+5+1=56 kph or 30+2+2 –> 50+3+3=56 kph or 30+5-1 –> 50+8-2 = 56 kph); the true answer is 54.7 kph, but that’s probably more precision than your car’s speedometer shows.

It’d be interesting to work out how well this works across reasonable transport speeds.

2. I always wondered *why* consecutive Fibonacci numbers converge to the golden ratio. Is there an explanation?

3. A full answer to your question would take a longer explanation, but here’s a high-level sketch. The Fibonacci numbers satisfy Fn+2Fn+1Fn = 0 and so the characteristic equation is x² − x − 1 = 0. One of these roots is the golden ratio φ and the other is its conjugate. These determine the limiting value of the ratios. The Fibonacci numbers have components due to both values (see Binet’s formula) but the component associated with the conjugate root decays to 0 and the golden ratio is the one that survives.

Shorter answer: Look up Binet’s formula, and the paragraph above explains where Binet got the idea for his formula.