# Kilometers and the golden ratio

Regular readers know I enjoy reading John D. Cook’s blog and often comment on it here. But I was a little creeped out by it a couple of days ago. It started off with something I’ve been thinking about a lot over the past several weeks, and it was as if he’d been reading my mind.

The post is about how the conversion factor between miles and kilometers, $1.609\dots$, is close to the golden ratio, $\varphi =1.618\dots$. To convert kilometers to miles, you can make a good estimate by multiplying by $\varphi$, which means that you can convert in the other direction by multiplying by $1/\varphi =0.618\dots$.

You may think multiplying by an irrational number is a pain in the ass, and you’d be right. Cook gets around this by approximating $\varphi$ by the ratio of consecutive Fibonacci numbers, so you can, for example, convert from miles to kilometers by multiplying by 21 and dividing by 13. Similarly, you can use consecutive Lucas numbers in the same fashion, multiplying, say, by 29 and dividing by 18.

The problem with these calculations is that I’m not John D. Cook, and I can’t do multiplications and divisions like this in my head without losing digits along the way. So my conversion method is much cruder: to go from miles to kilometers, I multiply by 16; and to go in the opposite direction, I multiply by 6. In both cases, I finish by shifting the decimal point to divide by 10. If I need more precision than this, I pull out my phone, launch PCalc, and use its built-in conversions.

By the way, the main reason I’ve been converting between miles and kilometers lately is that I recently switched the units I use in the Activity/Fitness app1 from miles to kilometers. I now often find myself in the middle of one of my walking routes wondering how long it is in kilometers. I know the lengths of all my routes in miles but haven’t memorized their lengths in kilometers yet. It was while doing conversions like this that I noticed that the conversion factor was close to $\varphi$ and started doing a lot of multiplication by 16.

If you’re wondering why I bothered switching units, it’s because I enter 5k races a few times a year and I like my time to be under 45 minutes. To keep myself in shape for this, every couple of weeks I push myself to do my first 5k in that time. It’s much easier to look at my watch and know that my pace should be about 9:00 per kilometer than 14:29 per mile. Also, it’s easier to know when I’m done—marking my time at 3.11 miles is as much a pain in the ass as multiplying by $\varphi$

1. Does it really have to have different names on the watch and the phone? I don’t get confused when I’m using them because the icons are the same, but I never know which name to use when talking about them.