Fibonacci kilometres: how the Mathematical Ninja converts (and how mere mortals can, too)

"You simply multiply by eight-fifths," said the Mathematical Ninja, as if that was the end of it.

The student looked hurt. "In my head?" He knew how to multiply by fractions, but much preferred writing things down. Fair enough, I thought.

I interrupted, in the way the Mathematical Ninja hates. "Say, I ran a half-marathon the other week1, and I noticed something - when I passed the five mile mark, my tracker said eight kilometres, when I passed the eight mile mark, it said 13 kilometres, and at 13 miles, it was 21 km."

"Fibonacci!" said the student, immediately, with a 'you see, I'm not a complete dolt' look at the Mathematical Ninja.

"Exactly," I said. "It's a nice coincidence: the ratio between consecutive Fibonacci numbers is about 1.62..."

"The Golden Ratio!" said the student, who had been paying attention. "And there are 1.6 kilometres to a mile."

"1.609," said the Mathematical Ninja, grudgingly.

"So you can use Fibonacci numbers to convert between miles and kilometres?" said the student. "Why don't they teach us this stuff? Let me try. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... so is 89 miles about 144 kilometres?"

"Try it," I said.

"143.201," muttered the Mathematical Ninja.

"That's about half a percent off," I pointed out. "In fact, almost exactly as far off as using the eight fifths."

"What if it's in between?" asked the student. "I mean, what if it's, I don't know, 64 miles?"

"You've got a couple of options," I said. "You can interpolate, or you can use other values you know about. Easiest would be to say 64 is $8 \times 8$, so in kilometres it'd be $8 \times 13$ - 104km. You could also estimate that it's a quarter of the way between 55 and 89, so you'd want to be quarter of the way between 89 and 144... about 103?"

"102.976," said the Mathematical Ninja, petulantly.

The student grinned. I grinned. The Mathematical Ninja... well, his way was better, you could tell by the way he scowled.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. To sponsor me for the Berlin Marathon, you can visit []


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