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Converting between units is often something you have to do on the fly, and it is made easier when you memorise some rough equivalents. For instance, an imperial pint is 568 millilitres, but I often think of it as being about half a litre to make conversions simpler. I know a yard is a bit less than a metre, there are 2.2 pounds in a kilogram and a foot is 30 centimetres (thanks to rulers).

But maths gives us a neat trick for one unit conversion, and it is related to Fibonacci numbers: a sequence that starts with 1, followed by another 1 and then each term is the sum of the previous two. So 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8 and so on.

These numbers are named after Fibonacci, a 13th-century mathematician, but were used by Indian mathematicians in 200 BC to study rhythmic patterns in poetry. Flowers (like the sunflower, pictured above) often have seeds arranged in Fibonacci numbers of spirals running in each direction, because it gives the most efficient use of space.

One way to study number sequences is to consider the ratios between successive terms. For the Fibonacci numbers, the sequence of ratios between pairs of terms – dividing the larger by the smaller – starts with 1 ÷ 1 = 1, then 2 ÷ 1 = 2, 3 ÷ 2 = 1.5, 5 ÷ 3 = 1.667, 8 ÷ 5 = 1.6 and then 13 ÷ 8 = 1.625. After a bumpy start, they get closer and closer to a particular value, somewhere around 1.62.

The exact value is (1 + √5) ÷ 2, or 1.618033988749, and is often called the Golden Ratio, or ϕ(phi). It has uses in geometry and architecture, but it is also close to another helpful ratio: 1.609344.

If you recognise this number, you may often convert miles to kilometres. A distance of 1 mile is 1.609344 km. While this isn’t exactly the Golden Ratio, it is close enough that we can use Fibonacci numbers to do a fun trick.

Since the ratio between pairs of consecutive Fibonacci numbers is getting closer and closer to somewhere near this number, if you want to convert a number of miles to kilometres, and the number of miles is a Fibonacci number, the number of kilometres will be close to the next number in the sequence. So, 5 miles is about 8 km (it is actually 8.04672 km), and 8 miles is about 13 km (it is 12.874752 km). The bigger the numbers, the more accurate the conversion will be.

It also works backwards: to convert kilometres to miles, just take the previous Fibonacci number. With a good memory (or a cheat sheet), you can make a conversion just by looking at the next or previous number in the list. And if your number of miles isn’t a Fibonacci number, you can add these numbers: 15 miles is (2 + 13) miles, or about (3 + 21) = 24 km (actually 24.14016 km).

You can use this trick to impress your friends – or just enjoy a beautiful mathematical coincidence.

Katie Steckles is a mathematician, lecturer, YouTuber and author basedÌýinÌýManchester, UK. SheÌýis alsoÌýadviser for NewÌýScientist’s puzzle column,ÌýBrainTwister. FollowÌýherÌý@stecks

For other projects visit newscientist.com/maker

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