Miles to Km Converter

13 miles is 21 kilometres, and so is every other Fibonacci pair

Take any two neighbours in 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144: the smaller is a distance in miles and the larger is the same distance in kilometres. It works because consecutive Fibonacci numbers approach the golden ratio, 1.6180, and a mile is 1.609344 kilometres. Those are 0.54% apart.

The first pair is useless — 1 mile as 2 km is +24.27% out. From 5 miles upward every pair is inside one per cent. The error flips sign as it goes, because Fibonacci ratios oscillate above and below the golden ratio on the way to it. Then it settles at 0.54% and stays there, so the trick has a floor: it never gets better, and it never gets worse either.

The best pair in the sequence is the half marathon

One of those alternating overshoots lands closer to the mile than any other: 13 to 21, off by +0.38%, beating the pairs on both sides of it. Which happens to be the half marathon. The race is 21.0975 km, or 13.1094 miles, and both figures round to the Fibonacci pair. The full marathon is 42.195 km and 26.2188 miles — the same pair doubled. Both races are defined in kilometres, incidentally, which is why the mile figures are the untidy ones and always get rounded in the telling.

The trick is not limited to Fibonacci numbers. Every whole number is a sum of non-adjacent Fibonacci numbers in exactly one way, so break it up, shift each piece one place along, and add:

The pieces always add back to the number they came from. Accuracy suffers when the breakdown needs a small term, because the small pairs are the bad ones — that is why 30 (which needs a 1) does worse than 100 (which does not). Anything whose largest piece is 13 or more stays inside about one and a half per cent, which is close enough for a road sign and not close enough for anything else.

Exact one way, never exact the other

Since 1959 the yard has been defined as exactly 0.9144 m, so a mile — 1,760 of them — is exactly 1609.344 m. That is a definition, not a measurement, which is why the factor stops after six decimal places instead of running on. As a fraction it is 25146/15625, and 15625 is 5⁶ — a power of five, so it divides cleanly into a decimal.

Going the other way the fraction flips, and now the awkward number is on the bottom: 25146 = 2 × 3 × 3 × 11 × 127. No power of ten clears a 3, an 11 or a 127, so kilometres to miles is a repeating decimal — 1 km is 0.621371192237… and it does not stop.

So every whole number of miles is an exact decimal number of kilometres. Whole kilometres almost never return the favour: in the first thirty thousand, exactly 2 do. The first is 12,573 km, which is exactly 7,812.5 miles. And the smallest pair that is a whole number on both sides is 15,625 miles = 25,146 km — the fraction itself, and the smallest in either direction.

The converter above rounds to five decimal places, which hides all of this. It is the right call for a road trip and worth knowing about if you are ever checking someone else's numbers: a kilometre figure that converts to a suspiciously round mile figure has usually been rounded twice.