When traveling in countries such as the UK or USA you may come across distances, or speed limits, expressed in miles instead of in kilometers, and sometimes it can be useful to have a quick method to convert from miles to km and vice versa. In these cases, the Fibonacci number trick with which to convert from miles to km you just need to go from one Fibonacci number to the next and vice versa, and to use it you just need to remember a few numbers. Let’s see the trick, how to use it to convert highway speed limits, and why whether it works is a question of the golden ratio.
The trick to convert miles to kilometers with Fibonacci
The trick is based on the numbers of Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55 …) and when the number to be converted appears in the list it works like this:
- To convert from miles to km take the next Fibonacci number, for example 8 miles is approximately 13 km,
- To convert from km to miles take the previous Fibonacci number, for example 8 km corresponds approximately to 5 miles
In case the number to be converted does not appear in the list then the method can be adapted with simple additions or multiplications depending on the case, here are some possibilities:
- If the number to convert is a easy multiple of a Fibonacci number just multiply, i.e. convert 80 miles in km we see 80 as 8×10, so we convert 8 miles in 13 km and we multiply 13 × 10, 80 miles is approximately a 130km.
- Similarly, if the number to be converted is an easy submultiple of a Fibonacci number, just divide, for example to convert 4 km into miles we look at the double of 4, i.e. 8, which is preceded by 5, so 8 km corresponds to 5 miles and by halving we find that 4 km corresponds to 2.5 miles.
- Finally, if the number to be converted is la sum (or difference) of two or more Fibonacci numberswe can convert those numbers and add the results, for example to find out how many km corresponds to 18 miles just look 18 as 5 + 13 and taking the subsequent fibonacci numbers we conclude that
18 miles ≅ 5 miles + 13 miles ≅ 8 km + 21 km ≅ 29 km.
These conversions, albeit approximate, they can be especially useful when we are driving in a country that uses miles for driving directions. For example in the UK on the motorway limit And 70 mph (miles per hour), but 70 = 7 × 10 and 7 = 2 + 5 so to convert we take the numbers 3 and 8 which in the sequence (1, 1, 2, 3, 5, 8, 13 …) follow the 2 and the 5 and add them: 3 + 8=11 which multiplied by 10 gives 110 km/h. Conversely, the Italian motorway limit is 130 km/ha multiple of 13 that we can easily convert to miles by taking the previous number, i.e. 8, and multiplying it by 10: the Italian limit of 130 km/h corresponds to approximately 80 mphwhich is much higher than the 70 mph in force in the UK.
But how accurate is this conversion method and why does it work?
This conversion method And approximate, and it doesn’t work well for all numbers, but the larger the numbers the more accurate it becomes. For example with this method 1 mile would correspond 2km awaybut in reality 1 mile coincides with 1,609 kilometers, and this number gives us the conversion factor between miles and kilometers: 1 mile = 1,609 km. Still according to our method, 2 miles would correspond to 3 km, a little short compared to the actual 3,218 km, but as we go forward things improve, and already when we convert approximately 3 miles into 5 km the conversion factor, between 3 and 5, turns out to be 1,666 which is very close to the correct 1,609.
Moving forwardtrying with 13 and 21, the conversion factor is even bettereven 1,615 against 1,609, and one wonders whether with increasingly larger numbers we could not actually reach 1,609. The answer is no, and we know it with certainty, because one of the characteristics of the Fibonacci sequence is that the ratio (the division) between a number and the previous one, as we take increasingly larger numbers, gets indefinitely closer to a number known as golden section which is worth approximately 1.618. And it is precisely this characteristic of the Fibonacci sequence that is why the conversion trick works, that is, the fact that, by a lucky coincidence, the ratio between two consecutive numbers of the Fibonacci sequence is very similar to the conversion factor between miles and km.
