Fibonacci numbers or Fibonacci series or Fibonacci sequence, named after Leonardo Fibonacci, are the numbers in the following sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55….
By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.
In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the relation
Fn = Fn-1 + Fn-2
with initial values Fo =0 and F1= 1
Here is how we can use this for our unit conversions from Miles to Kilometres.
Take two consecutive Fibonacci numbers, for example 8 and 13. Done. No seriously- there are 13 kilometers in 8 miles. To convert back, just go the other way – 8 miles in 13 km!
Fibonacci numbers 34 and 55. We can conclude that there are approximately 55 km in 34 miles and vice versa. (The exact answer is 54.7177 km.)
What if you want to convert a value which is not a Fibonacci number? Well, you just need to express this number as a sum of 2 or more Fibonacci numbers; individually convert them, and add them up.
How many kilometers are there in 81 miles? 81 is not a Fibonacci number. But, we can express 81 as the sum of Fibonacci numbers (55+21+5). 89 is the Fibonacci number following 55, the Fibonacci number following 21 is 34 and the Fibonacci number following that follows 5 is 8. Therefore the answer is 89+34+8 = 131 kilometers in 81 miles. This is less than 1% off from the precise answer 130.357km.
How many miles in 100 km? 100 is 89+8+3. Since we are going the opposite way now from miles to km, we need the preceding Fibonacci numbers. 55, 5 and 2. Therefore, there are 55+5+2= 62 miles in 100 km. The exact answer is 62.1371 miles.
Remember: To convert from km to miles, find the preceding Fibonacci number. But, to convert from miles to km, find the subsequent Fibonacci number.
If the distance you’re converting can be expressed as a single Fibonacci number, then for numbers greater than 21 the error is always around 0.5%.
Here’s why it works.
Fibonacci numbers have a property that the ratio of two consecutive numbers tends to the Golden ratio as numbers get bigger and bigger. The Golden ratio is a number and it happens to be approximately 1.618.
Coincidentally, there are 1.609 kilometers in a mile, which is within 0.5% of the Golden ratio.
Now, if we take two consecutive Fibonacci numbers, Fn+1 and Fn, we know that their ratio Fn+1/Fn is approximately 1.618. Since the ratio is also almost the same as kilometers per mile, we can write Fn+1/Fn = [mile]/[km]. It follows that Fn·[mile] = Fn+1·[km]
That is all the elegance that there is to this concept. Almost a pure coincidence that the Golden ratio is almost the same as kilometers in a mile.