Any Ideas?

How Fibonacci can help on your holiday to France

A practical use of the Golden Ratio

As August arrives, thoughts turn to Europe, and having to do all those conversions when on holiday.  No, not pounds to Euros but miles to kilometres, and vice versa.

The conversion rule that I was taught for miles to kilometres was to divide by five and multiply by eight.  Alternatively, if you double the miles and then knock off 20% that achieves the same thing.

But there is another method that saves all this mental arithmetic.  Just learn the Fibonacci sequence off by heart. Lest you’ve forgotten, the Fibonacci sequence begins 0 1, and after that each successive term is the sum of the previous two.  Hence you get:  0  1  1  2  3  5  8  13  21  34  55  89;.

And now for the conversion.  To convert 55 miles into kilometres, just look up the next number in the Fibonacci sequence: it’s 89, and sure enough 55 miles is 89 kilometres.  To convert from km to miles, find the previous value in the sequence.  34km is 21 miles.

Why does this work?

In the Fibonacci sequence, the ratio of successive terms is approximately equal to the Golden Ratio, which is about 1.6180.  And the further along the Fibonacci you go, the more accurate this approximation becomes.  The correct conversion from miles to km is 1.6093 (to four decimal places).  This is very slightly closer to the Golden Ratio than it is to the 1.6 conversion ratio that is generally taught.  And this means (for all values of 21 km or higher) that the Fibonacci sequence is a more accurate conversion to miles than the 8/5 rule.

There is only one snag in this scheme.  What if you are currently 56km from your destination?  Now you’ve got some tricky decimals to work out. 

At this point the best plan is just to sit tight.  By the time you’ve travelled another kilometre, you will be able to announce the distance in miles almost precisely.