## Fibonacci mileage

by I ran across a little tidbit about the Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21,34, 55, 89,…. It’s a mere coincidence, but still pretty cool:

As the list of numbers goes on, the ratio of successive terms gets closer and closer to the Golden Ratio ø≈1.618. And another number that’s close to 1.618 is the number of kilometers in a mile (1.609). This means that Fn miles is approximately Fn+1 kilometers, where Fn and Fn+1 are successive Fibonacci numbers.

In other words, if you’re driving at 13 miles per hour, that’s approximately 21 kilometers per hour (actual: 20.9). If you’re driving at 21 miles per hour, that’s approximately 34 kilometers per hour (actual: 33.8).

This way cool observation was found at Futility Closet, found via the Evil Mad Scientist Laboratories August Linkdump.

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### 7 Responses to “Fibonacci mileage”

1. mbork Says:

Well, this goes even further: if you know some 10 first Fibonacci numbers (and if you are a true geek, you should know at least 10;)), you can quickly convert miles into kilometers and vice versa. Assume that you want to know how many kilometers there are in 50 miles. Then, you first represent 50 as a sum of Fibonacci numbers (proving that it’s possible for each positive integer is an easy exercise): 50=34+13+3. Then, you just _shift_ each of them one place to the right in the sequence (for km->mi, one to the left, of course):
34->55, 13->21, 3->5. After summing up the shifted ones you’ve got kilometers: 50mi is approximately 55+21+5=81km (in this case, this gives you less than 1% error, since the true value is approx. 80.4672km). (I learned this trick from “Concrete Mathematics” by Graham, Knuth and Patashnik – yes _that_ Knuth;) – a really geeky book, very hard to read (for me), but also very interesting).

2. TwoPi Says:

Yowza. This is possibly the coolest stuff ever.

Well, okay, so multiplying 50 by 1.6 in your head is pretty simple too (double it 4 times and shift a decimal), which gives you 80km, which is slightly more accurate.

But still, to have this sort of application of representations of numbers as sums of Fibonacci numbers, that’s the sort of thing that makes me say “I love mathematics”.

3. TwoPi Says:

One factor I overlooked: Both of these methods are readily reversible.

Example: 50 kilometers: Using the Fibonacci method: 50 = 34 + 13 + 3. Replace each of those Fibonacci numbers with the previous Fibonacci number, and voila: an approximate conversion from km to miles! So 50 km becomes 21 + 8 + 2 = 31 miles.

Note though that one could also just divide by 1.6, by dividing by 2 four times, and shuffling a decimal: 50 -> 25 -> 12.5 -> 6.25 -> 3.125 -> 31.25 miles.

Seems like a toss-up.

4. jd2718 Says:

genuinely cool – I’ll teach this — thanks

5. Barry Leiba Says:

This post has been included in the 40th Carnival of Mathematics. Come check out the others.

6. Math in Bones « 360 Says:

[…] It’s right, too. (OK, not hard math, but for all we know Brennan used Fibonacci numbers to figure it out, since 200=144+55+1 [i.e. F12 + F10 + F2] so the miles would be approximately 89+34+1=124 [i.e. F11 […]

7. Dieting the MATH Way « 360 Says:

[…] calories?  The Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, … is more than just a fancy way to convert between miles and kilometers:  you can arrange your entire plan around these special numbers.  Break your eating into 3 meals […]