Tuesday, May 31, 2011

Conversion of miles to kilometers (and back) using addition only

Some time ago I came across the trick that allows easy conversion of miles and kilometers (and also mph and km/h) using addition only. The trick uses the fact that the ratio of the Fibonacci numbers is very close to the conversion factor between miles and kilometers.

This is an example of turning multiplication to addition in a similar manner to the use of logarithms in a slide rule.

Firstly, the Fibonacci sequence is pretty easy to remember or regenerate from scratch because it is generated by the rule "the next number is the sum of the preceding two numbers". The sequence starts from 0 and 1 and then proceeds 1, 2, 3, 5, 8, 13, 21, 34, ... Each number is the sum of the previous two numbers.

Secondly, the ratio of consecutive Fibonacci numbers gets closer and closer to 1.62 the larger the numbers get. For example, 3/2 = 1.5, 8/5 = 1.6, 13/8 = 1.63, 21/13 = 1.62, 34/21 = 1.62, ...

Lastly, there are 1.61 kilometers in a mile. If you take the ratio of Fibonacci numbers as 1.62 then the error in using that for miles/kilometers conversions is about 0.5%. For everyday usage that's pretty good.

To go from kilometers (or kph) to miles (or mph) find the number in the list of Fibonacci numbers and look at the preceding number. For example, 34km (or kph) is 21.1 miles (or mph); using Fibonacci you get 21 miles (of mph). 8km is 4.97 miles; using Fibonacci it's 5 miles.

To go the other way (miles to kilometers) simply go the other way along the line. So find the number (in miles) in the Fibonacci sequence and get the kilometers from the next number.

On a recent trip to France I tried this out. The speed limit on a French autoroute is 130 kph which is 80.78 mph. Using the Fibonacci sequence it comes out at 80 mph (you can drop the 0 on both numbers since you're dealing with a ratio and go from 13 to 8 in the Fibonacci sequence).

When it's raining the speed limit drops to 110 kph. Now 11 doesn't appear in the sequence (again I've dropped the zero and will put it back in later). But 11 can be decomposed into 8 and 3 (11 = 8 + 3) and the answer obtained by looking at the mph figures for 11 and 3 in the Fibonacci sequence and adding them together. Thus we get 5 + 2 = 7 and so 110 kph is 70 mph (it's actually 68.35 mph). The error comes about because the early Fibonacci numbers are not quite at the right ratio. You can remove the error by remembering that 3 km is 1.8 (not 2) and then you get 5 + 1.8 = 6.8 (giving 68 mph).

Note that you can also use subtraction. For example, to work out 16 kilometers in miles you could look at 13 and 3 (to get 8 + 1.8 = 9.8) or you could do 21 and 5 (to get 13 - 3 = 10).

You can also play around with other tricks. For example, returning to 110 km you can see that it is 55 * 2 and 55 is a Fibonacci number. So at once you have 110km is 34 miles * 2 = 68 miles.

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