Peteris Krumins has a nice post on using Fibonacci numbers to covert between miles and kilometers. The TL;DR is that if you take any two consecutive Fibonacci numbers, X and Y, there will be approximately Y kilometers in X miles (and vice versa, of course). The approximations are amazingly precise.
There’s more. The above works only if the number you want to convert is a Fibonacci number but you can express the number you do want to convert as a sum of Fibonacci numbers and sum up the corresponding pair numbers to get the conversion. Take a look at Krumins’ post for an example.
It’s not an accident that it works, of course. The secret is the ratio of consecutive Fibonacci numbers, \(F_{n+1} / F_{n}\) converges to the Golden Ratio, \(\frac {1+\sqrt{5}} {2} \approx 1.618034\). The key is that there are 1.609344 kilometers per mile, which is very close to the Golden ratio.
Serendipitously, John Cook, a Mathematician, has a recent post on the same subject. He makes an interesting observation. The higher the value of n the closer \(F_{n+1} / F_{n}\) gets to the Golden Ratio so, naively, one expects the kilometer-mile approximation to get better too. That works for a while, but for larger n, the \(F_{n+1} / F_{n}\) ratio gets closer to the Golden Ratio than the number of kilometers per mile. Cook calculates where the ratio is the best approximation to kilometers per mile and, surprisingly, it’s at 21/13, a lower number than I would have expected.
Cook also mentions Lucas numbers, which are very similar to Fibonacci numbers except for the starting values. It turns out that Lucas numbers provide an even better approximation but, really, both methods are back of the envelope approximations so who cares.
Perhaps I find this interesting only because I’m a Mathematician but I’m pretty certain the average Irreal reader will find it so too.