## Converting Miles to Kilometers – Fibonacci Numbers to the Rescue

October 9, 2014 at 12:45 am | Posted in Mathematics | Leave a commentTags: fibonacci, fibonacci series, kilometers, kms, mathematics, maths, maths tricks, maths trics, miles, miles to kilometers

A lot of times we encounter a situation where we want to convert miles into kilometers. If we go by the formula where 1 mile = 1.609 kms, we will probably reach the actual conversion, but multiplying a number with 1.609 is not which everyone wants to do.

This is where **Fibonacci series** comes to the rescue. A Fibonacci series is a mathematical number series in which every number is a sum of two preceding numbers. For example:

1,1,2,3,5,8,13,21,34,55,89…….

In the above series, every number is the sum of preceding 2 numbers. We can assume an invisible zero before the first 1.

If we calculate the ratio between two consecutive terms, we find out that the **ratio actually converges to 1.61803** after the first 14 terms.

Surprisingly this number is very close to the conversion ratio for miles to km conversion. If we consider anything between 1.6 – 1.65 as a decent conversion ratio for a back of the envelope conversion, we can safely say that once the series reaches 8, we consistently get the ratio between two consecutive terms to be well within the range.

**What does that signify?**

We have a very simple way of calculating the number of kilometers from miles (for kms > 5).* Simply take the next value in Fibonacci series and that will be a good approximation in kilometers. *

For example if we need to convert 13 miles to kms, simply take the next number in the series i.e. 21. In actual, 13 miles = 20.917 kms. Not bad.

Let us take another example:

55 miles = ? kms

Simple, from Fibonacci series, it should be close to 89 kms. By actual formula, it is 88.495 kms. Less than 1% error again. Superb.

What if the number is not a part of the Fibonacci series. Simple again. We all know about distributive law in Mathematics:

a x (b + c) = a x b + a x c

This essentially means we can add different Fibonacci numbers to reach the number we want to convert. For example, if we want to convert 26 to Fibonacci series, 26 = 21 + 5 so we take the next numbers from the Fibonacci series for 5 & 21 and add them:

Kms (26 miles) = Number next to 5 + Number next to 21

Kms (26 miles) = 8 + 34 = 42

If we use the formula, we get 41.834. Again percentage error is less than 0.5%.

Lastly, we take a big number say 145 and see how can we apply our analysis there:

141 = 144 – 3

Kms(141 miles) = Number next to 144 – Number next to 3

Kms(141 miles) = 233 – 5 = 228

By formula; its 226.869

Isn’t it Cool.

**Thanks for reading – Himanshu Joshi.**

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